Contact Geometry: Sasaki manifolds, Kenmotsu

Contact Geometry: Sasaki manifolds, Kenmotsu manifolds Gheorghe Piti¸s

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Contents Preface

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1 Contact structures 1.1 Almost contact manifolds . . . . . . . . . . . 1.2 Normal almost contact manifolds . . . . . . . 1.3 Contact manifolds . . . . . . . . . . . . . . . 1.4 K-contact manifolds . . . . . . . . . . . . . . 1.5 Sasaki manifolds . . . . . . . . . . . . . . . . 1.5.1 General properties of Sasaki manifolds 1.5.2 Deformations of Sasaki structures . . . 1.5.3 Sasaki potential . . . . . . . . . . . . . 1.6 Kenmotsu manifolds . . . . . . . . . . . . . . 1.7 Other almost contact structures . . . . . . . .

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1 1 7 11 30 32 32 41 45 47 53

2 Transformations and submersions 2.1 Contact transformations groups . . . . . 2.2 Harmonic maps . . . . . . . . . . . . . . 2.3 Almost contact Riemannian submersions 2.4 Contact bundles . . . . . . . . . . . . . 2.5 Almost contact fibered spaces . . . . . . 2.6 Orbit space of a contact manifold . . . .

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57 57 63 66 72 80 84

3 Curvature problems in contact manifolds 3.1 Curvature tensor of a contact manifold . . 3.2 Curvature tensor of a Sasaki manifold . . 3.3 Curvature tensor of a Kenmotsu manifold 3.4 F -sectional curvature . . . . . . . . . . . . 3.5 Sasaki space forms . . . . . . . . . . . . . 3.6 Kenmotsu space forms . . . . . . . . . . . 3.7 Bisectional curvature of Sasaki manifolds 3.7.1 F -bisectional curvature . . . . . . 3.7.2 F -bisectional curvature of order q 3.8 Einstein and η-Einstein manifolds . . . . . 3.8.1 General properties and examples .

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89 89 97 104 106 109 118 120 120 123 125 125

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CONTENTS

3.9

3.8.2 Einstein Sasaki manifolds . . . . . . . . . . . . . . . . . . 127 3.8.3 η-Einstein Sasaki manifolds . . . . . . . . . . . . . . . . . 134 Locally F -symmetric Sasaki manifolds . . . . . . . . . . . . . . . 137

4 Differential forms and topology 4.1 Harmonic forms . . . . . . . . . . . . . . . . . 4.2 C-harmonic forms on Sasaki manifolds . . . . 4.3 Betti numbers of a contact manifold . . . . . 4.4 Basic forms on Sasaki manifolds . . . . . . . 4.5 Basic cohomology of Sasaki manifolds . . . . 4.5.1 Basic de Rham cohomology . . . . . . 4.5.2 Basic Dolbeault cohomology . . . . . . 4.5.3 Spectral sequences of Sasaki manifolds 4.6 Basic Chern classes . . . . . . . . . . . . . . . 4.7 Positive Sasaki structures . . . . . . . . . . . 4.8 Morse theory on Sasaki manifolds . . . . . . . 4.9 Construction of contact manifolds . . . . . .

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143 143 148 156 160 163 163 166 167 170 174 177 182

5 Integral submanifolds 5.1 General properties . . . . . . . . . . . . . 5.2 Chern classes of integral submanifolds . . 5.3 Chen invariants of an integral submanifold 5.4 Legendre submanifolds . . . . . . . . . . . 5.5 Spherical type Legendre submanifolds . . 5.6 Maslov form of a Legendre submanifold . 5.7 Stability of integral submanifolds . . . . .

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191 191 196 200 207 211 214 219

6 Semi-invariant and slant submanifolds 6.1 Semi-invariant submanifolds . . . . . . . . . 6.2 Invariant submanifolds . . . . . . . . . . . . 6.3 Anti-invariant submanifolds . . . . . . . . . 6.4 Submanifolds of a Kenmotsu manifold . . . 6.4.1 Normal semi-invariant submanifolds 6.4.2 Semi-invariant submanifolds . . . . . 6.5 Parallel and cyclic parallel submanifolds . . 6.6 Slant submanifolds . . . . . . . . . . . . . . 6.7 Contact submanifolds . . . . . . . . . . . .

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223 223 230 236 238 238 241 248 257 262

Notations and Formulas

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Bibliography

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Index

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Preface Viewed so long time as the younger sister of the complex geometry, the geometry of contact manifolds had a remarkable development these last times, as well by the quantity, as by the worth of the obtained results. This is explained by its rich inside, but also by many applications in control theory and in various branches of the mathematical physics, like geometrical optics, mechanics of dynamical systems with time dependent Hamiltonian, thermodynamics and geometric quantization. Many special contact structures are studied, but the Sasakian structure turns out to be one of the most important, being the odd-dimensional analogue of the Kähler structure. In this book our interest is focused on these structures as well as on another contact structure, called the Kenmotsu structure. This was lately in the attention of geometers, many interesting and valuable results being obtained. I have in view two aims: first, to present an important part of the results already become classical in contact geometry , specially in Sasaki and Kenmotsu contact geometries, and second, to inform the reader about some new results considered to deserve attention by their worth and by some possible research directions that these opened. I tried to achieve this project in a reasonable number of pages, holding the idea that the reading of the book must incite the curiosity in order to learn more about the subject. For this reason for most results the proofs are complete and can be understand by the reader having a training corresponding to a general course on manifolds and who can use a part of this book as lecture notes. By worrying about this reader I have chosen some results whose proofs are presented without details. These details concern specially computation arguments which constitute good exercises. Finally, the third kind of results have no proofs. These are so few and generally concern improvements of others ones proved in the book. I notice them because these give answers to some new research directions. So I hope that the book is also useful for the research beginner of this important special differential geometry. Of course, it is very difficult to request from a book of this kind to be entirely self-contained, so much more the subject is a specialized one. So some background in complex geometry, algebraic and differential topology is required. However, whenever it is possible, notions or results which take easy the understanding are introduced in the text or in some footnotes. Also, formulas and v

vi

PREFACE

notations used in ours proofs can be found at the end of the book, in an Appendix. Briefly, the structure of this book is the following: In Chapter 1 we introduce almost contact, normal contact and K-contact structures on a manifold and the most important properties of them are proved. For all these structures, representative examples are discussed. The treatment of a part of this Chapter, as well as a part of Chapter 3 has its origins in the excellent books of S. Sasaki and D. E. Blair. We also study deformations of Sasaki structures and the Sasaki potential. The general properties of Kenmotsu manifolds are presented and some representative examples follow their study. We end Chapter 1 by a presentation of the most important generalizations of the above structures. A brief historical note so the author cold find during his travel among old and new papers dealing with several aspects of contact geometry is also included. The first part of Chapter 2 is dedicated to the study of groups of contact transformations and to harmonic maps preserving the contact structure. The main properties of almost contact Riemannian submersions are studied. The notions of contact bundle and almost contact fibered space are also presented. Special attention is payed to the orbit space of a contact manifold. The aim of the first part of Chapter 3 is to evidence the general properties of the curvature tensors for contact Riemannian, K-contact, Sasaki and Kenmotsu manifolds. Sasaki and Kenmotsu manifolds with constant F -sectional curvature are studied and then the F -bisectional curvature of order q as a generalization of the F -sectional curvature is analyzed. The last part of the Chapter is about the Einstein and η-Einstein Sasaki manifolds, for which we present many recent results. Basic results concerning the locally and globally F -symmetric Sasaki manifolds are also discussed. In the first part of the Chapter 4, harmonic and C-harmonic forms on a Sasaki manifold are studied and the correspondence between these forms is analyzed as well as important results about the Betti numbers of a contact manifold. The basic cohomology of a Sasaki manifold is defined and studied. Then the basic Chern classes are used in order to characterize positive, negative or null Sasaki structures. General construction methods of contact manifolds, such that the connected sum of two contact manifolds or the reduction of Sasaki manifolds are presented at the end of this Chapter. Thus we broach slowly the fascinating world of the contact topology. We also present few steps in the study of the Morse theory on Sasaki manifolds. The most important class of submanifolds of a contact manifold, namely the class of integral submanifolds, is studied in Chapter 5. Some characteristic classes of Chern type and the Chen invariants are associated to such a submanifold. The major part of this Chapter concerns the integral submanifolds of maximum dimension, called Legendre submanifolds, for which the Maslov form and the stability are specially studied. In the last Chapter we study invariant and anti-invariant submanifolds in

vii Sasaki and Kenmotsu manifolds, as well as the semi-invariant submanifolds, which generalize both these classes. Parallel and cyclic parallel submanifolds have important properties, noticed in this Chapter. Also, slant and semi-slant submanifolds in Sasaki manifolds, as well as others remarkable classes of submanifolds are studied. All these are submanifolds intensively studied the last twenty years. I cannot end this introductory part without express my gratitude to Academician Radu Miron, whose personality mainly influenced my mathematical career. Also I want to thank to my colleagues from the Faculty of Mathematics and Informatics of the University ”Transilvania” of Bra¸sov and specially to those from the Department of Differential Equations, who founded the secret to maintain a wonderful research and friendship atmosphere. Many friends from my country and from abroad have offered my much help by theirs papers. For me is extremely pleasant to thank to all them. Finally, the generosity of my wife Maria, my son Iulian, my daughter Cristina and the innocent purity of my granddaughter Karina were the real arguments to write this book. Bra¸sov, October, 2006

Gheorghe Piti¸s

viii

PREFACE

Chapter 1

Contact structures 1.1

Almost contact manifolds

All the manifolds considered afterward are assumed to be smooth, paracompact and without boundary, unless otherwise is specified. Let M be a manifold of dimension 2n + 1. If there are the tensor fields F ∈ T11 (M ), ξ ∈ X (M ), η ∈ F 1 (M ) such that F 2 = −I + η ⊗ ξ,

η(ξ) = 1

(1.1)

then we say that M admits the almost contact structure (F, ξ, η) and M is an almost contact manifold. ξ is the Reeb vector field or the fundamental vector field of the almost contact manifold M . The (local) transformations group generated by ξ is called the (local) Reeb group of M . Obviously the set Xξ (M ) = {f ξ; f ∈ F(M )} has a module structure over F(M ) and a Lie algebra structure. We call it the Lie algebra of Reeb vector fields. The subspaces Dx = {X ∈ Tx M ; η(X) = 0} of the tangent spaces to the almost contact manifold M define a distribution D of dimension 2n on M . We call it the contact distribution of M and remark that D = ker η = im F and that D has a vector bundle structure over M , with the fiber R2n . Denote also by D the module Γ(D) of sections in the bundle D. The quotient bundle T M/D over M is a line bundle and the following bijective map is defined Xξ (M ) −→ Γ(T M/D);

f ξ 7−→ f ξ + D

Let Dx0 ⊂ Tx∗ M be the annihilator of Dx . We obtain an oriented line bundle Do = ∪x∈M Dx0 over M . Then D0 = {f η; f ∈ F(M )} and D∗0 = ∪x∈M (Dx0 − 0 0 {zero}) has two components D+ and D− , corresponding to the vector charts for which the transition functions are increasing and decreasing, respectively. We say that the manifold M is co-oriented (or D is co-oriented )if D0 is oriented, that is if we choose one component of D∗0 . We also say that M is positively co0 oriented if we choose the component D+ , corresponding to increasing transition functions. 1

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CHAPTER 1. CONTACT STRUCTURES

Proposition 1.1.1. If (F, ξ, η) is an almost contact structure on the 2n + 1dimensional manifold M then: a) F ξ = 0; b) F 3 = −F ; c) η ◦ F = 0; d) rank F = 2n. Proof. a) From ( 1.1) it follows F 2 ξ = 0;

F 3 ξ = −F ξ + η(F ξ)ξ

(1.2)

We deduce F ξ = η(F ξ)ξ and if there is a point from M such that F ξ 6= 0 then η(F ξ) 6= 0. But this is impossible because, by applying F over the last equality, from (1.2) we deduce η(F ξ)F ξ = 0. b) follows from the second equality (1.2) by using a). c) can be obtained from the first equality 1.1 applied to vectors of the form F X and taking into account b). d) If ξ ∈ ker F then from (1.1) we obtain F 2 ξ = 0 = −ξ + η(ξ)ξ It follows ker F = hξi and then dim im F = rank F = 2n. Theorem 1.1.2. Let M be a manifold with the almost contact structure (F, ξ,η). There exists on M a Riemannian metric g with the property g(F X, F Y ) = g(X, Y ) − η(X)η(Y )

(1.3)

for all X, Y ∈ X (M ). Proof. M is paracompact, so that there exists a Riemannian metric g ∗∗ on M and then we define g by g(X, Y ) =

1 ∗ [g (F X, F Y ) + g ∗ (X, Y ) + η(X)η(Y )] 2

(1.4)

where g ∗ has the expression g ∗ (X, Y ) = g ∗∗ F 2 X, F 2 Y + η(X)η(Y ) It is easy to check that g given by (1.4) is a Riemannian metric and satisfies the condition (1.3). The manifold M with the almost contact structure (F, ξ, η) and the Riemannian metric g satisfying the condition (1.3) is called an almost contact Riemannian manifold and (F, ξ, η, g) is an almost contact Riemannian structure on M . Sometimes we say that g is a metric compatible with the almost contact structure (F, ξ, η). Some elementary but useful properties of such metrics are specified in the following Proposition 1.1.3. If g is a metric compatible with the almost contact structure (F, ξ, η) on the 2n + 1-dimensional manifold M then: a) η(X) = g(X, ξ) for all X ∈ X (M );

1.1. ALMOST CONTACT MANIFOLDS

3

b) on the domain U of each local chart from M there exists an orthonormal basis {X1 , X2 , . . . , Xn , F X1 , F X2 , . . . , F Xn , ξ} of the module X (U ); c) F + η ⊗ ξ and −F + η ⊗ ξ are orthogonal transformations with respect to the metric g; d) g(F X, Y ) = −g(X, F Y ) for any X, Y ∈ X (M ). Proof. a) follows from (1.1) and from Proposition 1.1.1, a) by setting Y = ξ in (1.3). b) Let X1 ∈ X (U ) be a unitary vector field with respect to the metric g. If X1 is orthogonal to ξ then from (1.3) and from a) it follows that F X1 is unitary and orthogonal to the vector fields X1 and ξ. If X2 ∈ X (U ) is unitary and orthogonal to X1 , F X1 , ξ then by the same argument we deduce that F X2 is unitary, orthogonal to X1 , F X1 , F X2 , ξ. By this method we obtain the desired basis. c) A straightforward calculus using (1.3), (1.1), a) and Proposition 1.1.1 shows that g[(F + η ⊗ ξ)X, (F + η ⊗ ξ)Y ] = g(X, Y ) for all X, Y ∈ X (M ) and a similar relation for −F + η ⊗ ξ. d) By applying again (1.3), (1.1) and Proposition 1.1.1, we have g(F X, Y ) = g(F 2 X, F Y ) + η(F X)η(Y ) = g(−X + η(X)ξ, F Y ) = = −g(X, F Y ) + η(X)η(F Y ) = −g(X, F Y ) and d) follows easily. The local basis {X1 , X2 , . . . , Xn , X1∗ = F X1 , X2∗ = F X2 , . . . , Xn∗ = F Xn , ξ} obtained above and denoted sometimes by {Xi , Xi∗ , ξ}i∈1,n , is called a F basis for the almost contact Riemannian structure (F, ξ, η, g). The existence of metrics compatible with an almost contact structure allow us to state the following characterization of almost contact manifolds by the structure group of the tangent bundle. Theorem 1.1.4. The structure group of the tangent bundle of an almost contact manifold of dimension 2n + 1 reduces to U (n) × 1. Conversely, if the structure group of the tangent bundle of a manifold reduces to U (n) × 1 then the manifold has an almost contact structure. Proof. Let g be a metric compatible with the almost contact structure (F, ξ, η) and consider two domains U, V of local charts with U ∩ V 6= ∅. Also, we ei , X ei∗ , ξ} the corresponding F -bases from denote by B = {Xi , Xi∗ , ξ}, Be = {X Proposition 1.1.3, b). The matrix (F ) of F with respect to these bases is 

0 (F ) =  In 0

−In 0 0

 0 0  0

4


e the column matrices For x ∈ U ∩ V and X ∈ Tx M we denote by (X), (X) e respectively. Then of components of the vector X with respect to B and B, e = P · (X), where (X)   A B 0 P = C D 0  0 0 1 and A, B, C, D ∈ Mnn (R). But P is orthogonal and commutes with the matrix (F ) (see Proposition 1.1.3, b)), thus we have D = A, C = −B and this proves that P ∈ U (n) × 1. Conversely, if the structure group of the tangent bundle of the manifold M reduces to U (n)×1 then there exists a covering {Uα }α∈I of M , for which we can choose the orthonormal local bases with the property that on the intersection Uα ∩Uβ these are transformed by the action of the group U (n)×1. With respect to such bases we can define the endomorphism Fα : X (Uα ) −→ X (Uα ) by the matrix (F ). But (F ) commutes with U (n) × 1, hence {Fα }α∈I determine a global endomorphism F : X (M ) −→ X (M ). In a similar way the tensor fields ξ ∈ X (M ), η ∈ F 1 (M ) are globally defined by the matrices of their components with respect to each open set Uα , namely ξ : (0, 0, . . . , 0, 1)tr ,

η : (0, 0, . . . , 0, 1)

Finally, it is easy to check that (F, ξ, η) is an almost contact structure. The determinants of the matrices from the proof of Theorem 1.1.4 are positive and then it follows Consequence 1.1.5. Any almost contact manifold is orientable. From Proposition 1.1.3, d) it follows that Ω defined by Ω(X, Y ) = g(X, F Y ) for all X, Y ∈ X (M ) is a 2-form on M . It is called the fundamental 2-form or the Sasaki 2-form of the almost contact Riemannian structure (F, ξ, η, g) and has the following obvious properties Ω(X, F Y ) = −Ω(F X, Y ), Ω(F X, F Y ) = Ω(X, Y ) (1.5) If ω i , ω i∗ , η is the dual basis of of the F -basis from Proposition 1.1.3 then the fundamental form Ω is locally given by

Ω = −2

n X

ω i ∧ ω i∗

i=1

We remark that rank Ω = 2n and then η ∧ Ωn (where Ωn is the exterior product of n copies of Ω) does not vanish nowhere on M . The converse of this result is also true, namely we have Theorem 1.1.6. Let M be a manifold of dimension 2n + 1 and η ∈ F 1 (M ). a) If there exists Ω ∈ F 2 (M ) such that η ∧ Ωn 6= 0 at each point of M then M has an almost contact structure.

1.1. ALMOST CONTACT MANIFOLDS

5

b) If η ∧ (dη)n 6= 0 on M then the manifold M has an almost contact Riemannian structure (F, ξ, η, g) whose fundamental form is dη and the Reeb vector field ξ is completely determined by the conditions η(ξ) = 1, ıξ dη = 0. Proof. a) Because the 2n + 1-form η ∧ Ωn does not vanish nowhere on M , it follows that the manifold M is orientable and then there is an atlas with the property that the coordinates transformations have positive determinant. Then it is easy to prove that on M the following tensor field of type (2n + 1, 0) is globally defined 1 (i1 i2 ...i2n+1 ) ei1 i2 ...i2n+1 = √ det h where h is a Riemannian metric on M and (i1 i2 ...i2n+1 ) is the signature of the permutation (i1 i2 . . . i2n+1 ). Moreover, as rank Ω = 2n, there exists a nowhere zero vector field X ∗ ∈ X (M ) with the local components eii1 ...i2n X (σ(i1 ) σ(i2 )...σ(i2n )) Ωσ(i1 )σ(i2 ) Ωσ(i3 )σ(i4 ) . . . Ωσ(i2n−1 )σ(i2n ) Xi = (2n)! Here the sum is taken over all the permutations σ of the set {i1 , i2 , . . . i2n }, x1 , x2 ,. . . , x2n+1 are the local coordinates in a given local chart and Ωij = ∂ ∂ Ω ∂x are the components of the fundamental form Ω in this chart. By i , ∂xj using the properties of permutations, a simple computation shows that Ωij X j = 0 and then Ω(X ∗ , Y ) = 0 for any Y ∈ X (M ). Hence we can consider the unitary vector field ξ and the 1-form η ∗ , given by the formulas X∗ ξ=p , η ∗ (Y ) = h(Y, ξ) h(X ∗ , X ∗ ) for any Y ∈ X (M ). But the restriction of the form Ω to the orthogonal comple⊥ ment hξi of the space hξi with respect to the metric h is a symplectic form and ⊥ ⊥ then there exists an endomorphism F : hξi −→ hξi with the property that ⊥ F 2 = −Ihξi⊥ and h(X, F Y ) = Ω(X, Y ) for all X, Y ∈ hξi . By extending F in ∗ direction ξ by F ξ = 0, it follows that (F, ξ, η ) is an almost contact structure on M . b) By setting Ω = dη and using the Riemannian metric h∗ defined by h∗ (X, Y ) = h(−X + η(X)ξ, −Y + η(Y )ξ) + η(X)η(Y ), we have η(X) = h∗ (X, ξ). Now, let g be a metric so that g|hξi = h∗ . If we consider the orthogonal complement of ξ with respect to g then, by the same argument as in a), the resulting almost contact structure (F, ξ, η, g) is Riemannian. The unicity of ξ follows from the imposed conditions taking into account dη(X, Y ) = g(X, F Y ). We remark that the almost contact structure given in Theorem 1.1.6, a) and the almost contact Riemannian structure from b) are, generally, not unique. However, for a fixed metric g this second structure is uniquely determined and in this case we say that (F, ξ, η, g) is the almost contact Riemannian structure associated to η.

6


Examples 1. Almost contact structure on the cylinder of an almost complex manifold Let J be the almost complex structure of M and denote by t the Cartesian coordinate in R. On the cylinder manifold M × R we consider the ∂ , the 1-form η = (0, dt) and the endomorphism F : vector field ξ = 0, ∂t ∂ X (M × R) −→ X (M × R), given by F X, f ∂t = (JX, 0) for X ∈ X (M ) and f ∈ F(M × R). Then the system (F, ξ, η) defines an almost contact structure on M × R. 2. Almost contact Riemannian structure on a parallelizable manifold Let M be a parallelizable manifold of dimension 2n + 1 and consider the vector fields X1 , X2 , . . . , X2n+1 so that {X1x , X2x , . . . , X2n+1x } is a basis of Tx M for each x ∈ M . Let g be the Riemannian metric defined by g(Xi , Xj ) = δij , set ξ = X2n+1 and let η be the 1-form dual of the vector field ξ, i. e. η(X) = g(X, ξ) for all X ∈ X (M ). Similarly, we denote by ω i the 1-forms dual of the vector ∗ fields Xi for each i ∈ 1, n and by ω i those dual of the fields Xi∗ = Xn+i . A simple verification shows that (F, ξ, η, g) is an almost contact Riemannian structure on M , with the tensor field F given by F =

n X

∗

ω i ⊗ X i∗ − ω i ⊗ X i

i=1

3. Almost contact structures on Bianchi-Cartan-Vranceanu spaces For any µ ∈ R, the set M3 = (x, y, z) ∈ R3 : 1 + µ(x2 + y 2 ) > 0 is a manifold of dimension 3 with the structure given by the trace of the standard structure on R3 . Moreover, on M3 is defined the following 1-parameter family of Riemannian metrics (see [Vr76], pg. 296–297) gλµ

2 λ ydx − xdy = 2 + dz + 2 1 + µ(x2 + y 2 ) [1 + µ (x2 + y 2 )] dx2 + dy 2

(1.6)

The manifold M3 equipped with a Riemannian metric gλµ is called a BianchiCartan-Vranceanu space and we denote it by M3λµ . An orthonormal frame in M3λµ is ∂ λ ∂ ∂ λ ∂ ∂ X1 = 1 + µ(x2 + y 2 ) − y , X2 = 1 + µ(x2 + y 2 ) + x , X3 = ∂x 2 ∂z ∂y 2 ∂z ∂z and its dual frame is given by the following 1-forms dy λ ydx − xdy , ω 3 = dz + 1 + µ(x2 + y 2 ) 2 1 + µ(x2 + y 2 ) Define the endomorphism F : X M3λµ −→ X M3λµ by F X1 = X2 , F X2 = ω1 =

dx , 1 + µ(x2 + y 2 )

ω2 =

−X1 , F X3 = 0 and take ξ = X3 , η = ω 3 . Then (F, ξ, η, g = gλµ ) is an almost

1.2. NORMAL ALMOST CONTACT MANIFOLDS

7

contact Riemannian structure on M3λµ . Moreover, ω 3 nowhere vanishes on M3λµ if and only if λ 6= 0 and with respect to the frame {X1 , X2 , X3 } the expression of the Levi-Civita connection ∇ of the metric gλµ is λ λ X3 , ∇X1 X3 = − X2 2 2 λ λ ∇X2 X1 = −2µxX1 − X3 , ∇X2 X2 = 2µxX1 , ∇X2 X3 = X1 2 2 λ λ ∇X3 X1 = − X2 , ∇X3 X2 = X1 , ∇X3 X3 = 0 2 2

∇X1 X1 = 2µyX2 , ∇X1 X2 = −2µyX1 +

(1.7)

Then it results [X1 , X2 ] = −2µyX1 + 2µxX2 + λX3 ,

[X2 , X3 ] = [X3 , X1 ] = 0

(1.8)

By a simple verification we deduce that ξ is a Killing vector field and it generates a globally Reeb group, whose action on M3λµ is simply transitive.

1.2

Normal almost contact manifolds

Let M be a manifold of dimension 2n + 1 with the almost contact structure (F, ξ, η) and consider its cylinder M ×R. Using the same notations as in Example 1, Section 1.1 we remark that the map J : X (M × R) −→ X (M × R),

∂ J(X, f ) = ∂t

∂ F X − f ξ, η(X) ∂t

is linear and J 2 = −I. This proves that J defines on M × R an almost complex structure. If J is integrable (i. e. if its Nijenhuis tensor NJ vanishes identically) then we say that (F, ξ, η) is a normal almost contact structure. This structure was defined by S. Sasaki, [Sa60] and by S. Sasaki and Y. Hatakeyama, [SH61]. Now, we present an expression depending on the Nijenhuis tensor for the condition that the almost contact structure (F, ξ, η) is normal. Theorem 1.2.1. The almost contact structure (F, ξ, η) is normal if and only if N (1) ≡ NF + 2dη ⊗ ξ = 0

(1.9)

Proof. Since the Nijenhuis tensor is antisymmetric, it is enough to compute d NJ ((X, 0), (Y, 0)) and NJ ((X, 0), (0, dt )) for X, Y ∈ X (M ). A straightforward calculus shows that NJ ((X, 0), (Y, 0)) = (NF (X, Y ) + 2dη(X, Y )ξ, (LF X η)Y − LF Y η)X) , d d NJ (X, 0), (0, ) = (Lξ F )X, (Lξ η)X , dt dt

8


hence the almost contact structure (F, ξ, η) is normal if and only if the condition (1.9) is satisfied and N (2) (X, Y ) ≡ (LF X η)Y − (LF Y η)X = 0, 1 N (3) X ≡ (Lξ F )X = 0, 2 N (4) X ≡ (Lξ η)X = 0.

(1.10)

Now, we prove that if (1.9) holds then the equalities (1.10) hold, too. Indeed, we have N (1) (X, ξ) = −[X, ξ] + η[X, ξ]ξ − F [F X, ξ] + X(η(ξ))ξ− − ξ(η(X))ξ − η[X, ξ]ξ = = [ξ, X] + F [ξ, F X] − ξ(η(X))ξ = 0

(1.11)

and applying η to the last equality we deduce η[ξ, X] − ξ(η(X)) = 0,

(1.12)

that is N (4) = 0. Now, replacing X by F X in (1.12) we have η[ξ, F X] = 0 and applying F to the last equality (1.11) we obtain N (3) = 0. Finally, applying η to the equality N (1) (F X, Y ) = 0 and taking into account the definition of the exterior differential and that of the Lie derivative, we obtain N (2) = 0. Replacing in the definition of the Nijenhuis tensor NF the brackets by theirs expressions (since the Levi-Civita connection is torsionless) [X, Y ] = ∇X Y − ∇Y X, from Theorem 1.2.1 it results Proposition 1.2.2. [Ta65] An almost contact Riemannian structure (F, ξ, η, g) on the manifold M is normal if and only if one of the following conditions is satisfied F (∇X F )Y − (∇F X F )Y − [(∇X η)Y ] ξ = 0, (∇X F )Y − (∇F X F )F Y + η(Y )∇F X ξ = 0

(1.13)

for all X, Y ∈ X (M ). √ √ Since the eigenvalues of F|D are + −1 and − −1, we deduce that the complexified Dc of D has the decomposition Dc = D10 ⊕ D01

(1.14) √ √ where D10 , D01 are the eigenspaces corresponding to + −1 and − −1, respectively. A simple argument shows that √ √ D10 = {X − −1F X; X ∈ D}, D01 = {X + −1F X; X ∈ D}

1.2. NORMAL ALMOST CONTACT MANIFOLDS

9

and extending to T c M the metric g by √ √ g c (X1 + −1X2 , Y ) = g(X1 , Y ) + −1g(X2 , Y ), √ √ g c (X, Y1 + −1Y2 ) = g(X, Y1 ) − −1g(X, Y2 ) we obtain a Hermitian metric g c on T c M . From Proposition 1.1.3, d) we deduce that with respect to this metric the decomposition (1.14) is orthogonal and to this one the following orthogonal decomposition of the complexified tangent bundle is associated c

c

T c M = Dc ⊕ hξi = D10 ⊕ D01 ⊕ hξi

(1.15)

c

where hξi = C ⊗ hξi. On the other hand, (T c M, g c ) is a Hermitian vector bundle over M and the natural extension ∇c of the Levi-Civita connection ∇ from M is a Hermic c tian connection in this bundle. Moreover, (Dc , g D = g|D c ×D c ) is a Hermitian c c c subbundle of (T M, g ), with the Hermitian connection ∇D induced by the following decomposition c

c

∇c Z = ∇D Z + AD Z c

(1.16)

c

c

where Z ∈ Dc , ∇D Z ∈ L (T c M, Dc ) , AD Z ∈ L (T c M, hξi ). A simple computation shows that c

AD X Z = −Ω(X, Y )ξ,

c

∇D F = 0,

c

hence ∇D is an almost complex connection in the complex bundle Dc . c Let g 10 be the restriction of the metric g D to D10 . Following the same argu c ment as above we deduce that D10 , g 10 is a Hermitian subbundle of Dc , g D , with the Hermitian connection ∇10 induced by the following decomposition c

∇D Z = ∇10 Z + A10 Z (1.17) where Z ∈ D10 , ∇10 Z ∈ L T c M, D10 , A10 Z ∈ L T c M, D01 . Another characterization of normal almost contact manifolds, due to S. Sasaki and C. J. Hsu, is the following Theorem 1.2.3. [SH62] Let M be a manifold equipped with the almost contact Riemannian structure (F, ξ, η, g). The following assertions are equivalent: a) the structure (F, ξ, η, g) is normal; c c b) the distributions D10 and D01 ⊕hξi (or D01 and D10 ⊕hξi ) are integrable and N (4) ≡ Lξ η = 0 Proof. Let J be the almost complex structure defined on M ×R at the beginning of this Section. From the definition of the tensor field NJ we deduce that the following conditions are equivalent: 1. NJ = 0;

10


2. the distributions √determined by the eigenspaces corresponding to the √ eigenvalues −1 and − −1 of J √are integrable.

√ d d The distributions D10 ⊕ ξ + −1 dt and D01 ⊕ ξ − −1 dt are complec mentary with respect to the metric g . Moreover, these distributions correspond √ √ to the eigenvalues −1 and − −1 of J, respectively. a) ⇒ b) By the proof of Theorem 1.2.1 we have N (4) = 0. On the other hand, let X, Y√∈ D. Then N (1) (X, Y ) = NF (X, Y ) = 0 and computing [X − √ −1F X, Y − −1F Y ], it follows that the distribution D10 is integrable. Now for Z ∈ D10 we have √ √ d d c ∈ D10 ⊕ ξ + −1 ∩ T c M = D10 ⊕ hξi [Z, ξ] = Z, ξ + −1 dt dt c

hence D10 ⊕ hξi is also integrable. c Similarly we prove the integrability of the distributions D01 and D01 ⊕ hξi . c b) ⇒ a) Assume that the distributions D10 and D01 ⊕ hξi are integrable. c Then D01 and D10 ⊕ hξi are obviously integrable and for Z ∈ D10 we have c [Z, ξ] ∈ D10 ⊕ hξi . But since Z⊥ξ we obtain η[Z, ξ] = −η (Lξ Z) = −Lξ (η(Z)) + (Lξ η)Z = 0 and then [Z, ξ] ∈ D10 . Therefore √ d √ d 10 10 = [Z, ξ] ∈ D ⊂ D ⊕ ξ + −1 Z, ξ + −1 d dt

√ d which proves that the distribution D10 ⊕ ξ + −1 dt is integrable. Similarly

√ d we prove that D01 ⊕ ξ − −1 dt is integrable and then the almost complex structure J is integrable, hence the almost contact structure (F, ξ, η, g) is normal. By direct verification and using a simple computation can be proved the following two results concerning the products of manifolds. Proposition 1.2.4. If the manifolds M1 , M2 are equipped with the almost Hermitian structure (F1 , g1 ) and the almost contact Riemannian structure (F2 , ξ2 , η2 , g2 ), respectively, then the tensor fields F, ξ, η, g, given by F (X, Y ) = (F1 X, F2 Y ), η(X, Y ) = η2 (Y ), ξ = (0, ξ2 ), g((X1 , Y1 ), (X2 , Y2 )) = g1 (X1 , X2 ) + g2 (Y1 , Y2 ) define an almost contact Riemannian structure on their product M1 × M2 . Proposition 1.2.5. If the manifolds M1 , M2 have the almost contact Riemannian structures (F1 , ξ1 , η1 , g1 ) and (F2 , ξ2 , η2 , g2 ), respectively, then the tensor field F given by F (X, Y ) = (F1 X − η2 (Y )ξ1 , F2 Y + η1 (X)ξ2 ) defines an almost Hermitian structure on the manifold M1 ×M2 , with the metric g from Proposition 1.2.4. This structure is Hermitian if and only if the almost contact Riemannian structures are normal.

1.3. CONTACT MANIFOLDS

11

The following formula is useful for the computation of the covariant derivative of F depending on the tensor fields N (1) , N (2) , in the case of arbitrary almost contact Riemannian structures. Proposition 1.2.6. Let (F, ξ, η, g) be an almost contact Riemannian structure on the manifold M . If ∇ is the Levi-Civita connection of the metric g then 2g ((∇X F )Y, Z) = 3dΩ(X, F Y, F Z) − 3dΩ(X, Y, Z)+ + g N (1) (Y, Z), F X + N (2) (Y, Z)η(X)+

(1.18)

+ 2dη(F Y, X)η(Z) − 2dη(F Z, X)η(Y ) for all X, Y, Z ∈ X (M ). Proof. The Levi-Civita connection of g is given by the formula (see for instance [KN69], vol. I, pg. 160) 2g(∇X Y, Z) = X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y ))+ + g([X, Y ], Z) + g([Z, X], Y ) − g([Y, Z], X)

(1.19)

and then, taking into account ∇g = 0, we have 2g ((∇X F )Y, Z) = 2g (∇X (F Y ), Z) + 2g(∇X Y, F Z) = X(g(F Y, Z)) + (F Y )(g(X, Z)) − Z(g(X, F Y ))+ + g([X, F Y ], Z) + g([Z, X], F Y ) − g([F Y, Z], X)+ + X(g(Y, F Z)) + Y (g(X, F Z)) − (F Z)(g(X, Y ))+ + g([X, Y ], F Z) + g([F Z, X], Y ) − g([Y, F Z], X) But g(X, F Y ) = Ω(X, Y ) and then the required formula follows by a straightforward computation, taking into account the expressions of the tensor fields N (1) , N (2) and dΩ.

1.3

Contact manifolds

If a 1-form η, satisfying the condition from Theorem 1.1.6, b) is given on the manifold M , that is if η ∧ (dη)n 6= 0 everywhere on M , then we say that η defines a contact structure or that M is a contact manifold and η is called the contact form of M . We remark that if f ∈ F(M ) nowhere vanishes on M then f η also is a contact form on M . Moreover, η and f η determine the same contact distribution, hence the authentic invariant of this change of contact forms is the contact distribution. For this reason it is more natural to define a contact structure by a 2n-dimensional distribution D on M , with the property that there exists a 1-form η ∈ F 1 (M ) so that ker η = D and η ∧ (dη)n nowhere vanishes on M . More generally, if D is a completely nonintegrable distribution of codimension 1 on M then the existence of the 1-form η with the above properties is guaranteed at least locally. The global existence of η is equivalent to the

12


coorientability of the distribution D (see Section 1.1). We generally consider cooriented contact structures and we specify such a structure by its contact form. When an almost contact Riemannian structure defined in Theorem 1.1.6, b) is fixed on the contact manifold M then we say that M is a contact Riemannian manifold. Remark 1.3.1. a) As the contact distribution D is defined by the equation η = 0 and since η ∧ (dη)n 6= 0 on M , from the Frobenius theorem it follows that the contact distribution of a contact manifold is never integrable. b) From the definition of the fundamental form and from Theorem 1.1.6, b) (see also the remark following its proof) it results that for a given contact Riemannian structure, the endomorphism F is uniquely determined by the 1form η and by the metric g. For the contact Riemannian manifold M we consider the contact distribution D with the natural vector bundle structure defined in Section 1.1. Taking into account Theorem 1.1.6, the restriction to D of the 2-form dη is non degenerate and then we can state the following Proposition 1.3.2. The contact distribution D of a contact Riemannian manifold has a symplectic vector bundle structure with the symplectic form dη|D . Denote by J (D) the space of almost complex structures on D, compatible with dη, that is the structures J : D −→ D with the properties J 2 = −ID ,

dη(J X, J Y ) = dη(X, Y ),

dη(J X, X) ≥ 0

(1.20)

for any X, Y ∈ D. This means that we consider on D only almost complex structures compatible with its symplectic bundle structure. We remark that if (F, ξ, η, g) is the almost contact Riemannian structure associated to the contact Riemannian structure defined in Theorem 1.1.6, b) on the manifold M then F|D ∈ J (D). For each J ∈ J (D) the map gJ , defined by gJ (X, Y ) = dη(J X, Y ),

X, Y ∈ D

(1.21)

is a Hermitian metric on D, that is it satisfies the condition gJ (J X, J Y ) = gJ (X, Y )

(1.22)

Moreover, if we denote by G(D) the set of all Riemannian metrics on D, satisfying the equality (1.22), it is easy to check that the map J ∈ J (D) 7−→ gJ ∈ G(D) is bijective. Since η nowhere vanishes on M , we denote by ξ a vector field with the property η(ξ) = 1 and extend J to an endomorphism F of X (M ) by setting F|D = J , F ξ = 0. Consider the decompositions X = X D + aξ, Y = Y D + bξ, where X D , Y D are the D components of the vector fields X and Y , respectively. Similarly, we extend gJ to a metric g on M by g(X, Y ) = gJ (X D , Y D ) + ab

(1.23)


13

for any X, Y ∈ X (M ). Taking into account (1.21) we can prove that dη(X, Y ) = g(X, F Y ), hence the contact structure on M is Riemannian. Moreover, (F, ξ, η, g) is an almost contact structure on M and then the set of almost contact Riemannian structures on M is in bijective correspondence with the set of almost complex structures of Hermitian type (J , gJ ), defined on the contact distribution D. The reader can compare this construction with the one given in the proof of Theorem 1.1.6. In the case when the manifold M has a contact Riemannian structure we use the notation gFD = gD and since (F|D , gD ) defines an almost Hermitian structure on the bundle D, we deduce Proposition 1.3.3. Any contact Riemannian manifold is a strongly pseudoconvex CR manifold1 . Let M be a contact Riemannian manifold and consider the product manifold M × R∗+ . Denoting by t the coordinate in R∗+ , we remark that the closed 2-form d(t2 η) is non degenerate and thus the manifold M ×R∗+ is symplectic. It is called the symplectic cone of the contact manifold M and we denote it by C(M ). On C(M ) we define the endomorphism C : X (C(M )) −→ X (C(M )) by C|D = F,

Cξ = −E,

CE = ξ

d and (F, ξ, η, g) is the associated almost contact Riemannian where E = t dt structure on M . We have C 2 = −I, hence C is an almost complex structure on C(M ). On the other hand, the metric G = dt2 + t2 g, defined on the symplectic cone C(M ), corresponds to the metric g of the contact Riemannian structure on M . From the definition of the almost complex structure C and taking into account 1.3 we deduce that the metric G is Hermitian and its fundamental 2form is exactly ΩC = d(t2 η). But ΩC is obviously closed, therefore C(M ) is an almost K¨ ahler manifold and then we can state the following

Theorem 1.3.4. If M is a contact Riemannian manifold then its symplectic cone C(M ) has an almost K¨ ahler structure. Otherwise, the above construction shows that to the contact Riemannian geometry of the manifold M , an almost Kähler geometry on its symplectic cone is corresponding. Let ∇ be the Levi-Civita connection of the contact Riemannian manifold M . By direct computation we obtain the following integrability condition for the distribution D10 (see also Theorem 1.2.3 for a similar result in the case of normal almost contact manifolds): Proposition 1.3.5. [Ta69d] The complex distribution D10 of the contact Riemannian manifold M is integrable if and only if the equality (∇X F )Y = g X + N (3) X, Y ξ − η(Y ) X + N (3) X (1.24) holds for any X, Y ∈ X (M ). 1 The manifold M is called a CR manifold if there exists a complex distribution T ( i. e. a subbundle of the complexified tangent bundle T M c ) so that [,T ] ⊂ T and T T ∩ T = 0.

14


Proposition 1.3.6. Let M be a contact Riemannian manifold and let (F, ξ, η, g) be the associated almost contact Riemannian structure. Then: a) N (2) = 0, N (4) = 0; b) N (3) = 0 if and only if ξ is a Killing vector field, i. e. Lξ g = 0; c) ∇ξ F = 0. Proof. a) A straightforward computation shows that N (2) (X, Y ) = 2dη(F X, Y ) − 2dη(F Y, X) and then the first equality a) results from dη(X, Y ) = g(X, F Y ) (see Theorem 1.1.6, b)). The second equality a) follows from the definition of the tensor field N (4) . Indeed we have (Lξ η) X = (dıξ η) X + (ıξ dη) X = dη(ξ, X) = 0 b) For all X, Y ∈ X (M ) we can write 0 = [LX , d]η(X, Y ) = (Lξ dη) (X, Y ) = = ξ(g(X, F Y )) − g([ξ, X], F Y ) − g(X, F [ξ, Y ]) = = (Lξ g) (X, F Y ) + 2g(X, N (3) Y ) Now, b) follows easily. c) results from 1.18 evaluated for X = ξ and using a). A more suitable form of the results from Proposition 1.3.6 and its proof is the following Proposition 1.3.7. Let M be a contact Riemannian manifold and let (F, ξ, η, g) be the associated almost contact Riemannian structure. Then: Lξ η = 0,

Lξ dη = 0,

(LF X η) Y = (LF Y η) X

for any X, Y ∈ X (M ). Another useful result is Proposition 1.3.8. On a contact Riemannian manifold the following formulas hold a) g N (3) X, Y = g X, N (3) Y ; b) ∇X ξ = −F X − F N (3) X; c) F N (3) = −N (3) F ; d) trace N (3) = 0, trace N (3) F = 0, N (3) ξ = 0, η N (3) X = 0; e) (∇X F ) Y + (∇F X F ) F Y = 2g(X, Y )ξ − η(Y ) X + N (3) X + η(X)ξ . Proof. We have 2g N (3) X, Y = g ((Lξ F )X, Y ) = g([ξ, F X] − F [ξ, X], Y ) = g (−∇F X ξ + F ∇X ξ, Y ) = η (∇F X Y ) + η (∇X (F Y )) = 2g(N (3) Y, X) + η[F X, Y ] + η[X, F Y ]


15

But from N (2) = 0 it follows η[F X, Y ] + η[X, F Y ] = 0 and then from the above equality we deduce a). b) By setting Y = ξ in (1.18) and taking into account Proposition 1.3.6 we obtain −2g (F ∇X ξ, Z) = g F 2 [ξ, Z] − F [ξ, F Z], F X − 2dη(F Z, X) = −2g F N (3) Z, F X − 2g(F Z, F X) Now, applying (1.3) and a) we deduce −g (F ∇X ξ, Z) = g N (3) X, Z − g(X, Z) + g(η(X)ξ, Z) that is − F ∇X ξ = −N (3) X − X + η(X)ξ

(1.25)

and b) follows by applying F to the last equality. c) By using a) and b), respectively, we get g N (3) F X, Y = g F X, N (3) Y = −g X, F N (3) Y = = g (X, ∇Y ξ) + Ω(X, Y ),

g FN

(3)

X, Y

= −g (∇X ξ, Y ) + Ω(X, Y )

On the other hand we have g (X, ∇Y ξ) − g (∇X ξ, Y ) = Y (η(X)) − X(η(Y )) + η[X, Y ] = −2dη(X, Y ) Now, c) follows easily. d) Let λ be a eigenvalue of N (3) and X be a corresponding eigenvector. From c) it follows that −λ also is an eigenvalue with F X as associated eigenvector and then we have trace N (3) = 0. The second equality follows similarly, while the third results if we take X = ξ in (1.25). The last formula follows applying a) for Y = ξ. e) From the definition of the tensor N (1) we deduce g N (1) (Y, Z), X + g N (1) (F Y, Z), F X = η [F Y, F Z] η(X)+ (1.26) + g ([η(Y )ξ, F Z] , F X) − g (F [η(Y )ξ, Z] , F X) On the other hand, applying Proposition 1.3.6, c) we have g ([η(Y )ξ, F Z] , F X) − g (F [η(Y )ξ, Z] , F X) = 2η(Y )g N (3) Z, F X and from a), b), c) we deduce η [F Y, XZ] = g ([F X, F Y ], ξ) = g (∇F Y (F Z), ξ) − g (∇F Z (F Y ), ξ) = = −g (F Z, ∇F Y ξ) + g (F Y, ∇F Z ξ) = = g F Z, F N (3) F Y − g F Y, F N (3) F Z = 0,

16


hence g N (1) (Y, Z), X + g N (1) (F Y, Z), F X = η(Y )g N (3) Z, F X

(1.27)

Now we compute the expression g ((∇X F ) Y, Z) + g ((∇F X F ) F Y, Z) with the use of the formula (1.18) and the equality e) follows from (1.26) taking into account a) and the third formula d). Finally, by putting into other words Theorem 1.1.6, b), we can assert that if η defines a contact structure on the manifold M then there exists an almost contact structure (F, ξ, η, g) with Ω = dη as fundamental form. Then it is natural to ask what kind can exist between the form η ∧ (dη)n and √ of relation 1 the volume form dV = det g dx ∧ dx2 ∧ . . . ∧ dx2n+1 of the Riemannian metric g. More exactly we have the following Theorem 1.3.9. Let M be a contact Riemannian manifold of dimension 2n+1 with the contact form η. The volume form with respect to the metric of M is given by dV =

1 2n n!

η ∧ (dη)n

(1.28)

Proof. Let us denote by x1 , x2 , . . . , x2n+1 the coordinates in a local chart with P2n+1 the domain U . Then η has the local expression η = i=1 ηi dxi and taking into account the definitions of the exterior differential and the inner product, we have ∂ηi 1 ∂ηj i j − j dη = Ωij dx ∧ dx where Ωij = (1.29) 2 ∂xi ∂x and η ∧ (dη)n = (2n)!λ dx1 ∧ dx2 ∧ . . . ∧ dx2n+1

(1.30)

where λ = (2n + 1)

X

(σ(1)σ(2)...σ(2n+1)) ησ(1) Ωσ(2)σ(3) . . . Ωσ(2n)σ(2n+1)

(1.31)

σ

Hence the condition η ∧ (dη)n 6= 0 is equivalent to λ 6= 0 everywhere on M . Moreover, if x e1 , x e2 , . . . x e2n+1 are the coordinates in a local chart with the domain V and U ∩ V 6= ∅ then it results from (1.30) that on U ∩ V we have i ∂x e λ (1.32) λ = det ∂e xj e is the function analogous to λ, but defined on V this time. Setting where λ eventually −x2n+1 instead of x2n+1 we can assume λ > 0 in any local chart of the manifold M .


17

In the local chart with the domain U we consider a F-basis B = {Xi , Xi∗ = F Xi , X2n+1 = ξ}. Denoting by {η1 , η2 , . . . η2n+1 } the dual basis of B we obtain Ωij =

n X

∗

∗

(ηia ηja − ηia ηja )

(1.33)

a=1

Moreover, taking into account (1.33) in (1.31) and using the elementary properties of permutations, we obtain X n(n+1) λ =(−1) 2 2n (2n + 1)n! (σ(1)σ(2)...σ(2n+1)) · σ (1.34) 2n+1 1 2 n 1∗ 2∗ n∗ ησ(2) . . . ησ(n) ησ(n+1) · ησ(1) ησ(n+2) . . . ησ(2n) ησ(2n+1) But λ > 0 everywhere, so that (1.34) shows that det(ηij ) has the same sign n(n+1) and this affirmation is also true for the sign of the determinant as (−1) 2 i det(Xj ) of components of the basis B with respect to the natural frame. Now, by considering the 1-forms ηek , locally given by ηek = ηik dxi for k ∈ 1, 2n + 1, it follows n(n+1) 2

∗

∗

ηe1 ∧ . . . ∧ ηen ∧ ηe1 ∧ . . . ∧ ηen ∧ ηe2n+1 = " X n(n+1) 1 2 n 2 (σ(1)σ(2)...σ(2n+1)) ησ(1) ησ(2) . . . ησ(n) · = (−1)

dV = (−1)

σ 1∗ 2∗ ·ησ(n+1) ησ(n+2)

i 2n+1 n∗ . . . ησ(2n) ησ(2n+1) dx1 ∧ dx2 ∧ . . . ∧ dxn

and then, taking into account (1.29), (1.30), (1.31) and (1.34), we deduce the announced formula (1.28). Let Fξ be the 1-dimensional foliation2 generated by the Reeb vector field ξ. It is called the characteristic foliation or the contact flow of the manifold M and it has the following properties deduced from the definitions: 2 A foliation F of dimension p on the m-dimensional M is a partition {L } α α∈Λ of M by connected subsets with the property that any point x ∈ M has a local chart (U, φ) with the local coordinates x1 , . . . , xp , y 1 , . . . , y q and such that φ(U ) = U1 × U2 , U1 ⊂ Rp , U2 ⊂ Rq , p + q = m, and the connected components of the set U ∩ Lα are characterized by equations of the form y 1 = constant, . . . , y q = constant. M endowed with the foliation F is called a foliated manifold and is denoted by (M, F ). The subsets Lα are connected submanifolds of dimension p of the manifold M and we call them the leaves of the foliation F . The vector field X ∈ (M ) is foliate if [X, Y ] ∈ Γ(F ) for all Y ∈ Γ(F ). If the foliated manifold (M, F ) is Riemannian with the metric g and if F ⊥ is the normal bundle of F with respect to g then we have the decomposition g = gF ⊕ gF ⊥ . On F ⊥ is defined the following connection  π ⊥ [X, Y ] for X ∈ Γ(F ) ∇F XY = π ⊥ (∇X Y ) for X ∈ Γ(F ⊥ )

where ∇ is the Levi-Civita connection of the metric g and π ⊥ is the natural projection of the tangent bundle T M on the normal bundle F ⊥ . g is a bundle-like metric for the foliation F if for any open set U ⊂ M the function g(X, Y ) is basic on U (i. e. is constant on the intersection of U with each leaf of F ) for all foliate vector fields X, Y ∈ X (U ) perpendicular

18


Proposition 1.3.10. The characteristic foliation of a contact manifold M is Riemannian, transversally orientable and its leaves are geodesics with respect to the bundle-like metric g. Proof. Assume that dim M = 2n + 1. It remains to prove that the characteristic foliation Fξ is transversally orientable. Indeed, from Theorem 1.3.9 it follows that the 2n-form Ωn ν= n 2 n! is a transversal volume form associated to the characteristic foliation Fξ of M and ∗ν = η. We will search an explicit local expression for the contact form of a contact manifold. For this purpose we first remark that, denoting by x1 , . . . , xn , y 1 , . . . , y n , z the Cartesian coordinates in R2n+1 , we can define the 1-form η0 = dz −

n X

y i dxi

(1.35)

i=1

and η0 ∧ (dη0 )n = (−1)

n(n+1) 2

dx1 ∧ . . . ∧ dxn ∧ dy 1 ∧ . . . ∧ dy n ∧ dz

This formula proves that R2n+1 is a contact manifold with the contact form η0 given by (1.35). But on arbitrary contact manifolds a converse assertion is also true at least locally, namely we have Theorem 1.3.11. Let η be the contact form of the 2n + 1-dimensional contact manifold M . At any point of M there exists a local chart with coordinates x1 , . . . , xn , y 1 , . . . y n , z such that the expression of η is η = dz −

n X

y i dxi

(1.36)

i=1

Proof. Since M is locally homeomorphic to R2n+1 , we can reason in R2n+1 and denote also by η the image of the contact form in R2n+1 . Now, it is sufficient to apply Darboux Theorem3 . to the leaves. F is a Riemannian foliation if ∇F X gF ⊥ = 0 for any X ∈ F . The connection ∇F is metrical (that is ∇F X gF ⊥ = 0 for any X ∈ X (M )) if and only if F is a Riemannian foliation and g is a bundle-like metric ([To88], pg. 53, theorem 5.11). The restriction to X ∈ F of the connection ∇F is the Bott connection in F ⊥ . For the study of foliations see [To88]. 3 In its general form, Darboux Theorem is the following: Theorem. By a suitable coordinates transformation in Rm , any 1-form η can be locally expressed under one of the following canonical forms η=−

p X k=1

x2k dx2k−1 + dx2p+1 ,

η=

p X

x2k dx2k−1

k=1

according to det(Ωij ) 6= 0 or to det(Ωij ) = 0, Ωij being given in (1.29). A proof can be given in [Vr47], pg. 31–35.


19

By means of these local coordinates we can characterize an important class of forms on a contact manifold. Denote by FBp (M ) the space of basic p-forms on the contact manifold M with respect to the characteristic foliation Fξ . These are p-forms ω ∈ F p (M ) satisfying the conditions ıξ ω = 0,

Lξ ω = 0

(1.37)

(see [To88], Chapter 9 for the general theory of basic forms on a foliated manifold). The form ω is called horizontal or invariant depending if the first or the second condition (1.37) is satisfied. We remark that an invariant form is invariant by the action of the Reeb group of M . To the decomposition (1.14) it correspond the following decompositions FBp (M ) ⊗ C = ⊕FBr,s (M ) where FBr,s (M ) is the space of basic forms of type (r, s), that is the basic forms which can be nonzero only when act on r vector fields from D10 and on s vector fields from D01 . The study of these forms in the case of an important class of contact manifolds will be approached in Chapter 4. Here we only mention the following elementary result; Proposition 1.3.12. Let ω be a p-form on the contact manifold M . The following affirmations are equivalent: a) ω is basic; b) ω and dω are horizontal forms; c) there are the local coordinates x1 , . . . , xn , y 1 , . . . , y n , z such that ω has the expression X ω= ωi1 ...ir ir+1 ...ip (xi , y i )dxi1 ∧ . . . ∧ dxir ∧ dy ir+1 ∧ . . . ∧ dy ip Proof. The verification is straightforward by using the local coordinates from Theorem 1.3.11 and the formulas 1.37. The diffeomorphism µ : M −→ M of the contact manifold M is called a contact transformation if there exists f ∈ F(M ) nowhere zero on M and such that µ∗ η = f η

(1.38)

If f ≡ 1 then µ is a strict contact transformation. This definition can be easily adapted to arbitrary maps between two (almost) contact manifolds. So, let M1 , M2 be two contact manifolds having the contact forms η1 and η2 , respectively. The differentiable map (diffeomorphism) µ : M1 −→ M2 is called a contact map (contact transformation) if there exists a function f : M1 −→ R∗ such that µ∗ η2 = f η1 If f ≡ 1 the map µ is called strict contact and if, moreover, µ is a contact diffeomorphism then the contact forms η1 and η2 are equivalent.

20


Proposition 1.3.13. The diffeomorphism µ of the contact manifold M is a contact transformation if and only if µ∗ D ⊆ D. Proof. Consider x ∈ M, Xx ∈ Dx and denote y = µ(x), Xy = µ∗,x Xx . Since Xy ∈ Dy , we have 0 = η(Xy ) = η(µ∗,x Xx ) = µ∗x η(Xx ) and therefore µ∗ η is collinear to η. On the other hand, by setting µ∗ ξ = aξ +bX, with a 6= 0 and X ∈ D, we have (µ∗ η)ξ = η(µ∗ ξ) = aη(ξ) hence the equality (1.38) is satisfied. The converse is obvious. From Theorem 1.3.11 we deduce that for any local charts (U, φ), (V, ψ) with non disjoint domains, the diffeomorphism ψ ◦ φ−1 is a contact transformation and then we can generalize the notion of contact structure in the following way: We say that the 2n + 1-dimensional manifold M has a contact structure in the wider sense if there exists an atlas A = {(Uα , φα )}α∈I with the property that for any α, β ∈ I such that Uα ∩ Uβ 6= ∅, the map φα ◦ φ−1 β is a contact transformation of R2n+1 equipped with the contact structure defined by the contact form η0 given in formula (1.35). As we already noticed, any contact manifold has a contact structure in the wider sense. Under some conditions the converse is also true. More precisely we have Theorem 1.3.14. Let M be a 2n + 1-dimensional orientable contact manifold in the wider sense. If n is even then M is a contact manifold. Proof. Let A be an atlas defining the contact structure in the wider sense of the manifold M . The form ηα = φ∗α η0 is the image in Uα of the contact form ∗ η0 of R2n+1 , and it also is a contact form and ηβ = (φβ ◦ φ−1 α ) ηα . Then the spaces Dx = {X ∈ Tx M : ηα (X) = 0} are the local fibers of a subbundle D of the tangent bundle T M . The manifold M is orientable so its tangent bundle is orientable and therefore its structure group reduces to the subgroup SO(n) of GL(n; R), (see [KN69], vol. II, pg. 314). But the subbundle D is orientable and then the linear quotient bundle T M/D is orientable, hence it has a global section s without zeros (see for instance [KN69], vol. I, pg. 57). On the other hand, the forms ηα define completely the vector fields Xα ∈ X (Uα ) by the equations ηα (Xα ) = 1 and ıXα dηα = 0. It follows that Xα is collinear with s, that is Xα = hα s, where hα : Uα −→ R∗ . Now, by putting η = hα ηα on each Uα , we obtain the 1-form η ∈ F 1 (M ) with the property that η ∧ (dη)n 6= 0. Conditions in order that a contact manifold in the wider sense can be orientable are given in Theorem 1.3.15. If n is odd then any 2n + 1-dimensional connected contact manifold in the wider sense is orientable.


21

Proof. Let ηα be the contact form corresponding to the local chart (Uα , φα ) in the atlas A (see the proof of Theorem 1.3.14). If Uα ∩ Uβ 6= ∅ then we have ηα = hαβ ηβ , with hαβ : Uα ∩ Uβ −→ R∗ and therefore n ηα ∧ (dηα )n = hn+1 αβ ηβ ∧ (dηβ )

(1.39)

For each 1-form ηα we construct the function λα like in the formula (1.31) and we can assume that λα > 0 (see the proof of Theorem 1.3.9). But from (1.32) and (1.39) it results i ∂x = hn+1 λα det αβ λβ ∂x ˜j i and then, since n is odd and λα > 0 for any α ∈ I, it follows det ∂∂xx˜j > 0 Let M be a positively co-oriented contact manifold with the contact distribution D and let g be a Lie algebra acting on M by Reeb vector fields. This means that we suppose the existence of a representation ρ : g −→ Xξ (M ) of g, i. e. ρ is a Lie algebra morphism. If g∗ is the dual of the algebra g considered as a module over F(M ) then the following map is well-defined 0 Ψ : D+ −→ g∗

;

Ψ(x, ηx+ )(X) = ηx+ (ρ(X)(x))

(1.40)

0 and X ∈ g. Ψ is the contact moment map relative to for any x ∈ M, η + ∈ D+ the representation ρ. Under this form, the notion was defined by E. Lerman, [Le03]. Now, we show that the notion of contact moment map is a natural adaptation to contact structures of the general notion of moment map4 . For this purpose we assume that the representation ρ results from the action (at left) of a Lie group G of transformations of M , so that the contact distribution and the co-orientation of the manifold M are preserved. This means that there is a differentiable map L : M × G −→ M , defined by L(t, x) = tx = Lt (x), where Lt are transformations of the manifold M and for any t ∈ G we have

(Lt )∗ D = D

;

0 0 (Lt )∗ D+ = D+

From these equalities it follows that if Φ : T ∗ M −→ g∗ is the moment map of the lift of the action of G over T ∗ M then Ψ = Φ|D+0 . On the other hand, the lift 4 Let ω be a symplectic form on the manifold M and assume that the Lie group G acts on M by an action · preserving ω. Also denote by : g∗ × g −→ R the pairing between the Lie algebra g of G and its dual g∗ . For each X ∈ g the equality

Xx• =

d exp(tX) · x dt |t=0

defines a vector field X • ∈ X (M ). Because ω is invariant by the action of G it follows that the 1-form ıX • ω is closed for all X ∈ g. A moment map for the action · on M is a map µ : M −→ g∗ such that for all X ∈ g we have d (< µ, X >) = ıX • ω where < µ, X > (x) =< µ(x), X >.

22


0 of the action of G to T ∗ M preserves D+ if and only if the action of G preserves D and its co-orientation and then any G-invariant 1-form α with the property that D = ker α is a G-equivariant5 section in the bundle with the total space 0 D+ and the base space M . Hence the composition map Φα = Φ ◦ α : M −→ g∗ is well-defined. Φα is G-equivariant and we call it the α-contact moment map. Taking into account (1.40), Φα has the expression

hΦα , Xi (x) = Φα (x)(X) = αx (ρ(X)(x)) = (ıρ(X) α)(x)

(1.41)

for any X ∈ g and x ∈ M . This is the classical notion of contact moment map, considered in C. Albert’s paper, [Alb89]. From (1.38), (1.40) and (1.41) we deduce that the contact moment map Ψ is invariant by contact transformations, while the α-contact moment map generally has not this property. Examples 1. Contact hypersurfaces Let S be a orientable hypersurface in the 2n + 2dimensional manifold M with the almost Hermitian structure (J, g). Since S is orientable and has odd dimension there exists a normal vector field nowhere vanishing on S. Denote by ~n such a vector field and assume it to be unitary with respect to g. Obviously, ξ = J~n is unitary and tangent to S. Moreover, for each vector field X ∈ X (S) the decomposition of F X into tangent and normal component JX = F X − η(X)~n

(1.42)

induces the endomorphism F : X (S) −→ X (S) and the 1-form η on S. Then the tensor fields F, ξ, η define an almost contact structure and denoting also by g the restriction to S of the Hermitian metric g and taking into account (1.42) we have g(X, Y ) = g(JX, JY ) = g(F X, F Y ) + η(X)η(Y ) hence (F, ξ, η, g) is an almost contact Riemannian structure on S. If the manifold M is Kählerian then the above result can be refined, but first we establish some formulas. Denoting by ∇ the Levi-Civita connection on M and by ∇S the induced connection on S, from the Gauss formula we get ∇X ξ = ∇SX ξ + h(X, ξ)~n

(1.43)

On the other hand we have ∇J = 0 and by using the Weingarten formula for the unitary normal vector ~n and the equality (1.42) we deduce ∇X ξ = ∇X (J~n) = J∇X ~n = −JA~n X = −F A~n X + η(A~n X)~n 5 Consider the manifolds M , M , a map µ : M −→ M and two right actions of the Lie 1 2 1 2 group G on these manifolds, each written as (x, a) 7−→ x · a. One says that µ is G-equivariant (or simply equivariant) if µ(x · a) = µ(x) · a for all x ∈ M1 and a ∈ G.


23

Now, by comparing (1.43) with the above equality we obtain ∇SX ξ = −F A~n X Similarly, taking the covariant derivative of (1.42) we get ∇SX F Y = g(h(X, Y ), ~n)ξ − η(Y )A~n X

(1.44)

(1.45)

for any X, Y ∈ X (S). If (F, ξ, η, g) is a contact Riemannian structure then S is called a contact hypersurface of the K¨ ahler manifold M . Proposition 1.3.16. The orientable hypersurface S of the K¨ ahler manifold M is a contact hypersurface if and only if A~n F + F A~n = 2F Proof. From (1.44) we obtain 2dη(X, Y ) = g ∇SX ξ, Y − g ∇SY ξ, X = = −g(F A~n X, Y ) − g(A~n Y, F X) = = −g ((F A~n + A~n F ) X, Y ) and since (F, ξ, η, g) is a contact Riemannian structure only when dη = Ω, we deduce the desired result. Let S be a contact hypersurface of the Kähler manifold M . Comparing (1.44) with the formula b) of Proposition 1.3.8 and taking into account the last equality d0 from the same Proposition, we obtain A~n X = X + N (3) X + η(A~n X − X)ξ

(1.46)

and then A~n ξ = (trace A~n − 2n)ξ,

trace A~n = 2n + η(A~n ξ)

(1.47)

From (1.46), (1.47) we deduce A~n X = X + N (3) X + (trace A~n − 2n − 1)η(X)ξ and taking into account (1.45) we have ∇SX F Y = g(X + N (3) X, Y )ξ − η(Y )(X + N (3) X)

(1.48)

(1.49)

On the other hand, we remark that this last equality holds on any 3-dimensional contact Riemannian manifold. it characterizes the integrability of the complex distribution D10 of S (see Proposition 1.3.5). The hypersurface M of the K¨ ahler manifold M is a Hopf hypersurface if the foliation determined by the integral submanifolds of the subbundle JT ⊥ S ⊂ T S is totally geodesic.

24


Proposition 1.3.17. A contact hypersurface of a K¨ ahler manifold is a Hopf hypersurface. Proof. The orientable hypersurface S of the Kähler manifold M is a Hopf hypersurface if and only if ξ is a principal curvature vector of S at any point and then by the first equality (1.47) we obtain the desired result. Proposition 1.3.18. [Ok66] Any contact hypersurface of a complex space form has constant mean curvature. Proof. If M (c) is a complex space form then its curvature vector is given by (see for instance [KN69], vol. II, pg. 167) c [g(Y, Z)X − g(X, Z)Y + 4 + g(JY, Z)X − g(JX, Z)JY + 2g(X, JY )JZ]

RS (X, Y )Z =

(1.50)

and for X = Z = ~n, Y = ξ we obtain RS (~n, J~n)J~n = c~n. On the other hand from the Codazzi equation and using the first equality (1.47) we get 0 = g(RS (~n, J~n)J~n, X) = X(trace A~n ) − (ξ trace A~n )η(X)

(1.51)

for X ∈ X (S). Hence d(traceA~n ) = (ξ traceA~n )η and applying d to this relation we deduce d(ξ trace A~n ) ∧ η + (ξ trace A~n )dη = 0 When this last equality acts on (F X, F Y ) we obtain ξ trace A~n = 0, hence by (1.51) it follows trace A~n = constant. In the case when the ambient manifold is the euclidean space R2n+2 with the standard K¨ ahler structure, M. Okumura obtained the following classification theorem Theorem 1.3.19. [Ok66] A complete simply connected contact hypersurface of R2n+2 is isometric to the unit sphere S 2n+1 or to Rn+1 × S n (4). This result will be proved in a more general case in Section 3.1 (see Theorem 3.1.7). Now, we use this method in order to construct an almost contact structure on the unit sphere S 5 . For this purpose we consider the sphere S 6 immersed into R7 identified with the space O of purely imaginary octaves and define the vector product u × v of two such octaves u and v as the imaginary part of their product uv. The following relations hold uv = −u · v + u × v, 1 u × v = (uv − vu), 2 where · is the usual scalar product in R7 .

u × v = −v × u, 1 u · v = − (uv + vu) 2


25

Let u ∈ S 6 , that is u · u = 1. By identifying the tangent space at u to S 6 7 7 with the subspace v ∈ R : u · v = 0 of R , the following endomorphism is well-defined (see for instance [KN69], vol. II, pg. 139–140) Ju : Tu (S 6 ) −→ Tu (S 6 ),

Ju (v) = v × u

We have Ju2 = −I, hence J is an almost complex structure on the sphere S 6 . Moreover, if g˜ is the metric induced on S 6 by the standard Riemannian metric on R7 then we have g˜(JX, JY ) = g˜(X, Y ), i. e. (J, g˜) is an almost Hermitian structure on S 6 . Let be the sphere S 5 ⊂ S 6 ⊂ R7 , defined by the equation x7 = 0 and let ~n = ∂z∂ 7 be its normal vector field. S 5 is a orientable hypersurface in S 6 and since S 6 has an almost Hermitian structure, by the above argument there exists on S 5 an almost contact Riemannian structure (F, ξ, η, g). By a straightforward computation, from the expression of J we obtain ξ = x1

∂ ∂ ∂ ∂ ∂ ∂ − x2 5 − x3 4 + x4 3 + x5 2 − x6 1 6 ∂x ∂x ∂x ∂x ∂x ∂x

On the other hand, η is the restriction to S 5 of the 1-form α = x1 dx6 − x6 dx2 + x5 dx2 − x2 dx5 + x4 dx3 − x3 dx4 and by computing α ∧ (dα)n , it follows that η is a contact form. Therefore, on S 5 is defined a contact structure, but it is not a contact hypersurface of S 6 because ∂ dη(X, Y ) = g X, 7 × Y 6= g(X, F Y ) ∂x 2. Contact structure on the real projective space P2n+1 (R) Proposition 1.3.20. Let S be a hypersurface of R2n+2 . If the tangent hyperplanes of S do not pass through the origin of the space R2n+2 then S has a contact structure. Proof. Consider on R2n+2 the 1-form α=

2n+1 X

xi dxi+1 − xi+1 dxi

i=1

where x1 , . . . , x2n+2 are the Cartesian coordinates and let X1 , . . . , X2n+1 be 2n+ ). 1 linearly independent vectors with the origin at the point x0 = (x10 , . . . , x2n+2 0 Also, we consider the vector w with the origin at x0 and having the components wj = ∗dxj (X1 , . . . , X2n+1 ), where ∗ is the Hodge operator and on the Riemannian manifold R2n+2 with the metric gij = δij . Since dj ∧ . . . ∧ dx2n+2 ∗dxj = (−1)j−1 dx1 ∧ . . . ∧ dx

26


(see Notations and Formulas in Appendix), it is easy to check that w is orthogonal to the vectors X1 , . . . , X2n+1 and [α ∧ (dα)n ] (V1 , . . . , V2n+1 ) =

2n+2 X

xj0 wj = x0 · w

j=1

Therefore, if the hyperplane spanned by the vectors X1 , . . . , X2n+1 and passing through the point x0 does not contains the origin then α ∧ (dα)n 6= 0. It follows that if ı : S ,→ R2n+2 is the immersion map of the hypersurface S into R2n+2 then η = ı∗ α is a contact form on S. Writing the equation of the hyperplane tangent to the real projective space P2n+1 (R) in a point we deduce that this hyperplane never contains the origin, therefore P2n+1 (R) has a contact structure. We also remark that Proposition 1.3.20 can be applied in order to construct a contact structure on the sphere S 2n+1 . 3. Contact structures on the sphere tangent bundle First, we define the Sasaki metric on the tangent bundle of a m-dimensional Riemannian manifold M with the metric g. We study the more general case of a vector bundle E = (E, π, M ) with the fiber of dimension k and under the hypothesis that a metric g ∗ and a metrical connection D (i. e. Dg ∗ = 0) are given on E. Denote by x1 , . . . , xm the local coordinates in a chart h = (U, φ) at the point x ∈ M and put q i = xi ◦ π. If {s1 , . . . , sk } is an orthonormal local basis of sections in the vector chart associated to h then the coordinates of the point (x, u) ∈ π −1 (U ) are q 1 , . . . , q m , u1 , . . . , uk , where u1 , . . . , uk are given by the Pk representation u = α=1 uα sα (x). For the vector field X ∈ X (M ) and for the section s ∈ Γ(E), locally given by X=

m X

Xi

i=1

∂ ∂ = Xi i , ∂xi ∂x

s=

k X

uα sα = uα sα

(1.52)

α=1

we have DX s = X i

∂uα β α + u µ βi sα ∂xi

where

D

∂ ∂xi

sβ = µα βi sα

The horizontal lift X H of the vector field X ∈ X (M ) and the vertical lift sV of the section s ∈ Γ(E) are defined by ∂ ∂ β α ∂ XH = Xi − u µ ; sV = uα α βi i α ∂q ∂u ∂u Then π∗ X H = X, π∗ sV = 0 and we can define the linear map K : T E −→ E by KX H = 0, (KsV )(x, u) = sx for any (x, u) ∈ E. It is called the connection map associated to D and has the following local expression ˜= X ˜ m+α + X ˜ i uβ µα sα KX βi


27

˜ i, X ˜ m+α are the components of the tangent vector X ˜ ∈ T(x,u) E with where X i α respect to the coordinates (q , u ). The equality ˜ Y˜ ) = g(π∗ X, ˜ π∗ Y˜ ) + g ∗ (K X, ˜ K Y˜ ), G(X,

˜ Y˜ ∈ X (E) X,

defines a Riemannian metric G on E, called the Sasaki metric. In the particular case when E is the tangent bundle of the manifold M , this metric was introduced by S. Sasaki, [Sa58]. See also the introductory part of Section 2.4. Other properties of the connection map and of the Sasaki metric are presented in [Ia83], pg. 65–70 and 101–104. Now, let E = T M and let D = ∇ be the Levi-Civita connection of the metric g on the Riemannian manifold M . The hypersurface of the manifold T M , whose elements are all unitary vector fields with respect to the Sasaki metric G on T M is called the sphere tangent bundle of M and we denote it by T1 M . In local coordinates T1 M is defined by the equation Gij v i v j = 1, where ∂ Gij are the components of the Sasaki metric G and X = v i ∂x i ∈ X (M ). By a method analogous to that one used in Example 1 it is easy to check that the almost complex structure J on T M given by JX H = X V , JX V = −X H , defines on T1 M the tensor fields ξ 0 , F 0 , η 0 with the expressions ξ 0 = −J~n = −v i J

∂ ∂xi

V

= vi

∂ ∂xi

H ;

JX = F 0 X + η 0 (X)~n,

where F 0 X, η 0 (X) vecn are the tangent component and the normal component of the vector field JX relative to the hypersurface T1 M . Moreover, denoting by g 0 the Riemannian metric induced by the Sasaki metric G on T1 M , it is easy to prove that (F 0 , ξ 0 , η 0 , g 0 ) is an almost contact Riemannian structure and it satisfies the condition g 0 (X, F 0 Y ) = 2dη 0 (X, Y ). Now, by putting η=

1 0 1 η , ξ = 2ξ 0 , F = F 0 , g = g 0 , 2 4

we obtain a contact Riemannian structure on T1 M , hence the sphere tangent bundle is a contact hypersurface of the tangent bundle endowed with the almost complex structure J. 4. Contact structure on T ∗ M × R Let M be a n-dimensional Riemannian manifold and denote by π : T ∗ M −→ M the canonical projection of its cotangent bundle T ∗ M . Also, we consider the local coordinates q 1 , . . . , q n , p1 , . . . , pn , where pi are the components of the 1-form ω with respect to the canonical cobasis {dx1 , . . . , dxn } and set q i = xi ◦ π. The 1-form β, locally given by Pn β = i=1 pi dq i , is globally defined on T ∗ M . Le t be the Cartesian coordinate in R and denote by p : T ∗ M × R −→ T ∗ M the first factor projection. A straightforward computation shows that η = dt − p∗ β defines a contact structure on T ∗ M × R.

28


Particularly, if M is the configuration space in the case of N particles from R3 then T ∗ M is the phase space, q i are the generalized coordinates, and pi are the generalized impulsions. Moreover, we can prove that there exists a coordinates transformation in T ∗ M , which also preserves the 2-form dβ. Such a transformation is exactly the ”canonical transformation ” in the sense of classical mechanics. For the notions from mechanics used here see for instance [Ar76]. More exactly, problems related to generalized coordinates and Hamilton equations are presented in Chapter 3 of Arnold’s book, the canonical transformations are discussed in Chapter 9, while the reader interested by the relation of the geometry with the mechanics find in pg. 351–374 a presentation of the contact structure, this being inspired from the study of the manifold of contact elements of the configuration space, as well as a lot of interesting results which derive from this approach. 5. Energy surface of Hamiltonians of a symplectic manifold Let M be a n-dimensional manifold and r ∈ R. We call a regular energy surface of f ∈ F(M ) a connected component Σr of f −1 (r) containing only regular points, i. e. Σr = {x ∈ M : f (x) = r, dx f 6= 0}. Since for regular x ∈ M , f −1 (r) is wholly a submanifold of M , it is natural to ask if this property remains valid for any oint x ∈ M and for any Σr . Indeed, we have Proposition 1.3.21. Let M be a n-dimensional manifold, f ∈ F(M ) and r ∈ R. Then any regular energy surface Σr of f is a connected hypersurface of M . Moreover, if M is orientable then Σr is also orientable. Proof. Any manifold is locally connected, hence Σr is both open and closed in M . Since f −1 (r) is closed in M , the set f −1 (r) − Σr is also simultaneously open and closed and taking into account that M is a normal space, there exists an open set O ⊂ M such that Σr ⊂ O and O ∩ f −1 (r) − Σr = ∅ and therefore O∩f −1 (r) = Σr . Now, r is a regular value of f|O , hence Σr is a n−1-dimensional submanifold of O and then of M . Before to define a contact structure on regular energy surfaces, we state some geometric properties, valid when the even-dimensional manifold M is symplectic with the symplectic form ω. For such a manifold the vector field X ∈ X (M ) is Hamiltonian if there is H ∈ F(M ) such that ıX ω = dH

(1.53)

In this case H is called a Hamiltonian for X and X itself is denoted by XH . Now, the announced properties of Σr are the following Theorem 1.3.22. Let M be a syplectic manifold and Σr a regular energy surface of H ∈ F(M ). a) Every orbit of XH that intersects Σr belongs wholly in Σr . b) There is a volume form on Σr , invariant by XH|Σr .


29

Proof. a) Let γ : (a, b) ⊂ R −→ H −1 (r) be a curve such that γ(a, b) ∩ Σr 6= ∅. Σr is a connected component of H −1 (r) and γ(a, b) is connected, therefore γ(a, b) ⊂ Σr . b) By a), XH|Σr is a vector field on Σr and from (1.53) we deduce LXH dH = dLXH H = 0 that is dH is an invariant form of XH . Then b) follows from the classical Hamilton-Jacobi theorem (see for instance [Abr67], pg. 105–106). Let M be a n-dimensional manifold, T ∗ M its cotangent bundle and β the 1-form defined on T ∗ M in Example 4. Then we have Proposition 1.3.23. Let Σr be a regular energy surface of H ∈ F(T ∗ M ). If β(XH ) is nowhere zero on Σr then ı∗ β defines a contact structure on Σr , where ı : Σr ,→ T ∗ M denotes the inclusion map. Its Reeb vector field is XH|Σr . Proof. ı∗ dβ has maximal rank since dβ is nondegenerate on T ∗ M and our first assertion follows because ı∗ β is nonzero on Σr . On the other hand, we have ıXH dı∗ β = dH and since on Σr the hamiltonian H is constant, it follows ıH|Σr ı∗ dβ = ıH|Σr dı∗ β = 0 But dH vanishes at every point of Σr , hence ıXH ı∗ β = 1 and we can apply Theorem 1.1.6, b). 6. Contact structure on M ×R, where M is a symplectic manifold Let M be a 2n-dimensional symplectic manifold whose symplectic form ω is exact, that is there exists θ ∈ F 1 (M ) such that ω = dθ. Also, let H ∈ F(M × R) be a nonzero function on M × R and denote by π : M × R −→ M the first factor projection. The 1-form ηH = π∗ θ + Hdt on M × R satisfies n

n−1

(dηH ) = (π∗ ω) n

n

∧ dH ∧ dt + (π∗ ω)

n

n

and then (dηH ) ∧ dt = (π∗ ω) ∧ dt, hence ηH ∧ (dηH ) is a volume form on M × R. Thus ηH defines a contact structure on M × R and we remark that this generalizes Example 4. Now, denote by XHt the Hamiltonian field on M , associated to Ht defined by Ht (x) = H(x, t), and consider the map XH : M × R −→ T M with XH (x, t) = XHt (x). Using the natural bundle isomorphism T (M × R) ≡ T M × T R, we ˜ H : M × R −→ T (M × R by X ˜ H (x, t) = define on M × R the vector field X ˜ ((t, 1), XH (x, t)). We remark that XH = XH +t , where t : M × R −→ T M × T R ≡ T (M ×R) is defined by t (x, s) = (0x , s, 1) and then by simple computation we get ıX˜ H dηH = ıX˜ H dπ∗ θ + ıX˜ H dH dt − ıX˜ H dt dH, ıX˜ H dηH (Y ) = 2dηH (XH +t , Y ) = 2ω(XH , Y ) = dH(Y ), ıX˜ H dH = dH(t ),

ıX˜ H dt = 1

30


˜ H is the Reeb vector Therefore we deduce ıX˜ H dηH = 0, ıX˜ H ηH = 1, that is X field of the contact structure defied by ηH . This contact structure is basic in the study of time depending mechanics (see for instance [Abr67], Chapter IV, pg. 132–153).

1.4

K-contact manifolds

A contact Riemannian manifold with the property that its Reeb vector field is Killing is called a K-contact manifold. From Propositions 1.3.6, b) and 1.3.8, b) it results immediately Proposition 1.4.1. A contact Riemannian manifold M is K-contact if and only if ∇X ξ = −F X

(1.54)

for any X ∈ X (M ). From the formula (1.54) it follows Proposition 1.4.2. On a K-contact manifold M the following equalities hold (∇X η) Y = g (∇X ξ, Y ) = Ω(X, Y );

(∇X F ) ξ = −X + η(X)ξ

(1.55)

for any X, Y ∈ X (M ). Using the same notations as in Section 1.2 we obtain a characterization of K-contact manifolds, namely: Theorem 1.4.3. Let M be a compact contact Riemannian manifold. The following assertions are equivalent: a) M is a K-contact manifold; b) the Reeb group of M is an isometries group; c) the metric gD is invariant with respect to the Reeb group of M ; d) the almost complex structure J on D is invariant by the Reeb group of M; e) F is invariant by the Reeb group of M . Proof. a) ⇔ b) follows from the definition of the K-contact manifolds. c) ⇔ d) ⇔ e) From b) and from Proposition 1.3.7 it results that η and dη are invariant by the Reeb group and then gD is invariant if and only if J is invariant, that is if and only if F is invariant by the Reeb group. The others implications follow easily. The above result was proved in [BG01] for the more general case when the manifold M is complete, renouncing to the compactness condition. Now, it is natural to ask about the relation existing between the geometry of the K-contact manifold M and the geometry of its symplectic cone. The following result holds

1.4. K-CONTACT MANIFOLDS

31

Theorem 1.4.4. The compact contact Riemannian manifold M is K-contact if and only if its symplectic cone C(M ) endowed with the metric dt2 + t2 g√is an √ ∂ almost K¨ ahler manifold and the complex vector field E − −1ξ = t ∂t − −1ξ 6 is holomorphic . Proof. If C is the natural almost complex structure on C(M ) (see Section 1.3) then we see that LE−√−1ξ C = 0 is equivalent to Lξ J = 0, and this last condition is equivalent to Lξ gD = 0. Then we apply Theorem 1.4.3. Obviously, any K-contact manifold is a contact manifold. P. Rukimbira proved the following converse. Theorem 1.4.5. [Ru95] Any almost regular compact contact manifold is Kcontact. E. Lerman, [Le03] and T. Yamazaki, [Ya99] find another condition in order that a contact manifold be K-contact, and they express this condition depending on the contact moment map defined in Section 1.3. Theorem 1.4.6. [Le03] The positively co-oriented compact contact manifold M is K-contact if and only if there exists an action ρ of a torus T on M , preserving the contact distribution D, and a vector X in the Lie algebra t of T , with the 0 property that the function hΦ, Xi : D+ −→ R, defined by hΦ, Xi (x, eta+ x) = + + 0 0 Ψ(x, ηx )(X) for any η ∈ D+ , is strictly positive, Ψ : D+ −→ t∗ being the contact moment map of the action ρ. Proof. Assume the existence of the action ρ with the properties from the theorem. Since ρ preserves D, the lifted action of T to T ∗ M preserves D0 . On the 0 . We can other hand, T is connected and then the lifted action also preserves D+ 0 assume that there is a T -invariant 1-form α with α(M )subsetD+ and consider −1 the 1-form η = hΦ ◦ α, Xi α. Since ıρ(X) α = hΦ ◦ α, Xi (see (1.41)), it follows ıρ(X) η = 1, hence we obtain the T -invariant decomposition T M = D ⊕ Rρ(X). Now, we define on M a metric g so that D and Rρ(X) are orthogonal and ρ(X) is unitary. Then ρ(X) is an unitary vector field normal to D and choose a T invariant almost complex structure J on D, compatible with dη|D , and define g|D (X, Y ) = dη|D (F X, Y ). g is T -invariant and therefore Lρ(X) g = 0 (see the formulas (1.21 and (1.23)). It follows that M is a K-contact manifold. Conversely, if M is K-contact then the transformations group {exp(t~n)} of the unitary vector field ~n, normal to D, is an isometries Lie group. Its closure 6 Let M be a complex manifold with the almost complex structure J and let X be a complex vector field of type (1,0) on M . X is called holomorphic vector field if Xf is a holomorphic function for any complex valued holomorphic function f on an open set from M . A classical result asserts that Proposition. The Lie algebra of all infinitesimal automorphisms of the structure J (i. e. the vector fields X ∈ X (M ) such that LX J = 0) is isomorphic to the Lie algebra of holomorphic vector fields by the isomorphism X 7−→ 21 (X − iJX). See for instance [KN69], vol. II, Proposition 2.11, pg. 129.

32


T = {exp(t~n)} is a compact connected abelian Lie group, hence a torus. If X ∈ t is the vector field with the property ρ(X) = ~n and η is its dual form then hΦ ◦ η, Xi = ıρ(X) η = g(~n, ~n) = 1, hence hΦ, Xi > 0.

1.5

Sasaki manifolds

1.5.1

General properties of Sasaki manifolds

The contact Riemannian manifold M is called a Sasaki (or Sasakian) manifold if the associated almost contact Riemannian structure (F, ξ, η, g) is normal. Otherwise, the almost contact Riemannian structure (F, ξ, η, g) is a Sasakian structure or a Sasaki structure if dη = Ω and N (1) = 0 (see Theorem 1.2.1). From Theorem 1.2.1 and from Proposition 1.3.6, b) we deduce immediately Theorem 1.5.1. Any Sasaki manifold is K-contact. The converse of this result is valid only for 3-dimensional manifolds, namely we have Proposition 1.5.2. Any 3-dimensional K-contact manifold is Sasakian. Proof. Denote by {e, F e, ξ} a F -basis around a point of the manifold M . Then we have g ((∇X F ) e, e) = 0,

g ((∇X F ) e, F e) = 0,

g ((∇X F ) e, ξ) = g(X, e)

We deduce (∇X F ) e = g(X, e)ξ for any X ∈ X (M ) and then (1.56) is satisfied for Y = e. Similarly one can verify (1.56) for Y = F e and Y = ξ, hence by Theorem 1.5.3 the 3-dimensional manifold M is Sasakian. A characterization of Sasaki manifolds by the Levi-Civita connection ∇ of the metric g is the following Theorem 1.5.3. The almost contact Riemannian structure (F, ξ, η, g) is Sasakian if and only if (∇X F ) Y = g(X, Y )ξ − η(Y )X

(1.56)

for any vector fields X and Y . Proof. If the structure (F, ξ, η, g) is Sasakian then the equality (1.18) reduces to g ((∇X F ) Y, Z) = g(X, Y )η(Z) − g(X, Z)η(Y ) = g(g(X, Y )ξ − η(Y )X, Z) and (1.56) results easily.

1.5. SASAKI MANIFOLDS

33

Conversely, by putting Y = ξ in (1.56) and using the well-known relation (∇X F )Y = ∇X (F Y ) − F (∇X Y )

(1.57)

we obtain F ∇X ξ = X − η(X)ξ and then, applying F we deduce that (1.54) is valid on M . Hence we have 2dη(X, Y ) = X(η(Y )) − Y (η(X)) − g ([X, Y ] , ξ) = g (∇X ξ, Y ) − g (X, ∇Y ξ) = 2g(X, F Y )

(1.58)

and this proves that (F, ξ, η, g) defines a contact Riemannian structure. Moreover, a straightforward computation in N (1) shows that N (1) = 0, hence the structure is also normal. Combining Theorem 1.5.3 with Proposition 1.3.5, we deduce Proposition 1.5.4. The K-contact manifold M is Sasakian if and only is the complex distribution D10 is integrable. Choosing a F -basis {ei } = {Xi , Xi∗ , ξ} in M , from (1.54) it results (∇ei η) ej = −g (∇ei ξ, ej ) = 0 and then from the definition of the codifferentiation operator δ we deduce δη = 0, hence we can state the following Proposition 1.5.5. The contact form of a Sasaki manifold is co-closed. Remark 1.5.6. Assuming that the elements of the basis {ei } are eigenvectors of the operator N (3) , by a similar argument it follows that Proposition 1.5.5 is valid for any contact Riemannian manifold. Another characterization of Sasaki manifolds by the almost Kähler structure of the symplectic cone is the following Theorem 1.5.7. The contact Riemannian manifold M is Sasakian if and only if its symplectic cone C(M ) is a K¨ ahler manifold. d ≡ Proof. By identifying X ≡ (X, 0) ∈ X (C(M )) for any X ∈ X (M ) and dt d C 0, dt ∈ X (C(M )) and denoting by ∇ the Levi-Civita connection of the metric G = dt2 + t2 g on C(M ), by a simple computation using the definition of the symplectic cone as a product manifold we give

d 1 = ∇Cd X = X, dt dt t d C ∇X Y = ∇X Y − tg(X, Y ) dt ∇Cd

dt

d = 0, dt

∇CX

and then ∇Cd dt = 0, dt

∇CX dt = tωX

(1.59)

34


where ωX is the dual form of the vector field X. It results that for any p-form ω ∈ F p (M ) identified with its image by the canonical projection C(M ) −→ M and for any X ∈ X (M ) we can write p ∇Cd ω = − ω, dt t

∇CX ω = ∇X ω −

p! dt ∧ ıX ω t

Now, by computing ∇C ΩC we obtain ∇Cd ΩC = 0, dt

∇CX ΩC = t2 (2ωX ∧ η + ∇X dη) + 2tdt ∧ (∇X η − ıX dη) (1.60)

From the last equality we deduce ∇CX ΩC (Y, Z) = t2 [g(X, Y )η(Z) − g(X, Z)η(Y ) + g (Y, (∇X F ) Z)] (1.61) d for any Y, Z ∈ X (M ) and ∇CX ΩC dt , Z = 0. Therefore, from (1.61) we remark that C(M ) is a K¨ ahler manifold (i. e. ∇C ΩC = 0) if and only if (1.56) is satisfied and then our affirmation follows by applying Theorem 1.5.3. Proposition 1.5.8. The angle between a geodesic γ and the Reeb vector field of a Sasaki manifold is constant along γ. Proof. From the formula (1.54) and from the condition (∇γ 0 g) (γ 0 , ξ) = 0 it results γ 0 (g(γ 0 , ξ)) = 0. Taking into account Proposition 1.4.1, this result is still valid for K-contact manifolds. A geodesic γ of the Sasaki manifold M is called a F -geodesic if γ 0 ∈ D for all the points of γ. From Proposition 1.5.8 we deduce that if γ 0 is orthogonal to ξ at some point then γ is a F -geodesic. Let M be a manifold with the almost contact structure (F, ξ, η) and suppose that there exists a pseudo Riemannian metric g and ∈ {−1, +1} such that η(X) = g(X, ξ),

g(X, F Y ) = −g(F X, Y )

(1.62)

for any X, Y ∈ X (M ). If besides this, the Levi-Civita connection ∇ of the metric g satisfies the condition (1.56), then M is a Sasaki manifold with pseudo Riemannian metric. This notion was introduced by T. Takahashi, [Ta69]. Depending on the index s of the metric g, we distinguish two types of Sasaki manifolds with pseudo Riemannian metric: a) = 1 and s = 2k; in this case the Reeb vector field ξ is spacelike and M is a spacelike Sasaki manifold ; b) a) = −1 and s = 2k + 1; in this case the Reeb vector field ξ is timelike and M is a timelike Sasaki manifold. In [Ta69] it is proved that there exists a bijective correspondence between these two kinds of Sasaki manifolds with pseudo Riemannian metric, hence we can assume that = +1.


35

A timelike Sasaki manifold M with the pseudo Riemannian metric g of index s = 1 is called a Lorentz Sasaki manifold. Then the endomorphism F satisfies the condition F X = ∇X ξ for X ∈ X (M ) and like for Sasaki manifolds, the cone M × R∗+ of the Lorentz Sasaki manifold M , with the metric G = −dt2 + t2 g is a pseudo Kähler manifold of signature (2, 2n), its fundamental 2-form being d(t2 η), where η(X) = g(X, ξ). Examples 1. Sasaki structure on R2n+1 Denote by x1 , . . . , xn , y 1 , . . . , y n , z the Cartesian coordinates in R2n+1 and consider the 1-form η and the vector field ξ defined by ! n X ∂ 1 dz − y i dxi ; ξ = 2 η= 2 ∂z i=1 Also, we consider the Riemannian whose local expressions are  δ + yi yj 0 1  ij 0 δij g: 4 −y j 0

metric g and the tensor field F of type (1,1),  −y i 0  1



0 F :  −δij 0

δij 0 yj

 0 0  0

where δij is the Kronecker symbol. We verify that (F, ξ, η, g) is an almost contact Riemannian structure on R2n+1 and this is associated with the contact structure defined by η. Moreover, we have N (1) ≡ 0, hence it is a Sasaki structure on R2n+1 . If n is even then we can consider the endomorphism F ∗ : X (R2n+1 ) −→ X (R2n+1 ) defined by, [CCFF00] ! n X ∂ ∂ ∂ F ∗ (X) = F ∗ Xi i + Yi i + Z = ∂x ∂y ∂z i=1 ! n X i+1 i = (−Xk+1 , Xk ) , (Yk+1 , −Yk ) , y Xi − y Xi+1 i=1

k∈1,n−1

We easily verify that (F ∗ , ξ, η, g) is another almost contact Riemannian structure on R2n+1 . Moreover, we have NF ∗ = 2dη ⊗ ξ, therefore according to Theorem 1.2.1 and taking into account Remark 1.3.1, b), this structure is not normal neither contact Riemannian. 2. Sasaki structure on S 2n+1 Let S be a orientable hypersurface in the 2n + 2-dimensional almost Hermitian manifold M . In Example 1, Section 1.3, we equipped S with an almost contact Riemannian structure (F, ξ, η, g) (see Proposition 1.3.16). In the case when the manifold M is Kählerian we have

36


Proposition 1.5.9. Let S be a orientable hypersurface of the K¨ ahler manifold M and denote by (F, ξ, η, g) the almost contact Riemannian structure defined above. (F, ξ, η, g) is a Sasaki structure if and only if there exists f ∈ F(S) such that the second fundamental form of S has the expression h = −g + f η ⊗ η. Proof. Let us denote by Ω and ΩS the fundamental 2-forms of the manifolds M and S, respectively. Since M is a Kähler manifold we have ∇Ω = 0 and then, by using the Gauss formula we obtain (∇SX ΩS )(Y, Z) = h(X, Y )η(Z) − h(X, Z)η(Y )

(1.63)

for any X, Y, Z ∈ X (S). If h = −g + f η ⊗ η then the equality (1.63) yields ∇SX ΩS (Y, Z) = −g(X, Y )η(Z) + g(X, Z)η(Y ) and since ∇SX ΩS (Y, Z) = g Y, ∇SX F Z , we deduce (1.56), hence S is a Sasaki manifold. Conversely, if S is Sasakian then from (1.63) and from (1.56) it follows h(X, Y )η(Z) − h(X, Z)η(Y ) = −g(X, Y )η(Z) + g(X, Z)η(Y )

(1.64)

For X = Y = ξ, the equality (1.64) becomes h(ξ, Z) = h(ξ, ξ)η(Z)

(1.65)

On the other hand, by putting Z = ξ in (1.64) we have h(X, Y ) − h(ξ, ξ)η(X)η(Y ) = −g(X, Y ) + η(X)η(Y ) and taking into account (1.65) we obtain h = −g + f η ⊗ η, where f = h(ξ, ξ) + 1. The sphere S 2n+1 is a totally umbilical hypersurface in R2n+2 and h = −g, therefore it satisfies the condition from Proposition 1.5.9. Moreover, by considering on R2n+2 the natural Kähler structure, from this Proposition it results that (F, ξ, η, g) is a Sasaki structure on S 2n+1 . 3. Sasaki structure on the sphere tangent bundle In Example 3, Section 1.3, we have equipped the sphere tangent bundle T1 M of a Riemannian manifold M with a contact Riemannian structure defined by the almost contact Riemannian structure (F, ξ, η, g). Necessary and sufficient conditions in order that this structure is K-contact or Sasakian were obtained by Y. Tashiro. Theorem 1.5.10. [Tas69] The contact Riemannian structure on the sphere tangent bundle T1 M of a Riemannian manifold M is K-contact if and only if M has constant sectional curvature equal to 1. In this case the structure is Sasakian. A proof of this result (which is not difficult but rather requires a hard computation) is given in [Bl02], pg. 143–145, where we can find many other properties of the structure (F, ξ, η, g) on T1 M .


37

4. Sasaki structures on Bianchi-Cartan-Vranceanu spaces From the formulas (1.8), (1.7) (see Example 3, Section 1.1) and from the definition of F we obtain λ dη(X, Y ) = λg(X, F Y ), (∇X F ) Y = [g(X, Y )ξ − η(Y )X] (1.66) 2 for any X, Y ∈ X M3λµ , hence the almost contact Riemannian structure (F, ξ, η, g) is not Sasakian. However for λ 6= 0 we can define another almost contact Riemannian structure on the Bianchi-Cartan-Vranceanu space M3λµ by the tensor fields F∗ = F

ξ∗ =

2 ξ, λ

η∗ =

λ η, 2

g∗ =

λ2 g 2

Then, expressing (∇∗X F ∗ ) Y by using the formulas (1.66), from Theorem 1.5.3 it follows that (F ∗ , ξ ∗ , η ∗ , g ∗ ) is a Sasaki structure on M3λµ . Equipped with the above structure we denote this space by M3∗ λµ . 5. Sasaki structures on Brieskorn manifolds For (a0 , a1 , . . . , an ) ∈ Nn+1 we consider the polynomial P (z) = z0a0 +z1a1 +. . .+znan , where z = (z0 , z1 , . . . , zn ) ∈ Cn+1 . We also denote by Z(a0 , a1 , . . . , an ) the set of all zeros of P . If we identify the element z = (z0 , z1 , . . . , zn ) ∈ Z(a0 , a1 , . . . , an ) with its correspondent (Re(z0 ), Re(z1 ), . . . , Re(zn ), Im(z0 ), Im(z1 ), . . . , Im(zn )) in R2n+2 then the set Σ(a0 , a1 , . . . , an ) = Z(a0 , a1 , . . . , an ) ∩ S 2n+1 (1) has a structure of real manifold of dimension 2n − 1 and we call it the Brieskorn manifold. Pn m ∂ In Cn+1 we consider the vector field ~a = k=0 ak zk ∂zk , where m is the lowest common multiple of the numbers a0 , a1 , . . . , an . Also we denote by J the standard almost complex structure on C given by √ √ ∂ ∂ ∂ ∂ −1 ; J = = − −1 k J k k k ∂z ∂z ∂ z¯ ∂ z¯ If we write the scalar product (with respect to the standard metric on Cn+1 ) Pn of ~a with a vector ~n = ¯kak −1 ∂z∂k , normal to the Brieskorn manifold k=0 ak z Σ(a0 , a1 , . . . , an ), then we observe that at all its points we have ~a · X = 0, hence at these points √ the vector field ~a is tangent to Σ(a0 , a1 , . . . , an ). We also remark that ξ = − −1~a = −J~a is a vector field tangent to Σ(a0 , a1 , . . . , an ). Now, by using the same method as in Example 1, Section 1.3, for each X ∈ Σ(a0 , a1 , . . . , an ) we can decompose the vector field JX under the form (1.42) and thus (F, ξ, η) is an almost contact structure. Moreover, a straightforward computation shows that N (1) ≡ 0, hence this structure is normal and the Riemannian metric g(X, Y ) = η(X)η(Y )+ 1 + [X · Y − η(X)(Y · ξ) − η(Y )(X · ξ) + η(X)η(Y )(ξ · ξ)] ~a · z

38


is compatible with this structure. Also, we have dη = Ω and thus the Brieskorn manifold Σ(a0 , a1 , . . . , an ) is Sasakian. The study of contact structures on Brieskorn manifolds is very rich in valuable results, its initial point being placed during the period 1975–1980, when were published many papers by S. Sasaki, [Sa75], S. Sasaki and C. J. Hsu, [SH76], S. Sasaki and T. Takahashi, [ST76], K. Abe and J. A. Erbacher, [AE75], K. Abe, [Abe79], I. Vaisman, [Va78]. 6. Sasaki structures on the Heisenberg group H3 (R) The 3-dimensional Heisenberg group H3 (R) is identified with the space R3 endowed with the group structure induced by the product xy 0 − x0 y (x, y, z)(x0 , y 0 , z 0 ) = x + x0 , y + y 0 , z + z 0 + 2 It is a Lie group and for the metric 1 dx2 + dy 2 + g= 4 4

ydx − xdy dz + 2

2

is a Riemannian manifold. With respect to g the vector fields X1 = 2

∂ ∂ −y , ∂x ∂z

X2 = 2

∂ ∂ +x ; ∂y ∂z

X3 = 2

∂ ∂z

form an orthonormal frame and its dual frame is composed by the 1-forms ω1 =

1 dx, 2

ω2 =

1 dy, 2

ω3 =

1 xdy − ydx dz − 2 4

and a simple computation shows that η = ω 3 is a contact form on H3 (R). Now, by setting ξ = X3 and defining the endomorphism F : X (H3 (R)) −→ X (H3 (R)) by F X1 = X2 , F X2 = −X1 , F X3 = 0, it follows that (F, ξ, η, g) defines an almost contact Riemannian structure on H3 (R). We also obtain [X1 , X2 ] = 2X3 , [X2 , X3] = [X3 , X1 ] = 0 and then we deduce the Levi-Civita connection ∇X1 X2 = −∇X2 X1 = X3 , ∇X1 X3 = ∇X3 X1 = −X2 , ∇X2 X3 = ∇X3 X2 = X1 By computing (∇X F ) Y with the above formulas, we can check the equality (1.56) and then from Theorem 1.5.3 it follows that (F, ξ, η, g) is a Sasakian structure on the Heisenberg group H3 (R). The map H : H3 (R) −→ GL(3, R), defined by   1 y z + xy 2  x H(x, y, z) =  0 1 0 0 1 is a Lie groups isomorphism from the Heisenberg group to the linear Lie group     1 y t  H3 (R) =  0 1 x  ; x, y, t ∈ R   0 0 1


39

and the image by H of the Sasaki structure (F, ξ, η, g) on H3 (R) is   0 1 0 1 dx2 + dy 2 F ∗ :  −1 0 0  ; g ∗ = + (dt − ydx)2 ; 4 4 0 y 0 1 ∂ (dt − ydx); ξ ∗ = 2 2 ∂z Now, by comparing with the first Example presented above, we deduce that the induced Sasaki structure on H3 (R) by that one of the Heisenberg group coincides with the standard Sasaki structure on R3 . This remark allows us to define the n-dimensional Heisenberg group as being the Lie group     1 xtr t  Hn (R) =  0 1 y  ; x, y ∈ Rn , t ∈ R   0 0 1 η∗ =

where xtr = (x1 , . . . , xn ) is the transposed of the matrix x. On Hn (R) we consider the left invariant metric g and the tensor fields ξ, η, F defined by g = dx · dx + dy · dy + (dt − x · dy)2 ;

ξ=

∂ ; ∂t

η = dt − xdy;

n X ∂ ∂ i ∂ i i F = +y ⊗ dy − i ⊗ dx ∂xi ∂z ∂y i=1 It is easy to prove that (F, ξ, η, g) is a Sasaki structure on Hn (R). 7. Sasaki structure on the universal covering space Let M be a connected and locally arcwise connected manifold. The connected space M ? is called a covering space of M if there exists a projection π : M ? −→ M with the property that any point x ∈ M has a connected open neighborhood U such that each connected component of π −1 (U ) is an open subset of M ? , homeomorphic with U by π. The covering spaces M1? and M2? of M are isomorphic if there exists a homeomorphism µ : M1? −→ M2? such that π2 ◦ µ = π1 , where π1 , π2 are the covering projections of M1? and M2? , respectively. Obviously, every covering space of M has an unique manifold structure for which π is differentiable. Moreover, M ? has the same dimension as the manifold M . If M ? is simply connected then it is called the universal covering space of M and it has the following properties: Proposition 1.5.11. Let M be a connected locally and arcwise connected manifold. Then: a) there is an universal covering space M ? of M , unique up to an isomorphism of covering spaces. If M is arcwise connected then M ? is simply connected. If M is a Riemannian manifold then on M ? there exists a Riemannian metric such that π is an isometric immersion.

40


b) M ? is a principal bundle over M , whose structure group is the first homotopy group of M and its projection is π. c) M ? is complete if and only if M is complete. For a proof of this classical result see for instance [Co01], pg. 33–36. Now, if M is a 2n + 1-dimensional manifold with the Sasaki structure (F, ξ, η, g) then we define the 1-form η ? ∈ F 1 (M ? ) and the vector field ξ ? by ηx?? = πx?? ηπ(x? ) ,

π?x? ξx?? = ξπ(x? )

for any x? ∈ M ? . Denote by g ? a Riemannian metric in M ? (see Proposition 1.5.11, a)). Since π is an isometric immersion, it follows that ξ ? is unitary and η ? (X ? ) = g ? (X ? , ξ ? ) for any X ? ∈ X (M ? ). On the other hand, if ∇? is the Levi-Civita connection of g ? then we define the endomorphism F ? : X (M ? ) −→ X (M ? ) by setting F ? X ? = −∇?X ? ξ ? for any X ? ∈ X (M ? ). Then by a straightforward computation it follows that the tensor fields F ? , ξ ? , η ? , g ? define a Sasaki structure on M ? . 8. Sasaki manifold with pseudo Riemannian metric [BD92] In R2n+1 with the Cartesian coordinates xi , y i , z, we consider the vector field ξ and the 1-form η given by ! n X ∂ 1 i i i ξ=2 , η= dz − y dx ∂z 2 i=1 ∗

∗

where s ∈ 1, n is a fixed number, a = a = 1 for a ∈ 1, s and a = a = −1 for a ∈ s + 1, n. We also define the endomorphism F and the pseudo Riemannian metric g of index s by theirs matrices   0 In 0 0 0  F :  −In 0 a y a 0  g:

 1  4 

−δab + y a y b ˜ −y a y b 0 0 ya

˜

−y a y b ˜ δa˜˜b + y a˜ y b 0 0 −y a˜

0 0 −Is 0 0

0 0 0 In−s 0

ya −y a˜ 0 0 1

     

˜, ˜b ∈ s + 1, n and 0 denotes the zero matrices whose dimenwhere a, b ∈ 1, s, a sions are easy to determine. A straightforward verification shows that R2n+1 endowed with this structure is a Sasaki manifold with pseudo Riemannian metric. The study of degenerate hypersurfaces in Sasaki manifolds with pseudo Riemannian metric was imposed by their importance for the relativity and for the electromagnetism. A survey of the known results is presented in C. Calin’s book, [Ca05].


1.5.2

41

Deformations of Sasaki structures

We consider deformations with the property that, starting from a Sasaki structure on the manifold M , other Sasaki structures on M can be produced. The simplest such a deformation is the following Proposition 1.5.12. [Ta69a] Let (F, ξ, η, g) be an almost contact Riemannian structure on the manifold M . Then for any α ∈ R∗ , the tensor fields F 0 , ξ 0 , η 0 , g 0 given by F 0 = F, ξ 0 = ξ, η 0 = η, g 0 = |α|g + (1 − |α|)η ⊗ η, = sgn α defines an almost contact Riemannian structure on M . If in addition the structure (F, ξ, η, g) is a Sasakian structure then (F 0 , ξ 0 , η 0 , g 0 ) is also Sasakian. The almost contact structure (F 0 , ξ 0 , η 0 , g 0 ) given in Proposition 1.5.12 in called the -deformation of (F, ξ, η, g). Now we study the following two types of deformations: A. those which deform the Reeb vector field ξ; B. those which deform the contact form η. The deformations of type A have the following characterization Theorem 1.5.13. [BGM04] Let (F, ξ, η, g) be a Sasaki structure on the manifold M . Then there exists f : M −→ R∗+ and ξ0 ∈ X (M ) such that the tensor ˜ η˜, g˜ given by the formulas fields F˜ , ξ, η˜ = f η,

ξ˜ = ξ + ξ0 ,

F˜ = F − F ξ˜ ⊗ η˜,

g˜ = d˜ η ◦ (F˜ ⊗ 1) + η˜ ⊗ η˜

define a Sasaki structure on M . ˜ = 1 is satisfied for Proof. The condition η˜(ξ) f=

1 1 + η(ξ0 )

(1.67)

˜ η˜, g˜) is also D, it is and because the contact distribution of the structure (F˜ , ξ, 2 ˜ ˜ ˜ easy to check that F X = −X for X ∈ D and F ξ = 0. If in addition we impose the condition that d˜ η is horizontal, i. e. iξ˜d˜ η = 0, then by computation we show ˜ η˜). Now, that g˜ is a metric compatible with the almost contact structure (F˜ , ξ, ˜ if the condition Lξ˜F = 0 is satisfied then M becomes a K-contact manifold. Likewise, a direct computation shows that under the above conditions we have ˜ η˜) N 1 ≡ NF˜ + 2d˜ η ⊗ ξ˜ = 0 and then, taking into account Theorem 1.2.1, (F˜ , ξ, is a Sasaki structure on M . Remark 1.5.14. a) The metrics g and g˜ from Theorem 1.5.13 are related by g˜(X, Y ) = η˜(X)˜ η (Y )+ +

h i 1 ˜ Y ) − g(ξ, ˜ X)˜ ˜ ξ)˜ ˜ η (X)˜ g(X, Y ) − η˜(X)g(ξ, η (Y ) + g(ξ, η (Y ) 1 + η(ξ0 )

42


b) From Theorem 1.5.13 and from its proof it follows that for ξ0 = aξ and a ∈ R∗ we obtain the deformed Sasaki structure F˜ = F,

1 ξ˜ = ξ, α

η˜ = αη,

g˜ = αg + α(α − 1)η ⊗ η

(1.68)

1 where α = 1+a = f > 0. This structure was studied by S. Tanno, [Ta68], who called it D-homothetic deformation with constant α of the structure (F, ξ, η, g).

Example. Denote by (F, ξ, η, g) the Sasaki structure defined in Example 2 on the sphere S 2n+1 . We can define another Sasaki structure on S 2n+1 by the following construction: 2n+1 Consider the sphere as a subset of Cn+1 with the usual Kähler strucn S o 2 2 2n+1 ture, i. e. S = (z1 , z2 , . . . , zn+1 ) ∈ Cn+1 : |z1 | + |z2 | + . . . + |zn+1 | = 1 and remark that for λ1 , λ2 , . . . , λn ∈ R the isometric deformation Φλt (z1 , z2 , . . . , zn+1 ) = (exp(λ1 t)z1 , exp(λ2 t)z2 , . . . , exp(λn t)zn , exp(t)zn+1 ) generates a vector field ξ λ =

dΦλ t dt |t=0 .

If ξ is the Reeb vector field of the Sasaki

structure constructed above then the vector field ξ0 = ξ λ − ξ has the following properties Lξ0 g = 0, [ξ0 , ξ] = 0, 1 + η(ξ0 ) > 0 and by Theorem 1.5.13 it is the Reeb vector field of a deformed Sasaki structure on S 2n+1 . Now, we study the deformations of type B of the Sasaki structure (F, ξ, η, g), that is those which conserve the Reeb vector field, but deform the contact form. For this purpose we consider a family of basic 1-forms {ηt }t∈[0,1] (see Section 1.3) with respect to the characteristic foliation Fξ of the Sasaki manifold M and assume that η0 = 0, and that ηt∗ = η + ηt are contact forms on M , i.e. ηt∗ ∧ (dηt∗ )n 6= 0

for any t ∈ [0, 1]

(1.69)

Then we can state the following Theorem 1.5.15. [BGM04] Let M be a manifold with the Sasaki structure (F, ξ, η, g) and let {ηt }t∈[0,1] be a family of basic 1-forms on M . If η0 = 0 and if the condition (1.69) is satisfied then the tensor fields Ft , ξ, ηt∗ , gt given by ηt∗ = η + ηt , Ft X = F X − ηt (F X)ξ, gt (X, Y ) = dηt∗ (F X, Y ) + ηt∗ (X)ηt∗ (Y )

(1.70)

define a Sasaki structure on M for any t ∈ [0, 1]. This structure is associated with the contact structure defined by η.


43

Proof. By direct verification it follows that (Ft , ξ, ηt∗ , gt ) is an almost contact Riemannian structure on M . Replacing in gt (X, Y ) the vector field F X by its expression from the second equality (1.70) we get gt (X, Y ) = dηt∗ (Ft X, Y ) + ηt∗ (X)ηt∗ (Y ). Now, replacing X by Ft X, we obtain dηt∗ (X, Y ) = gt (X, Ft Y ). If we compute the tensor field N (1) associated to Ft for couples of the form (X, Y ) with X ∈ D10 , Y ∈ D01 ; X, Y ∈ D10 or X, Y ∈ D01 and for couples of the form (X, ξ) with X ∈ D10 or X ∈ D01 , then we obtain N (1) ≡ 0, hence this new structure is Sasakian. Denote by S(ξ) the set of Sasakian structures obtained from the Sasaki structure F, ξ, η, g) by deformations described in Theorem 1.5.15 and we call them ξ-deformations. Remark that only the first class of deformations studied above preserves the contact distribution. However, the contact distributions have the following stability property: Theorem 1.5.16. [Gr59] Let M be a compact contact manifold and consider a family {Dt }t∈[0,1] of contact distributions on M such that D0 = D. Then there exists an isotopy µt : M −→ M with the property that µt∗ (D) = Dt for any t ∈ [0, 1]. Notice that an analogous result for the associated family {ηt }t∈[0,1] of contact forms is not true and this proves that the topology of the characteristic foliation Fξ is very sensitive to deformations of the contact forms [EGH00]. Linear deformations of Sasaki structures Since the next considerations occur only the contact distribution and the contact form we consider contact manifolds. Now, we shall solve, at least partially, a kind of converse problem of the one stated in Theorem 1.5.16, namely: If M is an oriented manifold of dimension 2n + 1 and F is a codimension 1 foliation, can we measure how ”far” is F from a one parameter family of contact structures on M ? We say that that F is deformable into contact structures if there exists a family {Dt }t∈[0,∞) of codimension 1 distributions such that D0 is the distribution associated to F and Dt is a contact distribution for each t > 0. denote by η0 a 1-form defining the foliation F and by ηt a contact form associated to the contact distribution Dt . The deformation {Dt }t∈[0,∞) of F is linear if there exists a 1-form η ∈ F 1 (M ) such that ηt = η0 + tη for any t > 0. Remark that the forms ηt satisfy the equality ηt ∧ (dηt )n = tn η0 ∧ (dη)n + tn+1 η ∧ (dη)n . We only deal with this particular case of deformations.

(1.71)

44


Theorem 1.5.17. [DR04] Let M be a compact oriented 2n + 1-dimensional manifold, η0 a closed 1-form on M and η another 1-form on M . The following affirmations are equivalent: a) η is a contact form and η0 (ξ) = 0, where ξ is the Reeb vector field of η; b) {Dt = Ker ηt }t∈[0,∞) is a linear deformation of F into contact structures on M , i. e. ηt = η0 + tη is a contact form for any t > 0. Proof. We fix a volume form dV on M and for a given form ω ∈ F 2n+1 (M ) we write ω > 0 if ω = f dV for some f > 0. a) ⇒ b) Assume that η ∧ (dη)n > 0. If g is a Riemannian metric compatible with the contact structure defined by η then we have dη(X, ξ) = 0 and considering a local basis of vector fields containing ξ, we see that η0 ∧ (dη)n = 0 and thus from (1.71) we get ηt ∧(dηt )n > 0, hence ηt is a contact form for each t > 0. b) ⇒ a) Since ηt is a contact form we have either ηt ∧ (dηt )n > 0 or ηt ∧ (dηt )n < 0. We prove that if ηt ∧ (dηt )n > 0 then η0 ∧ (dηt )n = 0. By (1.71) it is impossible to have η0 ∧ (dη)n < 0 and η ∧ (dη)n ≤ 0. Now, we assume η0 ∧ (dη)n < 0 and η ∧ (dη)n > 0 at some point x ∈ M and let {Xi }i∈1,2n+1 be a positively oriented basis of Tx M . For any t > 0 such that tη ∧ (dη)n (X1 , . . . , X2n+1 ) < |η0 ∧ (dη)n (X1 , . . . , X2n+1 )| the equality (1.71) yields [ηt ∧ (dηt )n ] (X1 , . . . , X2n+1 ) = = tn [η0 ∧ (dη)n (X1 , . . . , X2n+1 ) + tη ∧ (dη)n (X1 , . . . , X2n+1 )] < < tn [η0 ∧ (dη)n (X1 , . . . , X2n+1 ) + |η0 ∧ (dη)n (X1 , . . . , X2n+1 )|] = 0 We deduce tη ∧ (dη)n (X1 , . . . , X2n+1 ) ≥ |η0 ∧ (dη)n (X1 , . . . , X2n+1 )| for all t > 0. Then η0 ∧(dη)n = 0 at the point x ∈ M and this is also impossible. From the above argument it follows η0 ∧ (dη)n ≥ 0. But Z Z n η0 ∧ (dη) = − d η0 ∧ η ∧ (dη)n−1 = 0 M

M

hence η0 ∧ (dη)n = 0 and from (1.71) we deduce η ∧ (dη)n > 0, so that η is a contact form. Now, setting ω = η0 −η0 (ξ)η we remark that ω(ξ) = 0. If {X1 , . . . , X2n , ξ} is a basis of Tx M such that η(Xi ) = 0 for i ∈ 1, 2n then dη(ξ, Ei ) = 0 for i ∈ 1, 2n and ω ∧ (dη)n (ξ, X1 , . . . , X2n ) = 0, hence ω ∧ (dη)n = 0. We obtain η0 ∧ (dη)n = η0 (ξ)η ∧ (dη)n

(1.72)

and finally it follows η0 (ξ) = 0. From Theorem 1.5.17 it follows that if η0 (ξ) 6= 0 then there is t > 0 such that ηt is not a contact form. More precisely we have the following result


45

Proposition 1.5.18. [DR04] Let M be a compact contact manifold with the contact form η and the Reeb vector field ξ. If η0 is a closed 1-form such that η0 (ξ) does not vanish identically on M then there exists r > 0 with the property that the 1-form ηt = η0 + tη is not contact for neither t ∈ [0, r]. Proof. First we remark that (1.72) is true on any contact manifold and for any closed 1-form η0 and then, taking into account (1.71) we obtain ηt ∧ (dηt )n = tn [η0 (ξ) + t] η ∧ (dη)n On the other hand we have 0 = ıξ [η0 ∧ η ∧ (dη)n ] = η0 (ξ)η ∧ (dη)n − η0 ∧ (dη)n and by using (1.72) we deduce η0 (ξ)η ∧ (dη)n = η0 ∧ (dη)n = −d η0 ∧ η ∧ (dη)n−1 Now, applying Stokes theorem we get Z η0 (ξ)η ∧ (dη)n = 0

(1.73)

M

and then it follows that there exist the constants a < 0, b > 0 such that a ≤ η0 (ξ) ≤ b. Therefore for t ∈ [0, −a] the 1-form ηt is not contact on the set Ct = {x ∈ M : η0 (ξ)(x) = −t} ⊂ M . Hence we can take r = −a.

1.5.3

Sasaki potential

We study the local structure of a Sasaki manifold, as it was given by M. Godlinski, W. Kopczynski and P. Nurowski, [GKN00]. Let (F, ξ, η, g) be a Sasaki structure on the 2n + 1-dimensional manifold M . Taking into account the decomposition (1.15) of the complexified tangent bundle M , we can consider a orthonormal local complex basis defined on an open subset U from M and having the form {Zk , Zk¯ , ξ}k∈1,n , where Zk¯ = Z¯k n o ¯ and Zk ∈ D01 . With respect to this basis and to its dual ω k , ω k , η we k∈1,n

have g=2

n X

¯

ω l ⊗ ω l + η ⊗ η,

l=1

F =

√

−1

n X

¯ Z¯l ⊗ ω l − Zl × ω l ,

l=1 n X

√ dη = − −1

(1.74)

¯

ωl ∧ ωl

l=1

Moreover, by Theorem 1.2.3 the distribution D10 is integrable since M is Sasakian and then dω l ∧ω 1 ∧ω 2 ∧. . .∧ω n = 0 for l ∈ 1, n. These formulas show that we

46


can apply the complex version of Frobenius Theorem (see for instance [KN69], vol. II, pg. 322–324) and then there exist the functions flk , g k : U −→ C, k, l ∈ 1, n such that ω k = flk dg l

(1.75)

n o ¯ But the system ω k , ω k is linearly independent and thus on U we have dg 1 ∧ . . . ∧ dg n ∧ d¯ g 1 ∧ . . .d¯ g n 6= 0 and then, by setting z k = g k we can consider another local basis ∂z∂ 1 , . . . , ∂z∂n , ∂∂z¯1 , . . . , ∂∂z¯n , ξ in T c M and its dual is dz 1 , . . . , dz n , d¯ z 1 , . . . , d¯ z n , η . Now, by writing the Maurer-Cartan equations k for the 1-forms dz and d¯ z k (see Section 2.4 or [KN69], vol. I, pg. 77–78), it results ∂ ∂ ξ, k = ξ, k = 0. ∂z ∂ z¯ This proves that in U we can consider a real coordinate x, not depending on ∂ z 1 , . . . , z n , z¯1 , . . . , z¯n and such that ξ = ∂x . Then we have η = dx + pk dz k + pk¯ d¯ zk

(1.76)

By differentiating (1.76) and by identifying the terms of the same type and k taking into account (1.74) it also follows ∂p ∂x = 0, that is the functions pk do not depend on x and moreover, we have ∂pk ∂pl − k = 0, ∂z l ∂z n X √ ∂p¯l ∂pk ¯ fkm f¯lm − = −1 ∂ z¯l ∂z k m=1

(1.77)

¯ where f¯lm = flm . From the first set of formulas (1.77) it follows that on a connected subset of U there exists a complex valued function V such that

pk =

∂V ∂z k

(1.78)

and because pk and x are independent we can consider that the function V does not depend on x. From the second set of formulas (1.77) and from (1.78) we obtain n X ∂2K ¯ = fkm f¯lm ∂z k ∂ z¯k m=1

(1.79)

where K = 2√1−1 (V − V ) and L = 21 (V + V). By using (1.78) in (1.76) and making the transformation x 7−→ x + L it follows η = dx +

√

−1

n X ∂K k=1

∂z

dz k − k

∂K k d¯ z ∂ z¯k

(1.80)

1.6. KENMOTSU MANIFOLDS

47

Moreover, by using (1.75) in (1.74) and taking into account (1.79), by a straightforward computation we get g=2

F = −

n X

k,l=1 n X

∂2K dz k ⊗ d¯ zl, ∂z k ∂ z¯l

√

k=1 n X

√

∂ ∂K ∂ − k k ∂ z¯ ∂ z¯ ∂x

−1

∂ ∂K ∂ + k ∂z k ∂z ∂x

−1

k=1

⊗ d¯ zk −

(1.81)

⊗ dz k

From the formulas (1.80) and (1.81) it results that the function K locally determines the Sasaki structure (F, ξ, η, g). Moreover, if the function K is chosen so that g given by (1.81) is positively defined and ξK = 0 then ξ and the tensor fields η, g, F defined by these formulas determine a Sasaki structure on an open subset of Cn × R. The function K if called the Sasaki potential. It is easy to check that for any holomorphic function f (z k ), the transformations √ zk ) (1.82) K ∗ = K + f (z k ) + f¯(¯ z k ), x∗ = x − −1 f (z k ) − f¯(¯ do not perturb the Sasaki structure (F, ξ, η, g), so (1.82) are gauge transformations for the Sasaki potential. In [GKN00], Sasaki potentials are determined for Sasaki structures on S 2n+1 , q C × Cn × R and for a family of Sasaki structures in dimension 5.

1.6

Kenmotsu manifolds

A manifold with the almost contact Riemannian structure (F, ξ, η, g) is an almost Kenmotsu manifold if the following conditions are satisfied dη = 0;

dΩ = 2η ∧ Ω

(1.83)

From the first condition (1.83) it follows Proposition 1.6.1. The contact distribution of an almost Kenmotsu manifold is always integrable. We call a Kenmotsu manifold any normal almost Kenmotsu manifold. Theorem 1.6.2. [JV81] A manifold with the almost contact Riemannian structure (F, ξ, η, g) is a Kenmotsu manifold if and only if (∇X F ) Y = −η(Y )F X − g(X, F Y )ξ

(1.84)

Proof. If M is a Kenmotsu manifold then (1.84) follows from (1.18) taking into account (1.83) and the property of M to be normal.

48


Conversely, we suppose that the condition (1.84) is satisfied. From the definition of the exterior differential, from ∇g = 0 and taking into account (1.84), (1.57) we have X(Ω(Y, Z)) = X(g(Y, F Z)) = g (∇X Y, F Z) + g (Y, ∇X (F Z)) = = g (∇X Y, F Z) + g (Y, (∇X F ) Z + F (∇X Z)) = = g (∇X Y, F Z) + g (Y, F (∇X Z)) − η(Z)g(Y, F Z) − η(Y )g(X, F Z) Similar expressions for the terms Y (Ω(Z, X)), Z(Ω(X, Y )) used in dΩ, show that the second formula (1.83) is true. ξ is unitary and then we have η (∇X ξ) = g (∇X ξ, ξ) = 0

(1.85)

and 2dη(X, Y ) = X(η(Y )) − Y (η(X)) − η[X, Y ] = = g (X, ∇Y ξ) − g (Y, ∇X ξ) = = g (F X, F (∇Y ξ)) − g (F Y, F (∇X ξ)) = = g (F X, (∇Y F ) ξ) − g (F Y, (∇X F )) ξ Now, by application of (1.84) we obtain the first equality (1.83). Finally, by using (1.84) and (1.85) we deduce N (1) = 0 and then we can apply Theorem 1.2.1. Initially, the notion of Kenmotsu manifold was defined by K. Kenmotsu, [Ke72] as being an almost contact Riemannian manifold whose Levi-Civita connection satisfies the condition (1.84) and ∇X ξ = X − η(X)ξ.

(1.86)

But we remark that putting Y = ξ in (1.84) and taking into account (1.85), the formula (1.86) follows easily. We also notice that almost Kenmotsu and Kenmotsu structures defined here were introduced by D. Janssens and L. Vanhecke, [JV81]. Remark 1.6.3. In another context, S. S. Eum, [Eu69] was interested to find almost contact Riemannian manifolds M with the property that the leaves of the contact distribution are Kä hler hypersurfaces. He proved that this happens if the following condition is satisfied g ((∇X F ) Y, Z) = (∇X η) [η(Y )F Z − η(Z)F Y ]

(1.87)

But it is easy to prove that this condition is equivalent to (1.87), therefore M is a Kenmotsu manifold. By a straightforward computation it results


49

Proposition 1.6.4. On a Kenmotsu manifold the following equalities are true (∇X η) (Y ) = g(X, Y ) − η(X)η(Y ); Lξ g = 2(g − η ⊗ η); Lξ F = 0; Lξ η = 0 From Proposition 1.6.4 it follows that the Reeb vector field of a Kenmotsu manifold ξ cannot be Killing, hence such a manifold cannot be Sasakian and more generally, it cannot be K-contact. An interesting topological property of these manifolds is the following Theorem 1.6.5. A Kenmotsu manifold is never compact. Proof. For a F -basis {Xi , Xi∗ , ξ}i∈1,n we have div ξ = tr (Y 7−→ ∇Y ξ) =

n X

[g (∇Xi ξ, Xi ) + g (∇Xi∗ ξ, Xi∗ )] + g (∇ξ ξ, ξ)

i=1

and by using (1.86) it follows div ξ = 2n. If M is compact then by the Green Theorem (see for instance [KN69], vol. I, pg. 281–283) we have Z (div ξ)dV = 0 M

and this is impossible. We have the following local characterization of Kenmotsu manifolds: Theorem 1.6.6. [Ke72] Any point of a Kenmotsu manifold has a neighborhood isometric to the warped product7 (−, ) ×f V , where (−, ) is an open interval from R, f (t) = cet , c > 0 and V is a K¨ ahler manifold. Proof. Let x ∈ M . By Proposition 1.6.1 through the point x passes only one maximal integral submanifold M (x) of D. Denoting by J and by G the restrictions to M (x) of the tensor fields F and g, respectively, we remark that (J, G) is an almost Hermitian structure on M (x). If ∇0 is the Levi-Civita connection of the metric G on M (x) then taking into account the Gauss formula in (1.84) it follows (∇0X J)Y = 0,

h(X, JY ) − F h(X, Y ) = −g(X, JY )ξ

hence M (x) is a K¨ ahler manifold. Moreover, completing a local basis in M (x) with ξ, the metric g has the following local expression 1 0 gx = where x = (t, u) (1.89) 0 f 2 (t)Gu 7 Let us consider two Riemannian manifolds M and M with the metrics g , g , and let 1 2 1 2 f : M1 −→ R∗+ be a differential function. For (X1 , Y1 ), (X2 , Y2 ) ∈ T(x1 ,x2 ) (M1 × M2 ) the formula

g [(X1 , Y1 ), (X2 , Y2 )] = g1 (X1 , X2 ) + f 2 (x1 )g2 (Y1 , Y2 )

(1.88)

defines a Riemannian structure on the product manifold M1 × M2 . Equipped with this structure, the manifold M1 × M2 is called the warped product of M1 and M2 and we denote it by M1 ×f M2 . The notion was defined in [BN69].

50


therefore M (x) can be identified with a product of the form (−, ) × V and d , the second G|V , J|V is a K¨ ahler structure on V . Also, by putting ξ = dt 0 equality from Proposition 1.6.4 yields f = f , which proves that f (t) = cet . According to Proposition 1.6.1, the contact distribution D of a Kenmotsu manifold M defines a foliation on M and for this reason sometimes the maximal integral submanifolds of D are called the leaves of the distribution D. We prove some elementary properties of these leaves. Theorem 1.6.7. Let M be a Kenmotsu manifold. a) The contact distribution D of M defines a Riemannian foliation. b) Any leaf of D is totally umbilical, its mean curvature vector is equal to −ξ and has a natural K¨ ahler structure. Proof. a) A straightforward computation using (1.86) shows that g([X, ξ], ξ) = 0 ¯ in the normal bundle for any X ∈ D. On the other hand, the Bott connection ∇ Fξ of the foliation defined by D satisfies the condition ¯ X g (ξ, ξ) = 0 ∇ for any X ∈ D, hence D is a Riemannian foliation. b) The first two assertions result by computing the mean curvature vector and the last is stated in the proof of Theorem 1.6.6. By reconsidering the study of product manifolds (see Propositions 1.2.4 and 1.2.5), we notice the following construction given by D. Blair and J. Oubina, [BO90]: Let M1 , M2 be two almost contact Riemannian manifolds with the structures (Fi , ξi , ηi , gi ), i = 1, 2. On the product manifold M1 × M2 we define the endomorphism J : X (M1 × M2 ) −→ X (M1 × M2 ) by J (X1 , X2 ) = F1 X1 − e−2µ η2 (X2 )ξ1 , F2 X2 + e−2µ η1 (X2 )ξ2 Then J 2 (X1 , X2 ) = −(X1 , X2 ), hence J defines an almost complex structure on M1 × M2 . For σ = 0, he above almost complex structure was defined by K. Matsuo, [Mo95]. A straightforward computation shows that the metric G = e2σ g1 + e2τ g2 is Hermitian if and only if µ = 21 (σ − τ ). On the other hand, there is a local d chart in M2 such that at any point of its domain U we have ξ2 = dt . Now, taking into account Theorems 1.5.3 and 1.6.2 and the expression of the LeviCivita connection of the metric G depending on the Levi-Civita connections of the metrics g1 , g2 , by a long computation we obtain the following result: Proposition 1.6.8. [BO90] Let us consider two almost contact Riemannian manifolds M1 , M2 . There exists a constant k such that for σ = ln (k − e−t ), τ = −t the structure (J, G) defined on M1 × M2 is locally K¨ ahlerian if and only if the structure (F1 , ξ1 , η1 , g1 ) on M1 is Sasakian and U has a Kenmotsu structure induced by (F2 , ξ2 , η2 , g2 ).


51

In the case µ = σ = τ = 0 the almost Hermitian structure (J, G) was studied by A. Morimoto, [Mo63]. He also proved that if the structures on M1 , M2 are normal then (J, G) is Hermitian. Take it as an exercice! Moreover, the fundamental form Ω of M1 × M2 is given by Ω = Ω1 + Ω2 − 2(η1 , 0) ∧ (0, η2 ) and we easily can state the following Proposition 1.6.9. [Mo95] The Riemannian product M1 × M2 of the Sasaki manifold M1 with a Kenmotsu manifold M2 , endowed with the Hermitian structure (J, G) is locally conformal K¨ ahler. According to Proposition 1.2.4 it is natural to find conditions so that the almost contact Riemannian structure defined on M1 × M2 is Kenmotsu. The answer is the following: Theorem 1.6.10. [Ts90] Consider the manifolds M1 and M2 with the almost Hermitian structure (F1 , g1 ) and the almost contact Riemannian structure (F2 , ξ2 , η2 , g2 ), respectively. The product manifold M1 × M2 with the almost contact Riemannian structure (F, ξ, η, g) defined in Proposition 1.2.4 is a Kenmotsu manifold if and only if M1 is K¨ ahlerian and M2 is Kenmotsu. Proof. Suppose that M1 × M2 is a Kenmotsu manifold. If ∇1 , ∇2 , ∇ are the Levi-Civita connections of the metrics g1 , g2 and g, respectively, then we have g (F (X1 , X2 ) , (Y1 , Y2 )) ξ − η (X1 , X2 ) F (Y1 , Y2 ) = = (0, g2 (F2 X2 , Y2 ) ξ2 − η(Y2 )F2 X2 ) On the other hand, we can write ∇(X1 ,X2 ) F (Y1 , Y2 ) =

∇1X1 F1 Y1 , ∇2X2 F2 Y2

and taking into account Theorem 1.6.2 and by equalizing the second members of these two equalities it follows (∇X1 F1 ) Y1 = 0;

(∇X2 F2 ) Y2 = g2 (X2 , F2 Y2 )ξ2 − η2 (Y2 )F2 X2

that is M1 is K¨ ahlerian and M2 is a Kenmotsu manifold. The converse is trivial. Examples 1. Kenmotsu structure on the warped product R ×f M Taking into account Theorem 1.6.6, it is natural to consider the product R ×f M , where M is a manifold with the K¨ ahler structure (J, G) and the differentiable map f : R −→ R is given by f (t) = cet for some c ∈ R∗ . On R ×f M we define the tensor fields g, ξ, η by d 1 0 g(t,x) = ; ξ = ; η(X) = g(X, ξ) 0 f 2 (t)Gx dt

52


where (t, x) ∈ R × M, X ∈ X (R × M ) and Gx is the matrix of the metric G in a local chart at the point x ∈ M . Since (1.86) holds for the Levi-Civita connection of the metric g on the product R × M , we obtain N (4) = 0 and we can define on R × M the tensor field F by putting 0 0 F(t,x) = where F˜(t,x) = (exp(tξ))∗ Jx (exp(−tξ))∗ 0 F˜(t,x) Now, simple verification shows that (F, ξ, η, g) is an almost contact Riemannian structure on the warped product R ×f M . By identification of the vector fields X = (η(X)ξ, X0 ), Y = (η(Y )ξ, Y0 ) with the sums X0 + η(X)ξ and Y0 + η(Y )ξ, respectively, and denoting by D the Levi-Civita connection of the metric G on M , we have ∇X0 Y0 = DX0 Y0 − g(X0 , Y0 )ξ and then (∇X F ) Y = DX0 (JY0 ) − g(X0 , F Y0 )ξ + η(X)∇ξ (F Y0 )− − F [DX0 Y0 + η(Y )X0 + η(X)∇ξ Y0 ] Since ∇ξ F = 0, the above equality becomes (∇X F ) Y = DX0 (JY0 ) − g(X0 , F Y0 )ξ − η(Y )F X0 − F DX0 Y0 Now, taking into account the definition of the tensor field F , we have DX0 (JY0 ) = F DX0 Y0 , so that from Theorem 1.6.2 it results that the warped product R ×f M is a Kenmotsu manifold. 2. Kenmotsu structure on the hyperbolic space H 2n+1 Denote by H 2n+1 the hyperbolic space (x1 , . . . , x2n+1 ) ∈ R2n+1 : x1 > 0 , endowed with the Riemannian metric g = (x1 )−2 I2n+1 . D. Chinea and C. Gonzalez, [CG90], construct on H 2n+1 the almost contact Riemannian structure (F, ξ, η, g) with ∂ η(X) = g(X, ξ), ξ = x1 1 , ∂x ∂ ∂ ∂ ∂ ∂ F , F = 0, F = =− i ∂x1 ∂xi ∂xi∗ ∂xi∗ ∂x for i ∈ 2, n + 1. By a straightforward computation we determine the Levi-Civita connection of the metric g and we verify the equality (1.84), so that (F, ξ, η, g) is a Kenmotsu structure on H 2n+1 . 3. The complex space Cn with the usual metric, the complex projective space P n (C) with the Fubini-Study metric and the unit disc Dn ⊂ Cn with the Bergmann metric are Kähler manifolds of constant holomorphic curvature equal to 0, strictly positive and strictly negative, respectively (see for instance [KN69], vol. II, pg. 169–170). By applying Proposition 1.6.10 it follows that Cn ×H 2n+1 , P n (C) × H 2n+1 and Dn × H 2n+1 have Kenmotsu structures.

1.7. OTHER ALMOST CONTACT STRUCTURES

1.7

53

Other almost contact structures

Brief historical note S. S. Chern, [Che53] proved that the structure group of the tangent bundle of a contact manifold reduces to U (n)×1 and this property was used by J. W. Gray, [Gr59] as a definition of almost contact manifolds. S. Sasaki, [Sa60] defined the almost contact structure by the system (F, ξ, η) and observed the equivalence of this definition with the one given by J. W. Gray (see Theorem 1.1.4). S. Sasaki also defined the almost contact Riemannian structure (F, ξ, η, g). His definition of almost contact structure seems, at first sight, a formal analogous of the almost complex structure, fact noticed many times. However, the Sasaki’s argument was far from this one and in this way the commentary relative to the manner in which he was guided to the notion of almost contact structure defined by the tensor fields (F, ξ, η) is eloquent (see [Sa68], vol. 1, pg. 3.11–3.15). J. Bouzon, [Bou62] consider the almost contact structure by exploiting in a more direct manner the analogy with the almost complex structure. He called it the almost co-complex structure and defined it as follows: Let G be the subgroup of GL(2n + 1, R), whose elements are matrices of the form   A −B 0  B A 0  0 0 c with c > 0 and A, B ∈ Mn,n (R). Denote by L(M ) the principal bundle of frames of the 2n + 1-dimensional manifold M and consider an element u in the subbundle P (M, G) of L(M ). The frame u can be viewed as an isomorphism u : R2n+1 −→ Tx M (see [KN69], vol. I, Proposition 5.4, pg. 55) and then, using obvious identifications it follows that Tx0 = u(R), Sx = u(R2n ) define two distributions T 0 , S of dimensions 1 and 2n, respectively. Moreover, if we denote by F0 the endomorphism of R2n+1 given by the matrix J 0 0 −In where J = 0 0 In 0 then the map Fx = u ◦ F0 ◦ u−1 defines on M a tensor field F of type (1, 1), satisfying the conditions: F = 0 on T 0 and F 2 = −I on S. F is the almost co complex structure in J. Bouzon’s sense and it is easy to check its equivalence with the almost contact structure in S. Sasaki’s sense. P. Libermann, [Li59] defined the almost cosymplectic structure. This is given by two closed forms η ∈ F 1 (M ), Ω ∈ F 2 (M ) with the property that η ∧ Ωn 6= 0 everywhere on the 2n + 1-dimensional manifold M . If the almost contact structure given in Theorem 1.1.6 is normal then it is called a cosymplectic structure. This notion was defined by D. E. Blair [Bl67]. The quasi-Sasaki structure (see the definition below) was defined by D. Blair, [Bl67] and S. Kanemaki, [Ka77], [Ka84] gives a characterization analogous to the one presented in Theorems 1.5.3, 1.6.2 for Sasaki and Kenmotsu structures, namely

54


Theorem 1.7.1. The almost contact Riemannian structure (F, ξ, η, g) on the manifold M is quasi-Sasaki if and only if there exists a tensor field A of type (1,1), with the properties (∇X F ) Y = η(Y )AX − g(AX, Y )ξ; AF = F A; g(AX, Y ) = ±g(X, AY ) We observe that a Sasaki structure is obtained for A = −I, while for A = −F it is Kenmotsu. S. Kanemaki gives another characterization of Sasaki structures, namely he proves that a quasi-Sasaki manifold is Sasakian if and only if rank(A− η(Aξ)η ⊗ ξ) = 2n + 1. The trans-Sasaki structure was defined by G. Oubina, [Ou85]. His idea is to change the integrability condition of the almost complex structure J on the product manifold M × R (see Section 1.2) by a less restrictive condition, namely he requires that J is a locally conformal Kähler structure8 . So, we obtain the following characterization of these manifolds by the Levi-Civita connection Theorem 1.7.2. [Ou85] The manifold M with the almost contact Riemannian structure (F, ξ, η, g) is trans-Sasaki if and only if there exist α, β ∈ F(M ) such that (∇X F ) Y = α [g(X, Y )ξ − η(Y )X] + β [g(F X, Y )ξ − η(Y )F X] In this case too, we observe that the trans-Sasaki structure of type α = 1, β = 0 is Sasakian, and that one of type α = 0, β = 1 is a Kenmotsu structure. More generally, for the couples α 6= 0, β = 0 and α = 0, β 6= 0 we obtain the αSasaki and the β-Kenmotsu structures, respectively, introduced by D. Janssens and L. Vanhecke, [JV81], by the following definition: The manifold M with the normal almost contact Riemannian structure (F, ξ, η, g) is called α-Sasaki manifold (or α-Kenmotsu) if Ω=

1 dη α

(or dη = 0, dΩ = 2η ∧ Ω)

for some α ∈ R∗ . J. C. Marrero obtain the following classification of trans-Sasaki manifolds: Theorem 1.7.3. [Mar92] A trans-Sasaki manifold of dimension ≥ 5 belongs to one of the following classes: α-Sasaki, β-Kenmotsu or cosymplectic. We pointed out here only few marks in the development of the classical almost contact structures. There are other types of almost contact structures and we shall define the most important ones in the sequel. 8 Let M be a manifold with the almost Hermitian structure (J, g). M is a locally conformal K¨ ahler manifold if there exists a closed 1-form α such that dΩ = α ∧ Ω, where Ω(X, Y ) = g(X, F Y ) is the fundamental 2-form of the given almost Hermitian structure. For the study of these manifolds see the monograph [DO98], written by S. Dragomir and L. Ornea.

1.7. OTHER ALMOST CONTACT STRUCTURES

55

Special almost contact structures contact

→

contact Riemannian semi − Sasaki 1 η = 2n δΩ

K− contact

N (3) =0

→

N (1) =0

→

Sasaki

normal semi − Sasaki

N (1) =0

→

quasi − trans− Sasaki(a) almost trans− Sasaki(b)

trans− Sasaki(c)

N (1) =0

→

δη=0

→

quasi− Sasaki

nearly trans − Sasaki(d) almost Kenmotsu dΩ = 2η ∧ Ω; dη = 0

N (1) =0

→

semi − Kenmotsu (∇F X Ω)(F Y, Z) = −η(Y )Ω(X, Z); δΩ = 0, dη = 0

Kenmotsu

N (1) =0

→

normal semi − Kenmotsu

quasi− Kenmotsu(e) nearly Kenmotsu (∇X Ω)(X, Y ) = −η(X)Ω(X, Y ); dη = 0 almost trans− Kenmotsu(f) almost cosymplectic dΩ = 0; dη = 0 semi − cosymplectic δΩ = 0; δη = 0

N (1) =0

→

N (1) =0

→

cosymplectic normal semi − cosymplectic

quasi − K − cosymplectic (∇X F )Y + (∇F X F )F Y − η(Y )∇F X ξ = 0 nearly − cosymplectic (∇X F )X = 0

∇X ξ=0

→

nearly K − cosymplectic

56


The definitions of the above structures marked by an upper index are the following: (a) (∇X Ω)(Y, Z) + (∇F X Ω) (F Y, Z) + η(Y ) (∇F X η) Z = 1 = − [g(X, Y )δΩ(Z) − g(X, Z)δΩ(Y )]+ n + Ω(X, Y )ω(Z) − Ω(X, Z)ω(Y ) where ω(X) = (b)

1 n

[δΩ(F X) − η(X)δη]; dΩ = Ω ∧ ω, 1 dη = [δΩ(ξ)Ω − 2η ∧ F (δΩ)] ; 2n

(c) δΩ(F X) =0, 1 [g(X, Y )δΩ(Z) − g(X, Z)δΩ(Y )+ 2n +Ω(X, Y )η(Z)δη − Ω(X, Z)η(Y )δη] ;

(∇X Ω) (Y, Z) = −

(d) 1 [g(F X, F Y )δη + Ω(Y, X)δΩ(ξ)] , 2n 1 (∇X Ω) (X, Y ) = − [g(X, X)δΩ(Y ) − g(X, Y )δΩ(X) + Ω(Y, X)η(X)δη] ; 2n (∇X η) Y = −

(e) dη = 0, (∇X Ω) (Y, Z) + (∇F X Ω) (F Y, Z) = η(Y )Ω(Z, X) + 2η(Z)Ω(X, Y ); (f) dη = 0, X 1 (∇X Ω) (Y, Z) − 2η(X)Ω(Y, Z) − Ω(X, Y )δΩ(F Z) = 0. n cycl

Actually there are many specialized almost contact structures, whose geometry was intensively studied these last twenty five years. In this Section we only defined the most important known almost contact Riemannian structures and we present some subordination relations among them. Of course the reader interested by this subject can find by himself many others such relations or by consulting the references included in this book. Among these, we only notice the following [Ts92] Kenmotsu ⊂ nearly Kenmotsu; Kenmotsu ⊂ quasi-Kenmotsu

Chapter 2

Transformations and submersions of contact manifolds 2.1

Contact transformations groups

Let M be an almost contact Riemannian manifold and denote by A(M ) the set of its transformations (automorphisms). The sets Γc (M ) and Γs (M ) of contact and strict contact transformations of the (almost) contact manifold M , defined in Section 1.3, have group structures with respect to the usual composition. Γc (M ) is the contact transformations group and Γs (M ) is the strict contact transformations group of the manifold M . We remark that the functions f from the formula (1.38) have the property that Supp f ⊂ R+ or Supp f ⊂ R− and therefore we distinguish the following subgroup of Γc (M ) Γ+ (M ) = {µ ∈ Γc (M ); f = eg where g ∈ F(M )} . Obviously we have Γs (M ) ⊂ Γ+ (M ) ⊂ Γc (M ) ⊂ A(M ). If µ ∈ Γs (M ) then µ∗ dη = dη, so that the fundamental 2-form of a contact Riemannian manifold is invariant by µ∗ . Moreover, from Theorem 1.3.9 it follows Proposition 2.1.1. Let M be a contact Riemannian manifold and µ ∈ Γs (M ). Then µ∗ dV = dV , i. e. µ preserves the volume form of the manifold M . We know that any vector field X on a manifold M generates, at least locally, a 1-parameter group of transformations of M . A simple argument shows that in the case when M is a (almost) contact manifold these transformations are contact if and only if LX η = 0. Starting from this remark, we say that a vector field X on the contact manifold M is an infinitesimal contact transformation (infinitesimal strict contact transformation) if LX η = f η (LX η = 0). 57

58

CHAPTER 2. TRANSFORMATIONS AND SUBMERSIONS

Of course, the function f ∈ F(M ) generally depends on the field X. Denote by Lc (M ) (Ls (M )) the set of all infinitesimal contact transformations (infinitesimal strict contact transformations) of the contact manifold M . Taking into account Proposition 1.3.6, a) we have ξ ∈ Ls (M ) ⊂ Lc (M ). Moreover, we can state the following Theorem 2.1.2. For any contact manifold M the sets Lc (M ) and Ls (M ) are Lie algebras with respect to the bracket of vector fields. Proof. If X, Y ∈ Lc (M ) then L[X,Y ] η = [LX , LY ] η = LX (gη) − LY (f η) = (Xg − Y f )η hence [X, Y ] ∈ Lc (M ) and similarly for X, Y ∈ Ls (M ). Of course, generally the endomorphism F of the almost contact structure associated to a contact structure is not invariant by an infinitesimal transformation. However, S. Tanno found necessary and sufficient conditions so that this property is true. Theorem 2.1.3. [Ta62] Let M be a compact contact Riemannian manifold and X an infinitesimal transformation of M . The following affirmations are equivalent: a) X leaves F invariant, i. e. LX F = 0; b) X is an infinitesimal strict contact isometry. Afterward we present some characterizations of infinitesimal contact transformations and for their set Lc (M ). For this purpose denote by Fji the local components of the tensor field F , i.e. ∂ ∂ F = Fji i ∂xj ∂x where x1 , x2 , . . . , x2n+1 are the local coordinates in M . Proposition 2.1.4. Let M be a contact manifold and let X ∈ X (M ) be a vector field. a) If X ∈ Lc (M ) then there exists U ∈ F(M ) such that X = U ξ − XU ,

where

XUi = Fji g jk

∂U ∂xk

(2.1)

b) Conversely, any vector field X of the form (2.1) is an infinitesimal contact transformation. c) X ∈ Ls (M ) if and only if the function U satisfies the condition Lξ U = 0. Proof. a) Let X ∈ Lc (M ). If we express the condition LX η = f η in local coordinates then a simple computation shows that the function U = ηi X i satisfies (2.1). b) Likewise, by computing LX η for the vector field X given by (2.1) we ∂U obtain LX η = f η with f = ξ i ∂x i. c) follows from a) and b).

2.1. CONTACT TRANSFORMATIONS GROUPS

59

The function U = ηi X i is called the characteristic function of the infinitesimal contact transformation X. By using the formulas (2.1), an elementary coordinates computation shows that if UX and UY are the characteristic functions of the infinitesimal contact transformations X, Y ∈ Lc (M ) then the characteristic function U[X,Y ] of the bracket [X, Y ] ∈ Lc (M ) is U[X,Y ] = Fki g kj

∂UY j ∂UX j ∂UX ∂UY + UX ξ − UY ξ ∂xi ∂xj ∂xj ∂xj

(2.2)

The formula (2.2) suggests to define a new operation on the set F(M ) of real valued differentiable functions on the contact manifold M , namely [f, g] = Fki g kj

∂g ∂f ∂g ∂f + f j ξj − g j ξj ∂xi ∂xj ∂x ∂x

(2.3)

We call it the bracket of the functions f, g ∈ F(M ). By using the definition (2.3), a straightforward computation shows that this bracket satisfies the conditions [f, g] = −[g, f ],

[f, g + h] = [f, g] + [f, h]

and the Jacobi identity [f, [g, h]] + [g, [h, f ]] + [h, [f, g]] = 0 so that F(M ) becomes a Lie algebra. Moreover, as the characteristic function U of an infinitesimal strict contact transformation satisfies the condition Lξ U = 0 (see Proposition 2.1.4, c)), we can define the bracket [f, g]s on the subset F s (M ) = {f ∈ F(M ) : Lξ f = 0} by [f, g]s = Fki g kj

∂f ∂g ∂xi ∂xj

(2.4)

It is easy to prove that this bracket defines a Lie algebra structure on F s (M ). Now, it is natural to consider the following maps h : F(M ) −→ Lc (M ), hs : F s (M ) −→ Ls (M ),

h(f ) = U ξ − XU hs (g) = U ξ − XU

We remark that h and hs are morphisms with respect to the additive group structure of F(M ) and F s (M ), respectively and a simple computation using (2.3) and (2.4) shows that h[f, g] = [hf, hg],

s

hs [f, g]s = [hs f, hs g]

Particularly, if hf = 0 then we have ∂ ∂ U g ξ, j − F U =0 ∂x ∂xj and multiplying by ξ j and summing up on j we deduce U g(ξ, ξ) − (F ξ)U = 0. Hence U ≡ 0 and this proves that h is injective. Proceeding in the same manner with hs , we can state

60


Proposition 2.1.5. The maps h and hs defined above are isomorphisms of Lie algebras. Since dim F(M ) = ∞, from Proposition 2.1.5 it follows Theorem 2.1.6. The Lie algebra Lc (M ) of all infinitesimal contact transformations of the contact manifold M is infinite dimensional. The infinitesimal strict contact transformation X of the almost contact manifold M is an infinitesimal contact automorphism if LX F = 0. The diffeomophism µ of the almost contact manifold M is a contact automorphism of M if the following compatibility conditions with the almost contact structure (F, ξ, η) of M are satisfied F ◦ µ∗ = µ∗ ◦ F,

µ∗ η = η,

µ∗ ξ = ξ

(2.5)

Similarly we define the contact diffeomorphism between two almost contact manifolds or the contact automorphism. Clearly, each of the last two equalities in (2.5) follows from the other. By using the first of these formulas and from the definition of the Lie derivative, we deduce the following Proposition 2.1.7. Let M be a manifold with the almost contact structure (F, ξ, η). The vector field X ∈ X (M ) is an infinitesimal contact automorphism if and only if it generates a local one-parameter group of contact automorphisms of M . Proof. denote by {Φt }t∈R the local transformations group generated by X. then for any Y ∈ X (M ) we have (LX F ) Y = [X, F Y ] − F [X, Y ] = lim

t→0

F Φt∗ Y − Φt∗ F Y t

and we see that the first condition (2.5) is equivalent to LX F = 0. similarly we prove that the two last conditions (2.5) are equivalent to LX η = 0 and LX ξ = 0, respectively. Theorem 2.1.8. The group Γc (M ) of contact transformations acts transitively on the compact contact manifold M , i.e. for any x1 , x2 ∈ M there exists µ ∈ Γc (M ) such that µ(x1 ) = x2 . Proof. ([Sa68], vol. 2, pg. 18.8–18.11) Consider an arbitrary x ∈ M . According to Theorem 1.3.11, there is a local chart (U, φ) at the point x1 with local coordinates x1 , . . . , xn , y 1 , . . . , y n , z and satisfying the following conditions: 1. in U the contact form has the expression (1.36); 2. φ defines a diffeomorphism of U on an open cube V from R2n+1 with the edge equal to < n1 , the point x1 corresponding to the center of V . First we prove the local transitivity of the action of Γc (M ), namely we show the existence of a contact transformation which sends x to any point of U .

2.1. CONTACT TRANSFORMATIONS GROUPS

61

Since M is compact, we remark that any infinitesimal contact transformation generates a global group of contact transformations. Then we consider the group of contact transformations of M generated by an infinitesimal contact transformation h(f ), where h is the Lie algebras isomorphism defined above (see Proposition 2.1.5) and f ∈ F(M ) P is such that P U ⊂ supp f and over U the function f has the expression f = ai xi + bi y i + c with ai , bi , c real constants. Since the components of the Reeb vector field ξ in the chart (U, φ) are bi = 0, ai = 0, c = 1, we have X h(f ) = −bi , ai , aj xj + c and it generates a local group of contact transformations of the form µ = ∗ µi , µi , µ∆ i∈1,n , where µi (xi ) = xi − bi t, ∗

µi (y i ) = y i + ai t,   n n X t2 X aj bj µ∆ (z) = z +  aj xj + c t − 2 j=1 j=1

(2.6)

Since M is compact, these can be extended to contact transformations of M for any t ∈ R. The orbit in U of a point x0 (xi0 , y0i , z0 ) is xi = µi (xi0 ),

∗

y i = µi (y0i ), z = µ∆ (z0 ) (2.7) Pn Let x1 (xi1 , y1i , z1 ) ∈ U and put z10 = − 12 i=1 xi1 y1i . We observe that |z10 | < 1 2 0 i i 2 n < 2 , hence the points x2 (0, 0, z1 ) and x3 (x1 , y1 , 0) belong to U . If as above we write the orbit of the point x (see the formulas (2.7)), we remark that there is a contact transformation µz10 of type (2.6) which sends x at x2 . Now, if we consider a contact transformation ν of the form (2.6) with ai = y1i , bi = −xi1 , c = 0, then from (2.7) it follows that the orbit in U of the point x2 is xi = xi1 t, y i = y1i t, z = z10 (1 − t2 ); 0 ≤ t < t1 (2.8) where t1 = sup t : |xi1 t| < , |y1i t| < , |z10 (1 − t2 )| < . We observe that [0, 1] ⊂ [0, t1 ], so that the contact transformation ν1 (that is ν with t = 1) sends x2 in x3 . denote by λt the elements of the strict contact transformations group generated by ξ. We see that λz10 sends x to x2 and λz1 sends x3 to x1 , hence the contact transformation λz1 ◦ ν1 ◦ λz10 sends the point x at x1 . It follows that if x0 , x00 ∈ U then there exists a contact transformation of U which sends x0 at x00 . Now, we consider two arbitrary points x, x1 ∈ M and we prove the existence of of a contact transformation which sends x to x1 . For this purpose let γ be a curve which join them. As M is compact, the curve γ can be covered by a finite number of domains U1 , U2 , . . . , Uk of local charts having the same type as U constructed above and such that x ∈ U1 , x1 ∈ Uk and Ui ∩ Ui+1 6= ∅ for i ∈ 1, k − 1.

62


Consider the points y1 , y2 , . . . , yk−1 such that yi ∈ Ui ∩ Ui+1 . Like in the first part of the proof we can find the contact transformations µ1 , µ2 , . . . , µk , in a manner suggested in the following drawing: Uk−1 Uk U1 U2 U3 '$ '$ '$ '$ '$ µk−1 µ1- µ2- µ3- · · · - · µk- · · · · · · x y1 y2 y3 yk−2 yk−1 x1 &% &% &% &% &%

Finally, it is easy to see that the contact transformation µk ◦ µk−1 ◦ . . . ◦ µ1 maps x to x1 . Under more restrictive conditions an analogous result is valid for strict contact transformations. But before to state it, we define the notion of regular contact manifold. The 2n + 1-dimensional contact manifold M is called regular if each point x ∈ M has a local chart (Ux , φ) with the property that φ(Ux ) is an open cube in R2n+1 and a) each component of the intersection of Ux with an integral curve of the Reeb vector field ξ is given by the equations xi = constant for any i ∈ 1, 2n; b) any integral curve of ξ passes through Ux at most once. Intuitively, this means that each integral curve of ξ intersects in Ux at most once a transversal passing through x. The notion was defined by W. Boothby and H. C. Wang, [BW58] and was generalized by C. B. Thomas, [Th76]. Thus if in the above definition of the notion of regular manifold we replace the expression ”passes through Ux at most once” by ”passes through Ux at most r times”, we say that M is an almost regular contact manifold. Now, the intuitive representation is similar, namely: any point of an almost regular contact manifold has a neighborhood pierced by each orbit of the Reeb vector field in at most r points. Theorem 2.1.9. The group Γs (M ) of strict contact transformations acts transitively on the compact regular contact manifold M . The proof uses the same transformations µ and ν as in the proof of Theorem 2.1.8, but since ξ is regular and satisfies the conditions Lξ xi = Lξ y i = 0, these are strict contact transformations. For the details of this proof see [Hat66]. We remark that the set Ac (M ) of contact automorphisms of the manifold M has a group structure relative to the composition. It is called the contact automorphisms group of the almost contact manifold M and the following inclusions occur Ac (M ) ⊂ Γs (M ) ⊂ Γc (M ). Theorem 2.1.10. The contact automorphisms group Ac (M ) of the compact almost contact Riemannian manifold M is a Lie group. The proof is given in [Sa68], vol. II, pg. 21.2–21.4 and it uses some results from the theory of elliptic partial differential systems on a manifold.

2.2. HARMONIC MAPS

63

If the manifold M is equipped with an almost contact Riemannian structure (F, ξ, η, g) then any isometry which is also a contact automorphism is called a contact isometry of M . The set of all contact isometries of M is said the contact isometries group of M and it is denoted by Aci (M ). Obviously we have Aci (M ) ⊂ Ac (M ) and we can state the following Theorem 2.1.11. [Ta69a] Let M be a connected almost contact Riemannian manifold of dimension 2n + 1. Then dim Aci (M ) ≤ (n + 1)2 . Proof. Then any infinitesimal contact transformation X satisfying LX g = 0 generates around x ∈ M a local contact isometries group (see Proposition 2.1.7). On the other hand, from the proof of Proposition 2.1.7 we deduce LX F = 0,

LX η = 0,

LX ξ = 0

(2.9)

In local coordinates, the conditions (2.9) and LX g = 0 have the following expressions (LX η)a = X b ∇b ηa + ηb ∇a X b = 0, (LX ξ)a = X b ∇b ξ a − ξ b ∇b X a = 0, (LX g)ab = gbc ∇a X c + gac ∇b X c = 0, (LX F )ab = X c ∇c Fba − Fbc ∇c X a + Fca ∇b X c = 0

(2.10)

for any a, b, c ∈ 1, 2n + 1. Let {Xi , Xi∗ , X∆ = ξ} be a F -basis around x and ∗ denote by ω i , ω i , ω ∆ = η its dual basis. From the two first relations (2.10) it follows that if X(x) = 0 then with respect to the above bases we have ∇∆ X a = 0, ∇b X ∆ = 0. Hence the non-vanishing components are ∇a X b for a, b ∈ 1, 2n. But by the two last formulas (2.10) the set of these vector fields is contained in the Lie algebra of the group U (n), hence it is at most n2 -dimensional. On the other hand, the set of all vector fields X satisfying (2.10) and such that X(x) 6= 0 is at most 2n + 1-dimensional. Finally, we have dim Aci (M ) ≤ n2 + 2n + 1.

2.2

Harmonic maps on almost contact Riemannian manifolds

Let M1 , M2 be two manifolds of dimensions dim M1 = 2n1 + 1, dim M2 = 2n2 + 1, endowed with the almost contact Riemannian structures (F1 , ξ1 , η1 , g1 ) and (F2 , g2 , η2 , g2 ), respectively. Denote by ∇1 , ∇2 the Levi-Civita connections on M1 and M2 , respectively. The second fundamental form hµ of µ of a differentiable map µ : M1 −→ M2 is defined by hµ (X, Y ) = (∇X µ∗ ) (Y ) = ∇2µ∗ X (µ∗ Y ) − µ∗ ∇1X Y (2.11) for any X, Y ∈ X (M1 ). µ is a harmonic map if τ (µ) = 0, where τ (µ) = trace hµ is the tension field of µ and µ is called F -pluriharmonic if hµ (X, Y ) + hµ (F1 X, F1 Y ) = 0

(2.12)

64


for any X, Y ∈ X (M1 ). By choosing a F -basis in M1 , from (2.12) we deduce the following Proposition 2.2.1. Any F -pluriharmonic map between almost contact Riemannian manifolds is harmonic. If the condition (2.12) is satisfied only for X, Y belonging to the contact distribution of the manifold M1 then µ is called D-pluriharmonic. These notions as well as the next results were obtained by S. Ianus and A. M. Pastore, [IP93], [IP95]. From the formulas (2.12) and (1.15) it results Proposition 2.2.2. a) The map µ : M1 −→ M2 is F -pluriharmonic if and only if the following conditions are satisfied: ¯ = 0 for any Z ∈ D10 (M1 ); i) hµ (Z, Z) ii) hµ (X, ξ1 ) = 0 for any X ∈ D(M1 ); iii) h(ξ1 , ξ1 ) = 0. b) µ is D-pluriharmonic if and only if the condition ii) is fulfilled. Proof. a) If µ is F -pluriharmonic √ then i) follows from (2.12) since any Z ∈ D10 (M1 ) has the form Z = X − −1F X with X ∈ D. ii) and iii) are obvious. Conversely, if i), ii) and iii) are satisfied then (2.12) holds for X = Y = ξ, or X ∈ D(M √ the other hand, for X, Y ∈ D(M1 ) we put √ 1 ) and Y = ξ1 . On Z1 = X − −1F X, Z2 = Y − −1F Y and by applying i) to Z1 , Z2 and to Z1 + Z2 we obtain (2.12). b) is trivial. Remark that in the proof of Proposition 2.2.2 only the Riemannian structure of the manifold M2 occurs, but not its almost contact structure. The map µ : M1 −→ M2 is called F -holomorphic (or F -antiholomorphic) if µ∗ ◦ F1 = F2 ◦ µ∗

(or µ∗ ◦ F1 = −F2 ◦ µ∗ )

(2.13)

Proposition 2.2.3. Any F -holomorphic or F -antiholomorphic map between Sasaki manifolds is D-pluriharmonic. Proof. By applying successively Theorem 1.5.3 we obtain ∇1F1 X (F1 Y ) = −∇1X Y + [X, Y ] + F1 [F1 X, Y ] , ∇2µ∗ F1 X (µ∗ F1 Y ) = −∇2µ∗ X (µ∗ Y ) + [µ∗ X, µ∗ Y ] + F2 [F2 µ∗ X, µ∗ Y ] for any X, Y ∈ D1 and then from (2.11) we deduce hµ (X, Y ) + hµ (F1 X, F1 Y ) = = ∇2µ∗ X (µ∗ Y ) − µ∗ ∇1X Y + ∇2µ∗ F1 X (µ∗ F1 Y ) − µ∗ ∇1F X (F Y ) = = F2 [F2 µ∗ X, µ∗ Y ] − µ∗ F1 [F1 X, Y ] Now, taking into account (2.12), we obtain (2.13).

2.2. HARMONIC MAPS

65

Theorem 2.2.4. [IP93] Let M1 and M2 be two Sasakian manifolds and consider a non constant F -holomorphic (or F -antiholomorphic) map µ : M1 −→ M2 . Then there exists a ∈ R∗+ (or a ∈ R∗− ) such that µ∗,x (ξ1x ) = aξ2µ(x)

(2.14)

for any x ∈ M1 and µ is a D-homothetic immersion. Moreover, µ is an isometry if and only if a = 1 (or a = −1). Proof. If µ is F -holomorphic then from (2.13) it results F2 ◦ µ∗ (ξ1 ) = µ∗ ◦ F1 (ξ1 ) = 0 hence there exists a ∈ F(M1 ) such that µ∗,x (ξ1 ) = a(x)ξ2µ(x) for any x ∈ M1 . Then we deduce µ∗ η2 = aη1 and by differentiating we obtain µ∗ dη2 = dµ∗ η2 = da ∧ η1 + a ∧ dη1 From this last equality it follows ıξ (da ∧ η1 )(X) − (da ∧ η1 )(ξ, X) = 0 and then (Lξ a) η1 = da. Therefore (Lξ a) η1 ∧ dη1 = 0 and because η1 ∧ dη1 6= 0, we deduce Lξ a = 0, hence a = constant. On the other hand we have µ∗ Ω2 = aΩ1 and then g2 (µ∗ X, F2 µ∗ Y ) = ag1 (X, F1 Y )

(2.15)

for any X, Y ∈ X (M1 ). If in (2.15) we change Y by F1 Y then it results (µ∗ g2 ) (X, Y ) = ag1 (X, Y ) + a(a − 1)η1 (X)η2 (Y )

(2.16)

In particular, we have (µ∗ g2 ) (X, Y ) = ag1 (X, Y )

(2.17)

for X, Y ∈ D(M1 ), which proves that a ≥ 0. If a = 0 then from (2.14) it results µ∗ = 0. But this is impossible, hence a must be positive and from (2.16) we see that µ is D-homothetic. Moreover ker µ∗ ⊂ D(M1 ) and from (2.17) we obtain ker µ∗ = 0, hence µ is an immersion. Finally, (2.16) shows that µ is an isometry if and only if a = 1. We use the same argument if µ is F -antiholomorphic. Proposition 2.2.5. Any F -holomorphic (F -antiholomorphic) between Sasaki manifolds is harmonic. Proof. It follows immediately from Propositions 2.2.1 and 2.2.2. Theorem 2.2.6. Let µ : M1 −→ M2 be a non constant F -pluriharmonic map and assume that M1 , M2 are Sasaki manifolds. µ is F -holomorphic (F -antiholomorphic) if and only if µ∗ (ξ1 ) = ξ2

(µ∗ (ξ1 ) = −ξ2 )

66


Proof. If µ is F -pluriharmonic and F -holomorphic then from Proposition 2.2.2 and Theorem 2.2.4 we deduce 0 = hµ (X, ξ1 ) = ∇2µ∗ X (µ∗ ξ1 ) − µ∗ (∇1X ξ1 ) = = a∇2µ∗ X ξ2 + µ∗ F1 X = = −aF2 µ∗ X + µ∗ F1 X = (−a + 1)µ∗ F1 X hence a = 1. The converse assertion is obvious. A similar argument is used when µ is F -antiholomorphic.

2.3

Almost contact Riemannian submersions

Given the Riemannian manifolds M , M 0 with the metrics g and g 0 , respectively, we consider the submersion π : M −→ M 0 . The vertical distribution V(M ), defined by x ∈ M 7−→ Vx = ker pi∗,x is always integrable. Any vector from Vx , called vertical vector field, is tangent to the local fiber of the submersion. With respect to the metric g, the vertical distribution V admits an orthogonal distribution H(M ) : x ∈ M 7−→ Hx , where Hx is the orthogonal complement of the subspace Vx in Tx M and its elements are the horizontal vectors of the submersion. H(M ) is called the horizontal distribution and π is a Riemannian submersion if it satisfies the condition 0 gx (X, Y ) = gπ(x) (π∗,x X, π∗,x Y )

(2.18)

for any x ∈ M and for all X, Y ∈ Hx . The vector field X ∈ X (M ) is called basic for the submersion π : M −→ M 0 if it is horizontal and π-related with a vector field X 0 ∈ X (M 0 ), i. e. if X is so that for any x0 ∈ π(M ) and x ∈ π −1 (x0 ) = Mx0 we have π∗,x Xx = Xx0 0

(2.19)

We also say that X is the basic vector field associated to X 0 and we use the abridged notation π∗ X = X 0 . Assume that M is a manifold of dimension 2n + 1 with the almost contact Riemannian structure (F, ξ, η, g) and that M 0 is a Riemannian manifold with the metric g 0 . Moreover, we suppose that on M 0 there is an endomorphism F 0 : X (M 0 ) −→ X (M 0 ) and let π : M −→ M 0 be a F -holomorphic Riemannian submersion, that is it satisfies the condition F 0 ◦ π∗ = π∗ ◦ F

(2.20)

Under these assumptions, if (F 0 , ξ 0 , η 0 , g 0 ) is an almost contact structure on M 0 and if π∗ ξ = ξ 0 then π is called an almost contact Riemannian submersion of type I. If (F 0 , g 0 ) is an almost Hermitian structure on M 0 then π is an almost contact Riemannian submersion of type II. The almost contact Riemannian submersions were introduced by B. Watson, [Wa84], [Wa90], and by D. Chinea, [Chi85], [Ch84] under the generic name of almost contact metric submersions.

2.3. ALMOST CONTACT RIEMANNIAN SUBMERSIONS

67

Almost contact Riemannian submersions of type I Theorem 2.3.1. Let π : M −→ M 0 be an almost contact Riemannian submersion of type I and dim M 0 = 2n0 + 1. For any x0 ∈ M 0 we have: a) the fiber Mx0 is a closed invariant submanifold of dimension 2(n − n0 ) in M , i. e. F Tx Mx0 ⊂ Tx Mx0 for all x ∈ Mx0 ; b) if the almost contact structure of M is normal then Mx0 is a Hermitian manifold; c) if M and M 0 are Kenmotsu manifolds then Mx0 is K¨ ahlerian. Proof. a) From the equality (2.20) it results π∗ ◦ F X == F 0 ◦ π∗ X = 0 for any vertical vector field X ∈ V(M ) and then F V(M ) ⊂ V(M ). It follows that Mx0 is invariant. The other assertions are true for any submersion. b) Consider ξ1 ∈ V(M ) and ξ2 ∈ H(M ) so that ξ = ξ1 + ξ2 . Then η(ξ2 ) = g(ξ2 , ξ2 ) = g 0 (π∗ ξ2 , π∗ ξ2 ) = g 0 (ξ 0 , ξ 0 ) = 1 and this proves that ξ1 = 0, hence ξ ∈ H(M ). But any vector X tangent to Mx0 is vertical and then we have η(X) = 0, so that b) follows from Theorem 1.2.1. c) Taking into account that on a Kenmotsu manifold we have dη = 0 (see the formulas (1.83)) and N (1) = N (2) = 0, the formula (1.18) becomes 2g ∇M (2.21) X F Y, Z = 3dΩ(X, F Y, F Z) − 3dΩ(X, Y, Z) for any X, Y, Z ∈ X (M ), where Ω is the fundamental form of M . But we proved above that η vanishes on vertical vectors, hence dΩ = 0 and then from (2.21) it follows ∇M X F Y = 0 for all the vectors X, Y, Z tangent to the fiber Mx0 , which shows that it is a K¨ ahler manifold. In the proof of the assertion b) from Theorem 2.3.1 we observed that the Reeb vector field of the manifold M is normal to any fiber Mx0 of the submersion π : M −→ M 0 , hence Mx0 is an integral submanifold of the contact distribution of M and then there are no almost contact Riemannian submersions of type I with M a contact manifold (this is an immediate consequence of a result proved in Chapter 5, Theorem 5.1.1, a)). Then from Theorem 2.3.1 and from Theorem 5.1.1, b) we deduce that the dimensions of the manifolds of an almost contact Riemannian submersion of type I satisfy the condition n0 + 1 ≤ n ≤ 2n0 . Proposition 2.3.2. Let π : M −→ M 0 be an almost contact Riemannian submersion of type I and denote by Ω, Ω0 the fundamental 2-forms of the manifolds M and M 0 , respectively. Then: a) π ∗ η 0 = η; b) π ∗ Ω0 = Ω; c) the horizontal and vertical distributions are invariant by F .

68


Proof. a) and b) follow from the equalities π ∗ η 0 (X) = η 0 (π∗ X) = g 0 (ξ 0 , π∗ X) = g 0 (π∗ ξ, π∗ X) = g(ξ, X) = η(X), π ∗ Ω0 (X, Y ) = Ω0 (π∗ X, π∗ Y ) = g 0 (π∗ X, F 0 π∗ Y ) = = g 0 (π∗ X, π∗ F Y ) = g(X, F Y ) = Ω(X, Y ). c) The invariance by F of the vertical distribution was showed in the proof of Theorem 2.3.1, a). If X is horizontal and Y is vertical then F Y is vertical and we have 0 = g(X, F Y ) = −g(F X, Y ), hence F X is horizontal. Theorem 2.3.3. Let π : M −→ M 0 be an almost contact Riemannian submersion of type I. a) If M is a normal almost contact manifold then M 0 also is normal almost contact. b) If M is a Kenmotsu manifold then M 0 is Kenmotsu, too. Proof. a) Denote by h, v the horizontal and the vertical projections corresponding to the decomposition T M = H(M ) ⊕ V(M ). Let be X, Y ∈ X (M ) two basic vector fields and denote by X 0 , Y 0 ∈ X (M 0 ) theirs π−related vector fields. It is easy to verify that F X, h[X, Y ] and h[F X, Y ] are the basic vector fields π−related with the vector fields F 0 X 0 , [X 0 , Y 0 ], [F 0 X 0 , Y 0 ] and then hNF (X, Y ) is the basic vector field π−related to NF 0 (X 0 , Y 0 ). On the other hand, from the equality (2.18) we obtain g(X, Y ) = g 0 (X 0 , Y 0 ) ◦ π,

η(X) = η 0 (X 0 ) ◦ π

(2.22)

In order to simplify our writing, in the following computation we renounce to specify the composition with π in the formulas (2.22) and thus we have dη 0 (X 0 , Y 0 ) = Ω0 (X 0 , F 0 Y 0 ) = g 0 (X 0 , F 02 Y 0 ) = −g 0 (F 0 X 0 , F 0 Y 0 ) = = −g(F X, F Y ) = Ω(X, F Y ) = dη(X, Y ) But ξ is the basic field associated with the vector field ξ 0 and since ξ is horizontal, from the above argument we deduce that h [NF (X, Y ) + dη(X, Y )ξ] is the basic vector field associated with the vector field NF 0 (X 0 , Y 0 )+dη 0 (X 0 , Y 0 )ξ 0 and then our assertion follows from Theorem 1.2.1. The affirmation b) follows in a similarway noticing that the basic vector M field h ∇ F Y + η(Y )F X + g(X, F Y )ξ is associated with the vector field X 0

0 ∇M Y 0 + η 0 (Y 0 )F 0 X 0 + g 0 (X 0 , F 0 Y 0 )ξ 0 . F 0X0 F

The O’Neill tensors A, T of the submersion π : M −→ M 0 are given by the formulas M AX Y = v∇M hX hY + h∇hX vY,

M TX Y = h∇M vX vY + v∇vX hY

for all X, Y ∈ X (M ). Their fundamental properties are studied in [O’N66] or in [Ia83], pg. 130–142). Here we only recall the Gauss equation of the submersion


69

π : M −→ M 0 0

RM (X, Y, Z, U ) = RM (X 0 , Y 0 , Z 0 , U 0 ) − 2g (AX Y, AZ U ) + + g (AY Z, AX U ) + g (AZ X, AZ U )

(2.23)

where X, Y, Z, U ∈ X (M ) are horizontal vector fields. Now, we define the tensor field A∗ by A∗ (X, Y ) = AX F Y − AF X Y for any ˆ δˆ the fundamenhorizontal vector fields X, Y on M . We also denote by Ω, tal 2-form and the codifferential operator on the fibers of the almost contact Riemannian submersion π : M −→ M 0 of type I. Lemma 2.3.4. [Chi87] For any vector field X ∈ X (M ) the following equalities hold δΩ(X) = 2(n − n0 )g(H, F hX)+ 1 ˆ + δ 0 Ω0 (π∗ hX) + δˆΩ(vX) + g(trace A∗ , vX), 2 δη =2(n − n0 )g(H, ξ) + δ 0 η 0 ◦ π

(2.24)

where H is the mean curvature vector of a given fiber of the almost contact Riemannian submersion π : M −→ M 0 of type I. Proof. Let {e1 , . . . , en−n0 , f1 , . . . , fn0 , F e1 , . . . , F en−n0 , F f1 , . . . , F fn0 , ξ} be a F -basis of the almost contact Riemannian manifold M , with the property that {f1 , . . . , fn0 } are basic fields and {e1 , . . . , en−n0 } are vertical. From the definition of the codifferential operator we have δΩ(X) = −

0 n−n X

∇M ei Ω (ei , X) + (∇ei Ω) (F ei , X) −

i=1 0

−

n X

M ∇M fi Ω (fi , X) + ∇F fi Ω (F fi , X) − (∇ξ Ω) (ξ, X)

i=1

and then we deduce 1 ˆ δΩ(vX) = δˆΩ(vX) + g(trace A∗ , vX), δΩ(hX) = 2 = 2(n − n0 )g(H, F hX) + δ 0 Ω0 (π∗ hX) Now, the first equality (2.24) follows from the decomposition δΩ(X) = δΩ(hX)+ δΩ(vX). The second formula formula (2.24) can be proved by a similar argument. D. Chinea, [Chi87], calls the formulas (2.24) the structure equations of the almost contact Riemannian submersion of type I and these are used for the study of the fibers in the case when the almost contact Riemannian structures of the manifolds M and M 0 satisfy certain conditions (especially the minimality

70


or the existence of some particular almost complex structures). For this kind of results see [Chi87] and [Ts95]. When the total space M of the submersion is a Kenmotsu manifold we obtain the following Proposition 2.3.5. [Ts91], [Ts90] Let π : M −→ M 0 be an almost contact Riemannian submersion of type I. If M is a Kenmotsu manifold then the fibers are minimal submanifolds of M . Proof. By direct computation we obtain the following equalities TvX F hY = F TvX hY,

TvX F vY = F TvX vY,

AhX F hY = F AhX hY,

AF hX vY = AhX F vY

for any X, Y ∈ X (M ). Let {Ui , F Ui } be a local basis of the fiber Mx0 . From the above relations it results TF U F U = F (TF U U ) = F (TU F U ) = F 2 TU U = −TU U for any vector U tangent to the fiber and then 0

n−n X 1 [TUi Ui + TF Ui F Ui ] = 0 H= 2(n − n0 ) i=1

hence each fiber is minimal. Almost contact Riemannian submersions of type II Proposition 2.3.6. Let π : M −→ M 0 be an almost contact Riemannian submersion of type II. Then: a) the Reeb vector field of the almost contact Riemannian manifold M is vertical; b) the horizontal and vertical distributions are invariant by the almost contact structure F of M ; c) any horizontal vector field X is orthogonal to the Reeb vector field of M ; d) if Ω0 is the fundamental 2-form of the almost Hermitian manifold M 0 then π ∗ Ω0 = Ω. Proof. a) and the first assertion b) follow easily from the formula (2.20). If X ∈ H(M ) and Y ∈ V(M ) then g(F X, Y ) = −g(X, F Y ) = 0, hence F X ∈ H(M ) and H(M ) is invariant by F . c) For X ∈ H(M ) we have 0 = g(X, ξ) = η(X). d) If X, Y are basic vector fields then from (2.22) we deduce π ∗ Ω0 (X, Y ) = Ω0 (π∗ X, π∗ Y ) = Ω0 (X 0 , Y 0 ) = g 0 (X 0 , F 0 Y 0 ) = g(X, F Y ) and the proof is complete.


71

Proposition 2.3.7. [IP93] Let π : M −→ M 0 be an almost contact Riemannian submersion of type II and dim M = dim M 0 + 1. π is harmonic if and only if the Reeb vector field ξ of M is geodesic, i. e. ∇M ξ ξ = 0. Proof. Obviously we have π∗ (ξ) = 0, hence V(M ) = hξi and H(M ) = ker η = D. Denote by n the dimension of the manifold M 0 and let {e0i , F 0 e0i }i∈1,n be an orthonormal local basis adapted to the almost Hermitian structure on M 0 . Then there are the basic vector fields e1 , . . . , en , e∗1 , . . . , e∗n , π−related with vector fields of the above basis, i. e. π∗ (ei ) = e0i , π∗ (e∗i ) = F 0 e0i . Moreover, {ei , e∗i , ξ}i∈1,n is a local basis and τ (π) =

n X

[hπ (ei , ei ) + hπ (e∗i , e∗i )] + hπ (ξ, ξ)

i=1

On the other hand, from the definition of the vector fields ei , e∗i and taking into (2.18) we have hπ (ei , ei ) = hπ (e∗i , e∗i ) = 0, hence τ (π) = account

M −π∗ ∇M ξ ξ . We deduce that τ (π) = 0 if and only if ∇ξ ξ ∈ V(M ). But ξ

is unitary and then ∇M ξ ξ ∈ H(M ). From the above argument we deduce that τ (π) = 0 if and only if the Reeb vector field of M is geodesic. By using a similar argument we also obtain Proposition 2.3.8. [IP93] Let π : M −→ M 0 be an almost contact Riemannian submersion of type II. The following assertions are equivalent: a) π is F -pluriharmonic; b) ∇M ξ = 0; c) ξ is a Killing vector field and dη = 0. From Proposition 2.2.1 it follows that an almost contact Riemannian submersion of type II satisfying the condition b) or c) from Proposition 2.3.8 is harmonic. From Proposition 2.3.6, a), b) it follows that the fibers of an almost contact Riemannian submersion of type II are invariant submanifolds of M , tangent to the Reeb vector field and it is easy to check that the restrictions to fibers of the almost contact Riemannian structure of M define natural almost contact Riemannian structures. Theorem 2.3.9. If the total space M of an almost contact Riemannian submersion π : M −→ M 0 of type II is a Sasaki or a Kenmotsu manifold then the fibers are minimal submanifolds of M . Proof. If M is a Kenmotsu manifold then the O’Neill tensor T satisfies the conditions TvX F hY = F TvX hY,

TvX F vY = F TvX vY,

Tξ ξ = 0

Afterward the proof is identical with that of Proposition 2.3.5.

72


In the case when the total space is Sasakian we compute the mean curvature vector in a local F -basis of the fiber. We also remark that this assertion is trivial by Proposition 6.2.3 (see Section 6.2). By an argument similar to the one used in the proof of Theorem 2.3.3 we obtain Theorem 2.3.10. [Ts90] If the total space M of the almost contact Riemannian submersion π : M −→ M 0 of type II is a Kenmotsu manifold then its base space M 0 is K¨ ahlerian. The following result, gives a simple method to construct almost contact Riemannian submersions. Theorem 2.3.11. [Ts90] Let M1 be an almost Hermitian manifold, M2 be an almost contact Riemannian manifold and consider their product M1 × M2 with the induced almost contact Riemannian structure (see Proposition 1.2.4). Then: a) the projection π2 : M1 × M2 −→ M2 , π2 (x1 , x2 ) = x2 is an almost contact Riemannian submersion of type I; b) the projection π1 : M1 × M2 −→ M1 , π1 (x1 , x2 ) = x1 is an almost contact Riemannian submersion of type II. Proof. a) Obviously the maps π1 and π2 are Riemannian submersions and we have π2∗ F (X1 , X2 ) = F2 X2 = F2 π2∗ (X1 , X2 ), π2∗ ξ = π2∗ (0, ξ2 ) = ξ2 hence π2 is an almost contact Riemannian submersion of type I. Similar argument is used for b).

2.4

Contact bundles

Principal bundles Let P and M be two manifolds, G a Lie group and π : P −→ M a differentiable map. The third form (P, M, G, π) or simply (P, M, G) is a differential principal bundle if the following conditions are fulfilled: 1) M has an open covering {Uα }α∈I such that for each α ∈ I there exists a diffeomorphism φα : π −1 (Uα ) −→ Uα × G with the property that π ◦ φ−1 α : Uα × G −→ Uα is the projection on the first factor; 2) there is a (differentiable) right action P × G −→ P , given by Ra u = ua for any a ∈ G and u ∈ P , simply transitive on π −1 (x) = Px for each x ∈ M ; 3) for each α ∈ I the map φα is equivariant with respect to this action and to the right action (Uα × G) × G −→ Uα × G denoted by ((x, a), b) 7−→ (x, ab) (see the footnote ?, Section 1.3). The pair (Uα , φα ) is called a fibered chart of (P, M, G) and the closed submanifold Px is the fiber at the point x ∈ M of the bundle and it is diffeomorphic

2.4. CONTACT BUNDLES

73

to G. P is the total space, M is the base space, G is the structure group and π is the natural projection of the principal bundle (P, M, G). By 1) we have φα (u) = (π(u), φ∗α (u)) for some φ∗α : π −1 (Uα ) −→ G and then by 3) the map φ∗α satisfies φ∗α (u · a) = φ∗α (u) · a. Moreover, if u ∈ π −1 (Uα ∩ −1 −1 −1 Uβ ) then φ∗β (u · a) (φ∗α (u · a)) = φ∗β (u) (φ∗α (u)) , hence φ∗β (u) (φ∗α (u)) is depending only on π(u) but not on u and thus we can define a map gβα : Uα ∩ Uβ −→ G by −1 gβα (π(u)) = φ∗β (u) (φ∗α (u)) The maps gβα are the transition functions of the bundle and the following relations hold gγβ (x) · gβα (x) = gγα (x)

(2.25)

for all x ∈ Uα ∩ Uβ ∩ Uγ . Conversely, let {Uα }α∈I be an open covering of the manifold M and G a Lie group. If {gβα : Uα ∩ Uβ −→ G}α,β∈I is a family of maps satisfying (2.25) then we can construct a principal bundle with M as base space, the group G as Sstructure group and gβα as transition functions. For this purpose let X = α Uα × G be the disjoint union of Uα × G with the sum topology. Each element of X is a triple (α, x, a) and the relation [(α, x, a) ≡ (β, y, b)] ⇔ [x = y ∈ Uα ∩ Uβ , b = gβα (a)] is an equivalence. Denote by P the quotient manifold M/ ≡ and remark that the group G acts on the right over P by the action [α, x, a] · c = [α, x, ac], where [α, x, a] is the equivalence class of the element (α, x, a). The projection π : P −→ M is well defined by π [α, x, a] = x and then (P, M, G, π) is a principal bundle having gβα as transition functions. We remarked that each fiber Px is a submanifold of P and then V = ∪x∈P Tu Pπ(u) is a subbundle of the tangent bundle T P . We have V = ker π∗ and, like in the case of vector bundles and more general like in the case of a submersion (see Section 2.3), it is the vertical subbundle of the principal bundle (P, M, G). A section X ∈ Γ(V) is a vertical field (or a vertical vector) on P . We say that a distribution H on the manifold P defines a connection ∇(H) in the principal bundle (P, M, G) if: a) Hua = (Ra )∗ Hu for every u ∈ P and a ∈ G; b) Tu P = Hu ⊕ Vu for any u ∈ P . H is called the horizontal distribution and its sections are horizontal fields. From the condition a) we deduce that the horizontal distribution is invariant under the right action Ra . In the case of vector bundles such a horizontal distribution is called a nonlinear connection. This designation, used in Finsler geometry, was introduced by W. Barthel [Ba63] and for a modern study of such connections see [MA97]. Denote by g the Lie algebra of G and by ad : G −→ g the adjoint representation of the group G, that is the map defined by (ad(a))A = (Ra−1 )∗ A∗ for all a ∈ G, A ∈ g, also denoting by Ra the right translation og G by the

74


element a ∈ G. For A ∈ g the right action Ra on P of the 1-parameter subgroup {at = exp(tA)}t∈R induces a vector field A∗ ∈ X (P ). From the condition 1) in the definition of the principal bundle it follows that the action of G sends each fibre of the bundle into itself for each u ∈ P , hence A∗u is tangent to the fibre at u. Moreover, A∗ nowhere vanishes on P and since each fibre is difeomorphic to G it follows that for any u ∈ P the map A ∈ g 7−→ (A∗ )u ∈ Tu Pπ(u) is an isomorphism (see [KN69], vol. I, pg. 40–42 and 51). We also remark that ∗ ad(a−1 ) A = (Ra )∗ A∗ Now, to the connection ∇(H) we associate a 1-form ω on P with values in g. Namely for X ∈ Tu P we set ω(X) = A, where A ∈ g is the unique vector field on G so that (A∗ )u is equal to the vertical component of X. Obviously, ω(X) = 0 if and only if X ∈ Hu , hence H = ker ω. ω is called the connection form of the connection H and it satisfies the following conditions: a) ω(A∗ ) = A for any A ∈ g; b) (Ra )∗ ω = ad(a−1 )ω, that is ω ((Ra )∗ X) = ad(a−1 )ω(X) for any a ∈ G and X ∈ X (P ). Conversely, if the 1-form ω on P satisfies the conditions a) and b) then it uniquely defines a connection whose connection form is exactly ω. The curvature form of the connection ∇(H) is the 2-form Ω on P with values in the Lie algebra g, defined by Ω(X, Y ) = dω(hX, hY ), where h : Tu P −→ Hu is the natural projection and X, Y ∈ Tu P . then for each u ∈ P and X, Y ∈ Tu P we have 1 dω(X, Y ) = − [ω(X), ω(Y )] + Ω(X, Y ), 2

1 or dω = − [ω, ω] + Ω 2

(2.26)

The connection 1-form ω in the principal bundle (P, M, G) is called fat (in Weinstein’s sense, [We80]) at µ ∈ g ∗ if for any point u ∈ P the bilinear map ω ω ω µ ◦ Ωω u : Hu × Hu −→ R is non degenerate, Ω being the curvature form of the ω ω connection form ω and H : u 7−→ Hu = ker(ωu : Tu P −→ g) is the associated horizontal distribution. ω is fat (in Weinstein’s sense) if it is fat for each µ ∈ g. For a given connection ∇(H) in the principal bundle (P, M, G) and for X ∈ X (M ) there exists an unique vector field X H ∈ X (P ) such that XuH ∈ Hu and π∗,u (XuH ) = Xπ(u) for any u ∈ P . This is called the horizontal lift of the field X (see the case of vector bundles in Section 1.3). More generally, the horizontal lift of the curve γ : (a, b) −→ M is a curve γ H : (a, b) −→ P with the property that π ◦ γ H = γ and its tangent vectors at any point γ H (t) belong to Hγ H (t) . Let t0 , t1 ∈ (a, b) and u0 ∈ Pγ(t0 ) . There is an unique horizontal lift γ H of the curve γ, passing through the point u0 and we put u1 = γ H (t1 ) ∈ Pγ(t1 ) . If the point u0 run through the fiber Pγ(t0 ) then we obtain the bijective map Pγ : u0 ∈ Pγ(t0 7−→ u1 ∈ Pγ(t1 ) , called the parallel transport along the curve γ. Finally we apply some of these results to the case of principal bundle with the total space L(M ) of linear frames of the manifold M and G = GL(mR). Considering u ∈ L(M ) as a linear map from Rm to Tπ(u) M , we can define on L(M ) a 1-form theta with values in Rn by setting θ(X) = u−1 (π(X)) for all


75

X ∈ Tu L(M ). It is the canonical form of L(M ). A connection in the principal bundle (L(M ), M, GL(n, R)) is just a linear connection usually denoted by ∇ and the Rm -valued 2-form Θ on L(M ), defined by Θ(X, Y ) = dθ(hX, hY ) for all X, Y ∈ Tu L(M ), is called the torsion form of nabla. Then 1 dθ(X, Y ) = − [ω(X), θ(X)] + Θ(X, Y ) 2

(2.27)

(2.26) and (2.27) are known as the structure equations and these can be written in a more explicit this purpose let {e1 , . . . , em } be the natural basis of P form. For P Rm . Then θ = θi ei , Θ = Θi ei . On the other hand, denoting by Eji = (akl ) the m × m matrix with aij = 1 and akl = 0 for other values of k and l, we see that Eji is a basis of the Lie algebra gl(m, R) of GL(m, R) and then ω = ωji Eij ,

Ω = Ωji

Thus the structure equations have the following form i

dθ = −

m X

ωki

k

i

∧θ +Θ ,

dωji

k=1

=−

m X

ωki ∧ ωjk + Ωij

(2.28)

k=1

We also remark that if (U, φ) is a local chart and {X1 , X2 , . . . , Xm } is a local basis of vector fields on U then the local components ωij of the connection form ω of ∇ are given by ∇X Xi =

m X

ωij (X)Xj

(2.29)

j=1

and the curvature form Ω = Ωij is locally given by Ωij =

m 1 X i Rjkl dxk ∧ dxl 2

(2.30)

k,l=1

i where Rjkl are the components of the curvature tensor R. For a deep study of principal bundles see one of the monographs [Hu66], Chapter II, Section 1 or [KN69], vol. I, Chapter I, Sections 4 and 5.

Contact bundles Let (P, M, G) be a principal bundle and consider the contact manifold S with the contact form η. If the group G acts on S by the differentiable left action G × S −→ S, (a, p) 7−→ ap then on the product manifold P × S the following right action is defined (P × S) × G −→ P × S, ((u, p), a) 7−→ (ua, a−1 p), where ua is the right action on P . Let P × S/G be the quotient space of P × S by this action and denote it by E = P ×G S. Obviously, the elements from E are classes of pairs (u, p) ∈ P × S, namely [u, p] = (ua, a−1 p), a ∈ G and from

76


the condition 2) in the definition of principal bundles it follows that a map πE : E −→ M is well-defined by πE [u, p] = π(u). If (Uα , φα ) is a fibered chart of the principal bundle (P, M, G) then by condition 3) of the same definition the map −1 ψ α : πE (Uα ) −→ Uα × S;

ψα [u, p] = (π(u), φ0α (u)p)

−1 is well-defined. It is also bijective and induces on πE (Uα ) a structure of manifold diffeomorphic to Uα × S. We easily obtain on E a manifold structure and the projection πE becomes differentiable. If for any two pairs (Uα , ψα ), (Uβ , ψβ ) −1 for which Uα ∩ Uβ 6= ∅ the map ψβ ◦ ψα|{x}×S : S −→ S is an element of the group Γ+ (S) (see Section 2.1) then the principal bundle (E, P, M, G), equipped with the contact manifold S, is called the contact bundle associated to the principal bundle (P, M, G). It is denoted by (E, P, M, G, S) or by (E, P, M, S). This notion is natural taking into account the definition of contact manifolds in the wider sense (defined in Section 1.3) and it was introduced by E. Lerman [Le03], as well as a part of the next results concerning the contact bundles.

Proposition 2.4.1. Let (E, P, M, S) be a contact bundle. For any x ∈ M the fiber Px has a contact structure. Proof. Let (Uα , ψα ) be a pair as above and x ∈ Uα . If pr2 : Uα × S −→ S is the projection on the second factor then we can define the 1-form ηα = (pr2 ◦ψα )∗ η ∈ F 1 (π −1 (Uα )) and we easily verify that its restriction ηα|Px to the fiber Px is a contact form. Moreover, if (Uβ , ψβ ) is another pair with x ∈ Uβ then we have ker ηα|Px = ker ηβ|Px

(2.31)

therefore the contact structure constructed above on the fiber Px is well-defined. Denoting by Du the contact distribution of this contact structure on the fiber Pπ(u) we obtain the subbundle DV = ∪u∈P Du of the vertical bundle V. Proposition 2.4.2. Let (E, P, M, S) be a contact bundle. There exists a 1-form α ∈ F 1 (P ) such that ker α ∩ V = DV . Proof. Let M be a manifold and {Uα }α∈I be a locally finite open covering whose open sets Uα have the type of those from the proof of Proposition 2.4.1 and consider a partition of unity {fα }α∈I , subordinated to this covering. For each P α ∈∗ I denote by ηα the 1-form defined in the same proof. Then α = α∈I (π fα ) ηα is a 1-form on the manifold P and taking into account the equality (2.31), it satisfies the condition of our statement. The following converse of Proposition 2.4.2 can be proved: Theorem 2.4.3. [Le03] Let (E, P, M, S) be a contact bundle and α ∈ F 1 (P ) such that Ker α is a co-oriented 1-codimensional distribution on the manifold P . If for each x ∈ M there is on the fiber Px a contact structure whose contact


77

distribution is ker α∩T Px then there exists a natural connection ∇(Hcn (α)) such that the parallel transport with respect to ∇(Hcn (α)) (when it exists) preserves (at least locally) the contact structure on the fibers and their co-orientation. Proof. Since α|Px are contact forms on Px for each x ∈ M , it results that the 2-form dα|DV is non degenerate and we put Hcn = X ∈ ker α : dα(X, Y ) = 0 for any Y ∈ DV Therefore Ker α = DV ⊕ Hcn

(2.32)

and if α0 is another 1-form with ker α0 = ker α then there exists f ∈ F(P ) such that α0 = exp(f )α. It follows dα|0 ker α = exp(f )dα| ker α , hence Hcn does not depend on the choice of the form α and therefore it defines a natural connection on the bundle (E, P, M, S). Now, for X ∈ X (M ) denote by X H ∈ X (P ) its horizontal lift with respect to the connection ∇(Hcn ). For Y ∈ Γ(DV ) we have [X H , Y ] ∈ Γ(V) and taking into account the definition of Hcn we obtain dα(X h , Y ) = 0. Since ıY α = 0, ıX H α = 0, we deduce [X H , Y ] ∈ ker α, so that [X H , Y ] ∈ Γ(V) ∩ ker α = Γ(Dv ). This proves that the parallel transport relative to H preserves the distribution DV . Let Pγ : Pγ(0) −→ Pγ(1) be the parallel transport along the curve γ : [0, 1] −→ P . We have (Pγ )∗ ker α|Pγ(0) = ker α|Pγ(t) for any t ∈ [0, 1] and then there exists nowhere zero ft ∈ F(Pγ(0) ) such that (Pγ )∗ α|Pγ(0) = ft α|Pγ(0) and, obviously, ft depends continuously on t. But f0 = 1, therefore ft > 0 for any t ∈ [0, 1] and this proves that the parallel transport preserves the co-orientation of the fibers. The bundle (E, P, M, S) satisfying the conditions from the first part of Theorem 2.4.3 is called a strict contact bundle and the connection ∇(Hcn ) is a contact connection. Remark 2.4.4. Any strict contact bundle is, of course, a contact bundle. In fact, in [Le03] E. Lerman understand by contact bundle a strict contact bundle. We have seen that the fibers Px of the strict contact bundle (E, P, M, S) have a structure of contact manifold. Then it is natural to ask if the total space P of the bundle has a contact structure. The answer is related to the curvature form Rnc of the contact connection ∇(Hnc ). For X, Y ∈ Tx M and U, V ∈ X (M ) such that U (x) = X, V (x) = Y the form Rnc is defined by the formula Rxnc (X, Y ) = [U H , V H ] − [U, V ]H ∈ X (Px ) where

H

(2.33)

denotes the horizontal lift with respect to Hnc .

Lemma 2.4.5. Let (E, P, M, S) be a strict contact bundle and let ∇(Hnc (α)) be a contact connection. The 1-form α from Theorem 2.4.3 is a contact form

78


on the totalh space P if iand only if for any u ∈ P and for any ω ∈ (Ker αu )0 − 0 nc the map ω Rπ(u) (·, ·)u : Tπ(u) M × Tπ(u) M −→ R, defined by h i h i nc nc ω Rπ(u) (·, ·)u (X, Y ) = ωu Rπ(u) (X, Y )u is non degenerate. Proof. Taking into account the definition of the contact structure it is enough to prove that the 2-form dα| ker α is non degenerate. Or, by the last part of the proof of Theorem 2.4.3 the 2-form dα|DV is non degenerate and by using the decomposition (2.32) we have to prove just that the 2-form dα|Hnc is non degenerate. For this purpose we consider the vectors X, Y ∈ Hunc and the vector fields U, V ∈ X (M ) such that U H (u) = X, V H (u) = Y . Since α(U H ) = 0 for any U ∈ X (M ), we have 2dα(X, Y ) = 2dα(U H , V H ) = = U H (α(V H )) − V H (α(U H )) − α[U H , V H ] = = −[U H , V H ] + [U, V ]H Now, taking into account the formula (2.33) we obtain h i nc 2dαu|Hnc (X, Y ) = 2dαu (X, Y ) = −αu Rπ(u) (π∗ X, π∗ Y )u u 0

But for any ω ∈ (ker αu ) − 0 there exists f = 6 0 so that ω = f αu and then from the above equality we deduce that dαu|Hnc is non degenerate if and only if the u maps in the statement are non degenerate. Taking into account Proposition 2.4.1, we denote by Dx the contact distri0 bution on the fiber Px . Then Dx = ker α ∩ T Px and for u ∈ Px , η ∈ (Dx )u , X ∈ Xξ (Px ) we put h(u, η), Xi = η(Xu ) ∈ R. On the other hand by the connection ∇(Hnc ) we can identify (ker α)0 with ∪x (Dx )0 and then (ker α)0+ = ∪x (D)0+ . ˜ nc the pull-back of the vector bundle Hnc over P by the projection Denote by H 0 ˜ nc ≡ (p ◦ π)∗ T M . Then we p : (ker α)+ −→ P . Since Hnc ≡ π ∗ T M it follows H nc ˜ nc can define the following skew-symmetric form σ on the vector bundle H nc σ(u,η) (X, Y ) = h(u, η), Rxnc (X, Y )i = hη, Rxnc (X, Y )(u)i

for u ∈ P , η ∈ (ker α)0+ under the form

u

, X, Y ∈ Tπ(u) M . Thus Lemma 2.4.5 can be written

Lemma 2.4.6. α is a contact form on P if and only if σ nc is a symplectic form ˜ nc . on the vector bundle H Another case when on the manifold P there exists a contact structure is the following:


79

Theorem 2.4.7. Let S be a co-oriented contact manifold and let G be a Lie group whose left action on S preserves the contact distribution D and its co0 orientation. Denote by Ψ : D+ −→ g∗ the contact moment map induced by the representation ρ : g −→ Fξ . If (P, M, G) is a principal bundle and ω is a connection 1-form on P whose induced connection is ∇(H) then there is a codimension 1 co-oriented distribution E in the total space of the associated bundle (E = P ×G S, P, M, G, S) such that E = H⊕(P ×G D) and the distribution ker α∩T Px is isomorphic to D for any x ∈ M . Moreover, there is on E a contact structure with the contact distribution E if and only if the connection form ω is 0 fat in Weinstein’s sense at the points of Ψ(D+ ). Proof. The action of G on M preserves the co-orientation of the contact distribution D of S, hence DV = P ×G D is an oriented subbundle of the vertical bundle V of the associated bundle of (P, M, G) and V ≡ P ×G T S. On the other hand the connection form ω defines a horizontal distribution H in T E, therefore the distribution H ⊕ (P ×G D) is co-oriented. We also remark that for each u ∈ P there is an embedding ıu : S ,→ E, defined by ıu (p) = [u, p], where [u, p] ∈ E is the image in E of the element (u, p) ∈ P × S, that is [u, p] is the orbit of (u, p). V V Then dıu∗ (D) = D|P and we have E ∩ T Pπ(u) = D|P ≡ D. From Lemma π(u) π(u) 0 0 2.4.6 it follows that it is enough to prove that for any [u, p, η] ∈ P ×G D+ = E+ nc nc and x = π [u, p] the form σ[u,p,η] = h[u, p, η] , Rx (·, ·) [u, p]i : Tx M × Tx M −→ R is non degenerate. G acts on S by contact transformations and then there is a representation ρ : g −→ Xξ (S). Then, taking into account (1.40) and since ρ and Ψ are equivariant, these induce the following maps of associated bundles 0 Ψ : P ×G D+ −→ P ×G g∗ ,

ρ : P ×G g −→ P ×G Xξ (S), ω

Ψ [u, p, η] = [u, Ψ(p, η)] ρ [u, X] = [u, ρ(X)] ω

On the other hand the curvature form Ω : H ×Hω −→ g of ω defines a 2-form ω Ω on M with values in the bundle P ×G g. This is given by the formula ω H H ∈ P ×G g Ωx (X, Y ) = u, Ωω u Xu , Yu ω

for any u ∈ P and for all X, Y ∈ Tx M , x = π(u). Moreover, we have ρ ◦ Ω = Ωnc , where Ωnc is a 2-form on M with values in P ×G Xξ (S) and the moment 0 map Ψ : D+ −→ g∗ is the adjoint of ρ, i. e. h(p, η), ρ(X)i = hΨ(p, η), Xi for all 0 p ∈ S, η ∈ D+ and X ∈ g. Hence E D ω nc σ[u,p,η] (X, Y ) = h[u, p, η] , Ωnc u (X, Y )i = [u, p, η] , ρ ◦ Ωx (X, Y ) = D E ω = Ψ [u, p, η] , Ωx (X, Y ) = hΨ(p, η), Ωω u (Xu , Yu )i 0 It follows that ω is fat on Ψ D+ if and only if σ nc is non degenerate. The above argument suggests a relatively simple construction of strict contact bundles, namely:

80


Theorem 2.4.8. [Le03] Let S be a contact manifold and G be a Lie group whose action on the manifold S preserves the contact distribution D and its co-orientation. Assume that H ⊂ T P defines a G-invariant connection in the principal bundle (P, M, G) and let H be the induced distribution in E = P ×G S. Then the distribution H ⊕ (P ×G D) defines on (E, P, M, G, S) a strict contact bundle structure.

2.5

Almost contact fibered spaces

Let (M, M 0 , π, g) be a fibered space, that is M is a n-dimensional Riemannian manifold with the metric g, M 0 is a manifold of dimension n0 and π : M −→ M 0 is a submersion. We suppose, in addition, that all fibers Mx0 = π −1 (x0 ) are connected. At any point x ∈ M the metric g induces the orthogonal decomposition (see also the beginning of Section 2.3) Tx M = Tx Mπ(x) ⊕ Hx (M ) and we denote by h and v the corresponding projections. Let (U, xi ), i ∈ 1, n and (U 0 , y a ), a ∈ 1, n0 be two local charts in M and M 0 , respectively, and assume that π(U ) = U 0 . Then the projection π can be locally expressed by equations of the form y a (π(x)) = y a (xi (x)) or in abridged form y a = y a (xi ) a and rang ∂y = n0 for any x ∈ U . On U we define the 1-forms ∂xi E a = Eia dxi =

(2.34)

∂y a i dx ∂xi

and remark that these are linearly independent in F 1 (M ). Now, let Mx0 be a fiber such that Mx0 ∩U 6= ∅. Then in Mx0 ∩U we can choose the local coordinates {z α }α∈n−n0 ,n such that (y a , z α ) is a coordinates system in U . With respect to these coordinates the submanifold Mx0 ∩ U admits a parametric representation of the form xi = xi (y a , z α ) and we consider on U the vector fields Cα = Cαi

∂ ∂xi ∂ = ∂xi ∂z α ∂xi

Then {Cα }α∈n−n0 ,n is a basis in the fiber Mπ(x) for any x ∈ Mx0 ∩U and Eia Cβi = 0. The components of the metric induced by g on Mx0 are g αβ = gij Cαi Cβj and then C α = Ciα dxi = gij g αβ Cβj dxi are 1-forms on U . Moreover, {E a , C α } is a local basis of 1-forms over the open set U . A simple computation shows that we can choose the vector fields Ea such that {Ea , Cβ } is a local basis, whose dual basis is {E a , C α }. We also remark that Ea and E b are horizontal vector fields and horizontal 1-forms respectively,

2.5. ALMOST CONTACT FIBERED SPACES

81

while Cα and C β are vertical vector fields and vertical 1-forms, respectively. With respect to these bases the tensor fields F and g have the following local expressions α

F = Fãb E a ⊗ Eb + F β C β ⊗ Cα ,

g = gab E a ⊗ E b + g αβ C α ⊗ C β

(2.35)

Expressions of the form (2.35) can be obtained for any tensor field T defined on the base space M . For instance, if T is a tensor field of the type (1, 2) on M , having the components Tijk with respect to the local chart (U, xi ) then, [Tak93] γ c α a a T = Tãb E a ⊗E b ⊗Ec +Tab E ⊗E b ⊗Cα +. . .+Tαβ C α ⊗C β ⊗Ea +T¯αβ C α ⊗C β ⊗C γ

where c Tãb = Tijk Eai Ebj Ekc ,

α Tab = Tijk Eai Ebj Ckα , γ T¯αβ = Tijk Cαi Cβj Ckγ

a Tαβ = Tijk Cαi Cβj Eka ,

and Eai , Cαi is the inverse of the matrix (Eia , Ciα ). The terms c T˜ = Tãb E a ⊗ E b ⊗ Ec ,

γ T¯ = T¯αβ Cα ⊗ Cβ ⊗ Cγ

are the horizontal part and the vertical part of the tensor field T , respectively. T is projectable if ] ˜=0 L ¯T X

(2.36)

for any X ∈ X (M ). Taking into account the expression of the horizontal and vertical parts of a tensor field, we deduce that T is projectable if and only if c ∂ Tãb =0 α ∂z c i. e. if the functions Tãb are constant along the intersection of U with each fiber Mx0 . Of course, the notion of projectable tensor field can be easily extended to tensor fields of any type. In particular f ∈ F(M ) is projectable if and only if ¯ = 0 for all X ∈ X (M ). Then there is a unique function f ∗ ∈ F(M 0 ) such Xf that f = π ◦ f ∗ Conversely, for f ∗ ∈ F(M 0 ) the function π ◦ f ∗ ∈ F(M ) is projectable. f ∗ is the projection of f and we denote it by pf . We also remark that, conversely, for any f ∗ ∈ F(M 0 ) the function π ◦ f ∗ ∈ F(M ) is projectable. Now, we can associate another tensor field to the tensor field T 0 defined on M 0 by the following local expression in π −1 (U 0 ) ⊂ M 0c a T 0L = pTab E ⊗ E b ⊗ Ec

If the metric g is projectable then (M, M 0 , π, g) is called a Riemannian fibered space. If the total space M of the Riemannian fibered space (M, M 0 , π, g) has an almost contact Riemannian structure (F, ξ, η, g) and if F is projectable then

82


(M, M 0 , π, g) is an almost contact Riemannian fibered space and sometimes it is simply denoted by its total space M . This structure was defined and studied by Y. Tashiro and B. H. Kim, [TKi88]. From the above formulas we obtain the following Theorem 2.5.1. [TKi88] If the fibers of the almost contact Riemannian fibered space (M, M 0 , π, g) are invariant submanifolds (see Theorem 2.3.1, a)) and if its base space M 0 has even dimension then: a) the distribution spanned by {Ea } is invariant by F ; b ˜ b) J = Fa is an almost complex structure on the base space M of the fibered space; c) on each fiber there exists a vector field ξ and a 1-form η such that α F = F β C β ⊗ Cα , ξ, η, g = (g αβ ) is an almost contact Riemannian structure. From the formulas (2.35) it follows that g 0 = (gab ) defines a Riemannian metric on the base M 0 of the fibered space and thus we can state Theorem 2.5.2. [TKi88] Let (M, M 0 , π, g) be an almost contact fibered space whose fibers are invariant submanifolds. If M is a contact (or K-contact) manifold then (J, g 0 ) is an almost K¨ ahler structure on the base space M 0 and the almost contact Riemannian structure (F , ξ, η, g) on each fiber is a contact (or K-contact) structure. Let γ 0 be a curve passing through the point x0 ∈ M 0 and denote by X 0 the vector field tangent to γ 0 . There exists only one curve γ passing through the point x ∈ Mx0 and such that its tangent vector field is X 0L . If γ 0 join the points x01 , x02 ∈ M 0 then the curves γ at all the points of the fiber Mx01 define a map Φγ 0 : Mx01 −→ Mx02 in the following way: If γ is the curve associated to the curve γ 0 at the point x1 ∈ Mx01 then there is an unique point x2 ∈ γ ∩ Mx02 and we put Φγ 0 (x1 ) = x2 . The Riemannian fibered space (M, M 0 , π, g) is called a fibered space with isometric (conformal) fibers if the map Φγ 0 associated to any curve γ 0 of M 0 is an isometry (a conformal transformation). Taking into account the equality (2.36) it follows Proposition 2.5.3. The Riemannian fibered space (M, M 0 , π, g) has isometric fibers if and only if LX 0L g¯ = 0 for any X 0 ∈ X (M 0 ). The following construction due to B. H. Kim [Ki88] allows us to find a converse of Theorem 2.5.1. For this purpose let (M, M 0 , π) be a fibered space whose base space M 0 is a manifold with the almost complex structure J 0 and such that each fiber has an almost contact structure (F , ξ, η) depending, of course, on the point in the base space M 0 . But in the construction of the basis

2.5. ALMOST CONTACT FIBERED SPACES

83

{E a , C α } we used only the Riemannian fibered space structure, hence we can define the tensor fields 0 α α F = Ja0b E a ⊗ Eb + F β C β ⊗ Cα , η = (0, η α ), ξ = ξ α

where η α , ξ are the local components of the 1-form η and those of the vector field ξ with respect to the bases {C α } and {Cα }, respectively. By an elementary computation we prove that (F, ξ, η) is an almost contact structure on the total space M of the Riemannian fibered space (M, M 0 , π, g). Moreover, if M 0 is an almost Hermitian manifold with the metric g 0 and if the fibers are almost contact Riemannian manifolds then the system (F, ξ, η, g) with g given by the 0 formula (2.35) (where we replace gab by gab ), is an almost contact Riemannian structure on M , named the induced structure. Thus we obtain the following result analogous to Theorem 2.5.2: Theorem 2.5.4. [Ki88] Let (M, M 0 , π) be a fibered space with the base M 0 an almost complex manifold and with the fibers almost contact manifolds. If the almost contact Riemannian structure induced on the total space M is a contact Riemannian (K-contact) structure then: a) M 0 is an almost K¨ ahler manifold; b) each fiber is a contact (K-contact) manifold. The almost contact Riemannian fibered space (M, M 0 , π, g) is a Sasaki fibered space if the almost contact Riemannian structure (F, ξ, η, g) on the total space M is Sasakian. Example. [Ki88] In the plane R2 we consider the coordinates x1 , x2 and in the space R3 endowed with the standard Sasaki structure (see Example 1, Section 1.5) we denote by y 1 , y 2 , z the coordinates (in fact, the coordinates x1 , y 1 from the cited example are denoted here by y 1 , y 2 ). The vector fields C1 = (0, 0, 1, 0, 0), C2 = (0, 0, 0, 1, 0), C3 = (0, 0, 0, 0, 1), E1 = (2, 0, 0, 0, 2x2 ), E2 = (0, 2, 0, 0, 0) form a frame in R5 with the coordinates x1 , x2 , y 1 , y 2 , z and equipped with the standard Sasaki structure. We easily verify that R5 becomes a Sasaki fibered space with R2 as base space and the Sasaki manifold R3 as fiber. A similar argument shows that for any n > m the Sasaki manifold R2n+1 is a Sasaki fibered space with the base R2m and with the Sasaki manifold R2(n−m)+1 as fiber. Theorems 2.5.2 and 2.5.4 have the following forms for Sasaki fibered spaces: Theorem 2.5.5. [TKi88] If (M, M 0 , π, g) is a Sasaki fibered space with the fibers invariant submanifolds of M then the almost Hermitian structure (J, g 0 ) on the base space M 0 is K¨ ahlerian and the almost contact Riemannian structure (F , ξ, η, g) on fibers is Sasakian. Moreover, each fiber is a minimal submanifold of the manifold M .

84


Theorem 2.5.6. [Ki88] Let (M, M 0 , π, g) be a Riemannian fibered space with the fibers almost contact Riemannian manifolds and the base M 0 a manifold with the almost Hermitian structure (J, g 0 ). If the almost contact Riemannian structure induced on M is Sasakian then M 0 is a K¨ ahler manifold and the fibers are minimal submanifolds in M and have Sasaki structures.

2.6

Orbit space of a contact manifold

Let M be a manifold with the almost contact structure (F, ξ, η) and X ∈ X (M ). The maximal integral curve of the vector field X, passing through the point x ∈ M is called the orbit of X through x. All the orbits considered here are assumed to be oriented and connected. We have Proposition 2.6.1. The orbits of the Reeb vector field of a regular compact almost contact manifold are simple closed curves. Denote by Mξ the leaf space of the characteristic foliation Fξ of the almost contact manifold M . In fact, Mξ is the quotient space M/Fξ of M by the equivalence relation whose classes are the orbits of ξ. The natural projection π : M −→ Mξ associates to the point x ∈ M the orbit passing through x and defines on Mξ the quotient topology. Mξ endowed with this topology is called the orbit space of the manifold M . Moreover we have Proposition 2.6.2. Let M be a regular almost contact manifold of dimension 2n + 1. If the Reeb vector field is complete (i. e. if it generates a global group of diffeomorphisms of M ) then Mξ is a 2n-dimensional manifold and the projection π is a submersion. Proof. Let x0 ∈ M and let (U, φ) be a local chart in x0 such that φ(U ) ⊂ R2n+1 is an open cube centered at x0 . The manifold M is regular, hence in this chart we can consider the coordinates x1 , . . . , xn , y 1 , . . . , y n , z such that for ci = constant, i ∈ 1, 2n, the curve x1 = c1 , . . . , xn = cn , y 1 = cn+1 , . . . , y n = c2n is an integral curve of ξ in U . Then for any x ∈ U we have γx ∩ U = x0 ∈ U : xi (x0 ) = xi (x), y i (x0 ) = y i (x) for i ∈ 1, n where γx is the orbit passing through x. It results that π(U ) defines a local chart at the point π(x) ∈ Mξ , its local coordinates being x1 , . . . , xn , y 1 , . . . , y n . The compatibility of these local charts follows using the fact that ξ is complete. Example.

The orbit space of the Bianchi-Cartan-Vranceanu space M3λµ is M2µ = (x, y) ∈ R2 : 1 + µ(x2 + y 2 ) ≥ 0

−2 2 with the metric g 2 = 1 + µ(x2 + y 2 ) dx + dy 2 . The canonical projection π : M3λµ −→ M2µ , π(x, y, z) = (x, y) is a Riemann submersion with totally geodesic fibers.

2.6. ORBIT SPACE OF A CONTACT MANIFOLD

85

Theorem 2.6.3. Let M be a compact regular contact manifold. a) There exists a transformation η 0 = f η with f ∈ F(M ) nowhere zero and such that the diffeomorphisms group of M generated by the Reeb vector field ξ 0 of the contact form η 0 is a compact 1-dimensional Lie group, denoted by S1 . b) M is a principal bundle over Mξ with the structure group S1 . ˜ satisfies the c) Mξ is a symplectic manifold whose fundamental 2-form Ω condition ˜ dη = π ∗ Ω

(2.37)

Proof. a) If h is an arbitrary Riemannian metric on Mξ then g ∗ = π ∗ h + η ⊗ η is a Riemannian metric on M , which is generally not a metric compatible with the associated almost contact structure. From Proposition 1.3.6, a) it follows that η is invariant by the action of the Reeb group G. But π ∗ is invariant by this group and then the metric g ∗ is invariant by G and therefore ξ is a Killing vector field with respect to the metric g ∗ . Then its orbits are geodesics for the metric g ∗ . Let x ∈ M and denote by γx the orbit of ξ passing through x and let γ be an orbit sufficiently near from γx . Since M is regular, there exists an unique geodesic (always for the metric g ∗ ) starting from x and realizing the minimum of the distance from x to γ and denote by x0 their intersection point. But ξ is a Killing field for the metric g ∗ , hence G is a local isometry group and for any Φt ∈ G the image of the arc xx0 is orthogonal to γx and to γ. Now, consider the function µξ : M −→ R, defined by µξ (x) = inf {t : t > 0, Φt (x) = x}

(2.38)

We observe that 0 ≤ µξ (x) < ∞ and that µξ (x0 ) = µξ (x). Hence the function µξ is constant on M and denote by µ0 its value. If µ0 6= 0 then we put η 0 = µ10 η, ξ 0 = µ0 ξ and remark that Mξ0 = Mξ , µξ0 ≡ 1 and Φ0t = Φ0t+1 , hence the Reeb group of ξ 0 is generated by the transformations {Φt }t∈[0,1) , therefore it is a compact Lie group of dimension 1. If µ0 = 0 then, taking into account the manifold M is compact, it results that G is a compact Lie group of dimension 1 (see for instance [KN69], vol. II, pg. 236–244). b) Denote by S1 the Lie group founded in a) and let {(Uα , φα )}α∈I be an atlas on M with the domains Uα like in the definition of a regular manifold and π(Uα ) be the domains of the corresponding charts in the orbit space Mξ (see the proof of Proposition 2.6.2). For α ∈ I we define the map ψα : π(Uα ) × S1 −→ M,

ψα (˜ x, t) = Φα (sα (˜ x)), x ˜ ∈ π(Uα ), t ∈ S1

where sα : π(Uα ) −→ M associates to the point x ˜ of local coordinates x1 , . . . , xn , y 1 , . . . , y n , the point x of coordinates x1 , . . . , xn , y 1 , . . . , y n , z 2n+1 = c2n+1 = constant and t is identified with the transformation Φt . We can easily verify that Im ψα = π −1 (π(Uα )) and then the inverse diffeomorphisms ψα−1 : π −1 (π(Uα )) −→ π(Uα ) × S1 are well-defined. Hence (Uα , ψα ) is a fibered chart of (M, Mξ , S1 ). Moreover, if x ˜ ∈ Uα ∩ Uβ and t ∈ S1 then there is t0 such that

86


ψα (˜ x, t) = ψβ (˜ x, t0 ). It follows t0 = t + fαβ , where fαβ is a function defined on π(Uα ) ∩ π(Uβ ) and this proves that M is the total space of a principal bundle over Mξ , having S1 as structure group. c) The Lie algebra s1 of the group S1 is 1-dimensional, hence it can be identified (with respect to the vector space structure) with R and then we take d for a basis. Now, by putting η˜(X) = η(X)A for any X ∈ X (M ) we A = dt obtain the 1-form η˜ on M with values in s1 , invariant by the group S1 . We prove that η˜ is a connection form in the principal bundle (M, Mξ , S1 ), defined in b). For this purpose we verify the following conditions (see Section 2.4): 1) η˜(A∗ ) = A, where A∗ is the fundamental vector field corresponding to A; 2) (Rt )∗ η˜ = (ad (t−1 ))˜ η Since A∗ = ξ we have η˜(A∗ ) = η(A∗ )A = η(ξ)A = A, hence the condition 1) is satisfied. In our case we have Rt = Φt and since η˜ is invariant under the action of the group S1 , we have (Rt )∗ η˜ = η˜ and taking into account S1 is commutative, we obtain ad (t−1 ) = 1, therefore the condition 2) is also fulfilled. Now, let Ω∗ be the curvature 2-form of the connection defined by η˜. If we identify η˜ with η then for any x ∈ M and X, Y ∈ Tx M we have (see the structure equation (2.26)) 1 dη(X, Y ) = − [˜ η (X), η˜(Y )] + Ω∗ (X, Y ) 2 But S1 is commutative and then [˜ η (X), η˜(Y )] = 0, so that dη = Ω∗ . On the other hand, by Proposition 1.3.7 we have Lξ dη = 0, hence dη is invariant by the action of the group S1 and since dη(X, ξ) = 0 for any X ∈ X (M ), it follows that ˜ ∈ F 2 (Mξ ) such that Ω∗ = π ∗ Ω ˜ and then we obtain the equality there exists Ω (2.37). This last assertion can be easily obtained from Theorem 4.4.1, which will be proved in Section 4.4. ˜ = π ∗ dΩ ˜ By differentiating the equality (2.37) we obtain 0 = d2 η = dπ ∗ Ω ˜ is closed. Moreover, we and then from the same Theorem 4.4.1 it results that Ω ˜ n = (dη)n 6= 0, hence Ω ˜ n 6= 0. It follows that Ω ˜ is closed ˜ n ) = (π ∗ Ω) have π ∗ (Ω and non degenerate, therefore Mξ is a symplectic manifold. The group S1 from the above Theorem is called the circle group, its action on the manifold M is the circle action and the principal bundle (M, Mξ , S1 ) is said the Boothby-Wang fibration of the compact regular contact manifold M . Under the hypotheses of Theorem 2.6.3, W. H. Boothby and H. C. Wang, [BW58] also proved that the fundamental 2-form Ω determines an integral cocycle over Mξ (see the proof presented by S. Sasaki, [Sa68], part 1, pg. 14.6–14.8). Example. The sphere S 2n+1 with the Sasakian structure given in Example 2, Section 1.5 is compact and regular and its Boothby-Wang fibration coincides with the Hopf fibration (see [Hu66], pg. 54).

2.6. ORBIT SPACE OF A CONTACT MANIFOLD

87

Theorem 2.6.4. Let M be a compact regular almost contact Riemannian manifold. Then the orbit space Mξ has a natural structure of almost Hermitian manifold and the projection π is a Riemannian submersion. Moreover: a) If M is a contact Riemannian manifold then Mξ is an almost K¨ ahler manifold. b) If M is Sasakian then Mξ is K¨ ahlerian. ˜ x˜ ∈ Tx˜ Mξ there exists an Proof. Let x ∈ M and x ˜ = π(x). For any X H ˜ x˜ and by setunique horizontal vector Xx ∈ Tx M such that π∗,x XxH = X ˜ x˜ = π∗,x F XxH we obtanin the endomorphism ting J˜x˜ : Tx˜ Mξ −→ Tx˜ Mξ , J˜x˜ X J˜ : X (Mξ ) −→ X (Mξ ) and it is an almost complex structure on Mξ . More˜ Y˜ ) = g(F X H , F Y H ) is a Hermitian metric over, g˜ given by the equality g˜(X, ˜ on Mξ . Thus (J, g˜) is an almost Hermitian structure and because g˜ verifies the condition (2.18), we deduce that π is a Riemannian submersion. ˜ of the almost Simple computation proves that the fundamental 2-form Ω Hermitian manifold Mξ is given by ˜ x˜ (X ˜ x˜ , Y˜x˜ ) = Ωx (XxH , YxH ) Ω

(2.39)

and if M is a contact manifold then, as Ω is closed, it follows that Mξ is almost Kähler. ˜ x˜ , Y˜x˜ ∈ Tx˜ Mξ and denote by XxH , YxH the correspondb) Let us consider X ing horizontal vectors. M is a Sasaki manifold, hence N (1) (XxH , YxH ) = 0 or equivalent π∗,x N (1) (XxH , YxH ) = 0,

ηx N (1) (XxH , YxH ) = 0

(2.40)

Now, consider the vector fields X, Y defined on a neighborhood of the point x and such that at x these coincide with XxH and YxH , respectively. If in addition, we suppose that X, Y are invariant by left translations then π∗ [X, Y ] = [π∗ X, π∗ Y ] (see for instance [KN69], vol. I, pg. 42) and simple computation using the definition of J shows that at the point x we have π∗ N (1) (X, Y ) = ˜ Y˜ ) and then using (2.40) it results N ˜ = 0 and therefore from a) we NJ˜(X, J deduce that Mξ is K¨ ahlerian. In the case b), under the supposition that M is a Sasaki manifold, we can prove a more deep result, namely that the orbit space is a Hodge space, i. e. it ˜ is of type (1, 1) and is a compact K¨ ahler manifold whose fundamental 2-form Ω determines an integral cocycle [Hat63], [Mo64] (see also the affirmation which follows the proof of Theorem 2.6.3). A natural question is to find more general hypotheses in order to obtain a result similar to Theorem 2.6.4. From the general theory of foliations it is known that the leaf space has not a manifold structure, but it behaves as a manifold except that instead of considering open subsets of the model Rm , quotients of Rm by finite groups of diffeomorphisms are used (see for instance [Mol88], pg. 87–95). First, we define such a structure. For this purpose let O be an open

88


subset of Rm and denote by Γ a finite group of diffeomorphisms of O. We also denote by OΓ the space of orbits of Γ on O, equipped with the quotient topology and by p : O −→ O/Γ the natural projection. If (O0 , O0 /Γ0 ) is another pair as above, then a continuous map µ : O/Γ −→ O0 /Γ0 which lifts in a neighborhood of each point of O/Γ to a differentiable map from O to O0 , is an orbimorphism. Now, let V be a Hausdorff space and consider the family AV = {(Uα , φα )}α∈I where {Uα }α∈I is an open covering of V . If: a) for each α ∈ I the map φα : Uα −→ Oα /Γα is a homeomorphism from Uα to the quotient of an open subset Oα ⊂ Rm by a finite group of diffeomorphisms Γα ; b) for all α, β ∈ I with Uα ∩ Uβ 6= ∅ the map φβ ◦ φ−1 α : φα (Uα ∩ Uβ ) −→ φβ (Uα ∩ Uβ ) is an orbimorphism then AV is called a Satake atlas on V . The space V equipped with the Satake atlas AV is a m-dimensional orbifold (or Satake manifold or V-manifold ). The notion of orbifold was defined by I. Satake, [Sat57]. When we replace Rm by Cm and if the elements of Γα are holomorphic maps then V is a complex orbifold. The almost complex structure and the Hermitian metric on an orbifold can be defined alike for manifolds and we say that V is a K¨ ahler orbifold if the associate fundamental 2-form ΩV is closed and the almost complex structure is integrable. The compact Kähler orbifold V with the property that ΩV defines a cohomology class [ΩV ] ∈ H 2 (V, Z) is a Hodge orbifold. Now, replacing the regularity condition in the case b) of Theorem 2.6.4 by a weaker one, C. P. Boyer and K. Galicki prove Theorem 2.6.5. [BG00] Let M be a compact quasi-regular Sasaki manifold. ˜ defines The orbit space of M is a Hodge orbifold whose fundamental 2-form Ω ˜ ∈ H 2 (Mξ , Z). The fibers of the Riemannian submersion π an integral class [Ω] are totally geodesic submanifolds of M , diffeomorphic to the circle S 1 . The problem of characterization of a regular almost contact manifold by the space of its orbits was solved by D. E. Blair and L. Vanhecke, at least in a particular case. Theorem 2.6.6. [BV87] A complete regular almost contact manifold M is one of the products Mξ × R and Mξ × S 1 if and only if its contact form is closed. For a proof of this theorem see [BV87]. Assume that the 3-dimensional regular almost contact manifold M coincides with one of the products Mξ × R, Mξ × S 1 and if in Mξ we consider the curve γ parametrized by its arc length. Then by inverse image we obtain a surface S = π −1 (γ) ⊂ M . If M = Mξ × R (or if M = Mξ × S 1 ) then S is called a Hopf cylinder (or a Boothby-Wang cylinder ) over γ. A direct computation proves that if γ has the curvature equal to k then S is a flat surface with mean curvature equal to k/2. In the particular case when γ is a closed curve in Mξ , the surface S is called a Hopf torus over γ.

Chapter 3

Curvature problems in contact manifolds 3.1

Curvature tensor of a contact manifold

Curvature tensor of contact Riemannian manifolds Let M be a contact Riemannian manifold with the contact form η and denote by (F, ξ, η, g) the associated almost contact Riemannian structure (see Theorem 1.1.6). Let ∇ be the Levi-Civita connection of the metric g, denote by R(X, Y ) = [∇X , ∇Y ]−∇[X,Y ] the curvature 2-form of ∇ and by R the RiemannChristoffel curvature tensor, defined by R(X, Y, Z, U ) = g(R(Z, U )Y, X). We also recall the vector field N (3) defined by the formula N (3) X = 21 (Lξ F ) X (see Section 1.2). Proposition 3.1.1. On a contact Riemannian manifold of dimension 2n + 1 the following equalities hold: a) R(ξ, X, Y, Z) − R(ξ, X, F Y, F Z) + R(ξ, F X, Y, F Z) + R(ξ, F X, F Y, Z) = = 2η(Y )g(X + N (3) X, Z) − 2η(Z)g(X + N (3) X, Y ) − 2 (∇N (3) X Ω) (Y, Z); 2 b) R(X, ξ)ξ = X − η(X)ξ − N (3) X + F ∇ξ N (3) X; 2 c) R(ξ, X)ξ − F R(ξ, F X)ξ = 2 N (3) + F 2 X: 2 d) Ric(ξ, ξ) = 2n − trace N (3) . Proof. a) By applying the formula (1.18) and taking into account Proposition 1.3.6 in the case when the manifold has a contact Riemannian structure, by a computation rather laborious than complicated we obtain 2g ((∇X F ) Y, Z) = g N (1) (Y, Z), F X + 2g(X, Y )η(Z) − 2g(X, Z)η(Y ) 89

90 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS and then (∇X Ω)(F Y, Z) − (∇X Ω) (Y, F Z) = = −η(Y )g X + N (3) X, F Z − η(Z)g X + N (3) X, F Y Now, from the formula (3.1) and from Proposition 1.3.8 we deduce R(Y, Z)ξ = (∇Z F ) Y − (∇Y F ) Z + ∇Z F N (3) Y − ∇Y F N (3) Z

(3.1)

(3.2)

and then from the symmetry properties of R and from the definition of the fundamental 2-form Ω we get R(ξ, X, Y, Z) = (∇X Ω) (Y, Z)+ + g X, ∇Y (F N (3) )Z − g X, ∇Z (F N (3) )Y

(3.3)

On the other hand we have dΩ(X, Y, Z) =

X

(∇X Ω) (Y, Z) = 0

cycl

and by using (3.1) in the equality dΩ(X, Y, Z) + dΩ(F X, F Y, Z) + dΩ(F X, Y, F Z) − dΩ(X, F Y, F Z) = 0, after reducing the similar terms we deduce (∇F X Ω) (Z, F Y ) + (∇X Ω) (Z, Y ) = = 2η(Z)g(X, Y ) − η(Y )g X + N (3) X, Z − η(X)η(Y )η(Z)

(3.4)

Now, taking into account (3.3) in the left hand-side of a) and then by using the formula (3.4), by a straightforward computation we obtain a). b) From (3.2) we get R(X, ξ)ξ = (∇ξ F ) X − (∇X F ) ξ + ∇ξ F N (3) X − ∇X F N (3) ξ (3.5) But on a contact Riemannian manifold we have ∇ξ F = 0 (see Proposition 1.3.6, c)) and N (3) ξ = 0 (see Proposition 1.3.8, d)), so that by applying Proposition 1.3.8, b), c) we obtain (∇X F ) ξ = −X + η(X)ξ − N (3) X,

2 ∇X F N (3) ξ = N (3) X + N (3) X

Now, taking into account these equalities in (3.5) we obtain b). c) results from b) by using Proposition 1.3.8, c). d) is also a corollary of b).

3.1. CURVATURE TENSOR OF A CONTACT MANIFOLD

91

Before studying the flat contact structures we shall give local characterizations for the contact Riemannian manifolds whose curvature tensor vanishes on some vector fields. Such kind of problems were solved by D. E. Blair, [Bl77], E. Boeckx, [Boe00]. Theorem 3.1.2. [Bl77] If the curvature tensor of a 2n + 1-dimensional contact Riemannian manifold M satisfies the condition R(X, Y )ξ = 0 then M is flat for n = 1 and it is locally isometric to Rn+1 × S n (4) for n > 1. Proof. Following [Bl02], pg. 101–102 we present the details of this proof because it is illustrative for a family of similar results. From Proposition 3.1.1, c) it 2 results N (3) + F 2 = 0, hence the eigenvalues of N (3) are +1 and −1, each of them √ with the√same multiplicity order equal to n (since the eigenvalues of F are + +1 and − −1, with the same multiplicity orders). We denote by V−1 , V+1 the eigenspaces corresponding to these eigenvalues of N (3) and we prove that the distributions generated by V−1 ⊕ hξi and V+1 are integrable. Indeed, from Proposition 1.3.8, b) it follows ∇X ξ = ∇Y ξ = 0 for X, Y ∈ V−1 and then 0 = R(X, Y )ξ = −∇[X,Y ] ξ = F [X, Y ] + F N (3) [X, Y ]

(3.6)

But X ∈ V−1 and then from Proposition 1.3.8, c) we deduce that F X ∈ V+1 and therefore we have 1 dη(X, Y ) = − η[X, Y ] = Ω(X, Y ) = 0 2

(3.7)

From (3.6) and (3.7) it results [X, Y ] ∈ V−1 , hence the distribution generated by V−1 is integrable. On the other hand we have R(X, ξ)ξ = 0 for X ∈ V−1 and then ∇[X,ξ] ξ = 0, so that from the same Proposition 3.1.1, a) it follows 2F [X, ξ] + F N (3) [X, ξ] = 0,

ηN (3) [X, ξ] = 0,

η[X, ξ] = 0

(3.8)

Now, by applying F to the first equality (3.8) and taking into account the second and the third, we deduce [X, ξ] ∈ V−1 , hence the distribution generated by V−1 ⊕ hξi is integrable. Let X, Y ∈ V+1 . From the hypothesis and from Proposition 1.3.8, b) it follows 2 [∇X (F Y ) − ∇Y (F X)] = F N (3) [X, Y ] − F [X, Y ]

(3.9)

But F V−1 = V+1 and by using the symmetry properties of the RiemannChristoffel curvature tensor, from Proposition 1.3.8, b) it results (∇X Ω) (Y, Z) = 0 for X, Y, Z ∈ V+1 and then the equality a) from the same Proposition becomes g Z, ∇X F N (3) Y − g Z, ∇Y F N (3) X = 0 (3.10) From (3.10), (3.9) and from 2∇X (F N (3) )Y = 2F N (3) ∇X Y − ∇X (F Y )

92 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS we obtain

g Z, F N (3) [X, Y ] + F [X, Y ] = 0

and taking into account Proposition 1.3.8, c) it follows that N (3) [X, Y ] + [X, Y ] is orthogonal to V−1 . From Proposition 1.3.8, b), d) we obtain η[X, Y ] = 0, hence N (3) [X, Y ] + [X, Y ] also is orthogonal to ξ. Therefore the distribution generated by V+1 is integrable. Now we prove that the integral submanifolds of the distributions generated by V−1 ⊕ hξi and V+1 are totally geodesic. For this purpose we choose the ∂ ∂ ∂ local coordinates (u0 , u1 , . . . , u2n , u2n+1 ) in M such that ∂u 0 , ∂u1 , . . . , ∂un ∈ P n j ∂ V−2 ⊕ hξi and consider the vector fields Xi = ∂u∂i∗ + j=0 fi ∂u j , i ∈ 1, n, j where the functions f are so that X ∈ V . A simple computation shows that i +1 i ∂ 0, n and then from the hypothesis it follows , X ∈ V ⊕ hξi for any k ∈ i −1 ∂uk 0 = ∇[ ∂k ,Xi ] ξ = −2∇ ∂k (F Xi ). We deduce ∂u

∂u

∇F Xj (F Xi ) = 0

(3.11)

that is the second fundamental form h vanishes for any X, Y ∈ V−1 ⊕ hξi. Thus we have proved that any integral submanifold M− of the distribution V−1 ⊕ hξi is totally geodesic. For X ∈ V−1 , Y ∈ V+1 we have 0 = R(X, Y )ξ = −2∇X (F Y ) + F [X, Y ] + F N (3) [X, Y ] = = −2 (∇X F ) Y − 2F (∇Y X) − F [X, Y ] + F N (3) [X, Y ] Now, if we compute the scalar product of this equality with Z ∈ V+1 then we obtain g(F (∇Y X), Z) = 0, that is ∇Y X is orthogonal to V+1 and then from the Weingarten formula it follows h(Y, Z) = 0, hence any integral submanifold M+ of the distribution generated by V+1 is totally geodesic. We have proved above that the distributions generated by V−1 ⊕ hξi and V+1 are integrable and that theirs maximal integral submanifolds M− and M+ are totally geodesic. But these distributions are parallel relative to the Levi-Civita connection ∇ of the metric g on the manifold M and we can apply the local version of de Rham decomposition Theorem1 . Hence, at least locally, M is the Riemannian product M− × M+ and M− is locally isometric to the Euclidean space E n+1 because on M− the connection ∇ is flat (see the formula (3.11)). A straightforward computation shows that R(X, Y, F U, F Z) − R(X, Y, U, Z) = 4 [g(X, Z)g(Y, U ) − g(Y, Z)g(X, U )] 1 De Rham decomposition theorem (local version): Let M be a Riemannian manP (i) ifold, Tx M = kl=0 Tx be a canonical decomposition of Tx M and let T (i) be the integrable (i)

distribution on M obtained by parallel transport of Tx for any i ∈ 0, k. Consider y ∈ M and for i = 0, 1, . . . , k we denote by Mi the maximal integral submanifold of T (i) passing through y. Then the point y has an open neighborhood V = V0 × V1 × . . . × Vk with the property that any Vi is an open neighborhood of y in Mi and the restriction to V of the metric g is the product metric. For the proof of this classical theorem an for its global version see [KN69], vol. I, pg. 185–193.


93

for X, Y, Z, U ∈ V+1 . But for such vector fields we have F Z, F U ∈ V−1 and then R(X, Y, F U, F Z) = 0

(3.12)

R(X, Y, U, Z) = 4 [g(Y, Z)g(X, U ) − g(X, Z)g(Y, U )]

(3.13)

It follows

hence M+ is a space form of sectional curvature equal to 4 and then it is locally isometric to the sphere of radius 4. In the case n = 1 we have K(X, ξ) = 0 for X ∈ V−1 or X ∈ V+1 and K(X, Y ) can be determined only when X ∈ V−1 , X ⊥ ξ, Y ∈ V+1 and in this case we deduce K(X, Y ) = 0 because of the equality (3.12). A generalization of Theorem 3.1.2 was obtained by E. Boeckx, namely Proposition 3.1.3. [Boe00] Let M be a non-Sasaki contact manifold such that R(X, Y )ξ = k[η(Y )X − η(X)Y ] + µ[η(Y )N (3) X − η(X)N (3) Y ]

(3.14)

for some real constants k, µ. M is locally isometric (up to a D-homothetic deformation) to the sphere tangent bundle of a space form if and only if µ < √ 2(1 + 1 − k). Its proof is similar with the one of Theorem 3.1.2. It seems that in Proposition 3.1.3 we must impose the condition k ≤ 1, but this is not necessary according to Proposition 3.2.3, proved in the next Section. Concerning the existence of flat contact structures we have the following surprising result Theorem 3.1.4. [Bl76b] There are no flat contact Riemannian structures on manifolds of dimension 2n + 1 ≥ 5. Proof. We remark that, excepting the case n = 1, the argument used in the proof of Theorem 3.1.2 for n ≥ 2 is still valid. Hence from (3.13) we deduce g(X, Z)g(Y, U ) − g(Y, Z)g(X, U ) = 0 for X, Y, Z, U ∈ V+1 and setting X = Z = Xi , Y = U = Xj , it follows that the vectors Xi , Xj belonging to a F -basis are collinear, but this is impossible. However, in the 3-dimensional case there are flat contact manifolds. Indeed we consider the torus T 3 with the contact Riemannian structure defined by η=

1 (cos t dx + sin t dy) , 2

g : gij =

1 δij 4

where x, y, t are the Cartesian coordinates in R3 . This structure is flat. Moreover, the contact form η is invariant by the translations x0 = x + a, y 0 = y + b, t0 = t + 2kπ and by the motions (Rθ , t), where Rθ is the rotation of angle θ in the plane (x, y) and t is the translation x0 = x, y 0 = y, z 0 = z, t0 = t + 2π − θ. Such a flat contact Riemannian structure on the torus T 3 is fundamental for the following classification for 3-dimensional contact manifolds:

94 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Theorem 3.1.5. [Ru98] Let M be a compact contact Riemannian manifold of dimension 3. If M is flat then: a) there exist nontrivial parallel vector fields on M : b) M is isometric to the quotient of a flat 3-torus by a finite cyclic group of isometries of order 1, 2, 3, 4 or 6. In the more general case when M has constant sectional curvature, D. E. Blair and R. Sharma found the following result Theorem 3.1.6. [BS90a] A 3-dimensional contact Riemannian manifold M of constant sectional curvature K has either a) K = 1 and then it is Sasakian or b) K = 0 i. e. M is flat. Results similar to Theorem 3.1.2 were obtained by R. Sharma for another class of contact manifolds. Theorem 3.1.7. [Sh03] Let S be a contact hypersurface of the K¨ ahler manifold M . If its second fundamental form is a Codazzi tensor (i. e. (∇X h) Y = (∇Y h) X for any X, Y ∈ X (S)) then S is either a) Sasakian, totally umbilical and has constant mean curvature equal to 1 or b) locally isometric (up to a D-homothetic deformation) with the sphere tangent bundle of a real space form of sectional curvature 6= 1. In particular, if S is parallel (i. e. ∇h = 0) then the case b) becomes b’) S is locally isometric to Rn+1 × S n (4) for n > 1 and S is flat for n = 1. Proof. Using the same notations as in Example 1, Section 1.3 we have g ∇SX A~n ξ, Y = g ∇SY A~n ξ, X and using Proposition 1.3.8, b), the formulas (1.47) and the above relation for X = ξ, we deduce d(trace A~n ) = (ξ trace A~n )η, hence trace A~n = constant on S (see the proof of Proposition 1.3.18). From the Codazzi equation and because h is a Codazzi vector field we obtain RS (X, Y )ξ = RS (X, Y )J~n = JRM (X, Y )~n = 0 Then using (1.48), the first relation (1.47) and the Gauss equation it follows h i RS (X, Y )ξ = (trace A~n − 2n) η(Y )(X + N (3) X) − η(X)(Y + N (3) Y ) (3.15) Now, comparing this result with Proposition 3.1.1, b) and taking into account Proposition 1.3.8, d), e), we deduce 2 trace N (3) = 2n(1 − trace A~n + 2n) But N (3) is self-adjoint (see Proposition 1.3.8, a)) and then trace A~n − 2n ≤ 1. Moreover, the equality trace A~n − 2n = 1 holds if and only if N (3) = 0, that


95

is if S is Sasakian (see the formula (1.49) and Theorem 1.5.3). In this case by (1.48) we have A~n X = X, hence S is totally umbilical. If trace A~n − 2n < 1 then, taking into account (3.15), the hypothesis of Proposition 3.1.3 is fulfilled for k = µ = trace A~n − 2n < 1 and

N (3)

2

= (2n + 1 − trace A~n )(I − η ⊗ ξ)

(3.16)

If the second fundamental form h is parallel then by using (1.47), (1.48) and Proposition 1.3.8, b) we get 2 0 = ∇SX A~n ξ = N (3) F X − N (3) F X −(traceA~n −2n−1) F X + F N (3) X and then (trace A~n − 2n)N (3) F X = 0, hence either N (3) = 0 and thus S is Sasakian, or traceA~n = 2n. In this last case, from (3.15) it follows RS (X, Y )ξ = 2 0 and we can apply Theorem 3.1.2. Moreover, (3.16) yields N (3) = −F 2 and from (1.48) we deduce that the principal curvatures are 0 (with multiplicity n + 1) and 2 (with multiplicity n). We remark that from the Codazzi equation it follows that for M = R2n+2 the second fundamental form of the contact hypersurface S is a Codazzi tensor, hence Theorem 3.1.7 is a generalization of Theorem 1.3.19. Finally on an almost contact manifold we define two useful linear connections. Proposition 3.1.8. [AC92] Let M be a 2n + 1-dimensional manifold with the almost contact Riemannian structure (F, ξ, η, g) and denote by ∇ its Levi-Civita connection. The linear connection given by ∇∗X Y = ∇X Y + η(Y )X,

for X, Y ∈ X (M )

is semi-symmetric, i. e. its torsion is T ∗ (X, Y ) = η(Y )X − η(X)Y . The curvature tensor R∗ , the Ricci tensor Ric∗ and the scalar curvature ρ∗ of this connection are R∗ (X, Y )Z = R(X, Y )Z − α(Y, Z)X + α(X, Z)Y, Ric∗ (X, Y ) = Ric(X, Y ) − 2nα(X, Y ), ρ∗ = ρ − 2n trace α

(3.17)

where R, Ric, ρ are the corresponding tensor fields for ∇ and α(X, Y ) = (∇∗X η) Y = (∇X η) Y − η(X)η(Y ). Proposition 3.1.9. [SH76] [Ya70] Let M be a 2n+1-dimensional manifold with the almost contact Riemannian structure (F, ξ, η, g) and let ∇ be its Levi-Civita connection. The linear connection given by ∇∗∗ X Y = ∇X Y + η(Y )X − g(X, Y )ξ

96 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS is Riemannian, semi-symmetric and ∇∗∗ F = 0. The curvature tensor of ∇∗∗ is given by R∗∗ (X, Y )Z = R(X, Y )Z − α∗ (Y, Z)X+ + α∗ (X, Z)Y − g(Y, Z)β ∗ (X) + g(X, Z)β ∗ (Y ) 1 1 ∗ ∗∗ where α∗ (X, Y ) = (∇∗∗ X η) U − 2 g(X, Y ), β (X) = ∇X ξ − 2 X.

The proofs of these two last propositions are simple exercises. Curvature tensor of K-contact manifolds Proposition 3.1.10. Let M be a 2n + 1-dimensional K-contact manifold. The following assertions hold for any X, Y ∈ X (M ): a) R(X, ξ)ξ = X − η(X)ξ, R(X, Y )ξ = (∇Y F ) X − (∇X F ) Y ; b) the sectional curvature of any plane containing the Reeb vector field ξ is equal to 1; c) Ric(ξ, ξ) = 2n. Proof. b), c) and the first equality a) follow easily from Proposition 3.1.1, b), while the second equality a) can be obtained from (3.2). Now, we prove some converse assertions of c) and d) in Proposition 3.1.10 and so we obtain other characterizations of K-contact manifolds. Theorem 3.1.11. Let M be a contact Riemannian manifold of dimension 2n+ 1. M is K-contact if and only if Ric(ξ, ξ) = 2n. 2 Proof. If Ric(ξ, ξ) = 2n then from Proposition 3.1.1, d) it follows trace N (3) = 0. On the other hand we know that all eigenvalues of N (3) are real (see the 2 proof of Proposition 3.1.2) and then the eigenvalues of N (3) are ≥ 0, hence N (3) ≡ 0. The converse is the assertion c) from Proposition 3.1.10. Theorem 3.1.12. A contact Riemannian manifold is K-contact if and only if the sectional curvatures of all planes passing through ξ are equal to 1. Proof. If M is K-contact then multiplying b) from Proposition 3.1.1 by the unitary vector X orthogonal to ξ it follows K(X, ξ) = 1. The converse is exactly the assertion b) from Proposition 3.1.10. Theorem 3.1.13. Let M be a Riemannian manifold. M is K-contact if and only if there exists an unitary Killing vector field ξ such that the sectional curvatures of M are equal to 1 for all the planes containing ξ. Proof. Let g be the Riemannian metric on M and denote by ∇ its Levi-Civita connection. We define the 1-form η and the endomorphism F : X (M ) −→ X (M ) by η(X) = g(X, ξ);

F X = −∇X ξ

(3.18)

3.2. CURVATURE TENSOR OF A SASAKI MANIFOLD

97

Since ξ is an unitary Killing vector field, we have ∇ξ ξ = 0;

R(X, ξ)Y = ∇X ∇Y ξ − ∇∇X Y ξ

(3.19)

and then (F, ξ, η) is an almost contact structure whose compatible metric is g. Moreover we have 2dη(X, Y ) = g (∇X ξ, Y ) − g (∇Y ξ, X) = −2g (∇Y ξ, X) = 2g(X, F Y ) hence (F, ξ, η, g) is a contact Riemannian structure on M and we can use Theorem 3.1.12. Remark 3.1.14. In fact the condition ”M has sectional curvature K(X, ξ) = 1” is not essential in Theorem 3.1.13. We have only to require that the sectional curvature is a constant√c > 0. Indeed, in this case it is enough to replace g by cg, ξ by √1c ξ and η by cη.

3.2

Curvature tensor of a Sasaki manifold

Let M be a Sasaki manifold, let {Xi , Xi∗ , ξ} be a F -basis at the point x ∈ M and consider the tensor field P, defined by P(X, Y, Z, U ) = Ω(X, Z)g(Y, U ) − Ω(X, U )g(Y, Z)− − Ω(Y, Z)g(X, U ) + Ω(Y, U )g(X, Z) Then we have Proposition 3.2.1. Let M be a Sasaki manifold of dimension 2n + 1. The following assertions hold: a) R(X, Y )ξ = η(Y )X − η(X)Y ; b) R(X, ξ)Y = η(Y )X − g(X, Y )ξ c) R(X, Y, Z, F U ) + R(X, Y, F Z, U ) = P(X, Y, Z, U ); d) R(F X, F Y, F Z, F U ) = R(X, Y, Z, U ); R(X, F X, Y, F Y ) = R(X, Y, X, Y ) + R(X, F Y, X, F Y ) + 2P(X, Y, X, F Y ) for X, Y, Z, U orthogonal on ξ; e) Ric(F X, F Y ) = Ric(X, Y ) − 2nη(X)η(Y ); Ric(X, ξ) = 2nη(X); f ) the sectional curvatures at x are given by K(Xi , Xj ) = K(Xi∗ , Xj ∗ ), K(Xi , Xj ∗ ) = K(Xi∗ , Xj ), K(Xi , ξ) = K(Xi∗ , ξ) = 1. Proof. a) follows from Proposition 3.1.10 and Theorems 1.5.1, 1.5.3. b) By using a) we have g(R(X, ξ)Y, Z) = R(Z, Y, X, ξ) = R(X, ξ, Z, Y ) = = g(R(Z, Y )ξ, X) = η(Y )g(Z, X) − η(Z)g(Y, X) On the other hand, from the symmetry properties of the Riemann-Christoffel curvature tensor we have g(R(X, ξ)Y, Z) = g(R(Z, Y )ξ, X) and b) follows easily.

98 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS c) We have R(X, Y, Z, F U ) + (X, Y, F Z, U ) = g(R(X, Y )F U, Z)g (R(X, Y )F Z, U ) and then c) follows by a straightforward computation using the formula (1.56). d) results from c) and 1.5. The first equality e) is an immediate consequence of the first formula d), while the second follows from a) and from the definition of Ricci tensor. f) The two first equalities result from c) evaluated when X, Y, Z, U are elements of a F -basis and taking into account dη = Ω. The last two equalities are consequences of a). From Proposition 3.2.1, c) we deduce the following useful formula R(X, Y )F Z − F R(X, Y )Z = g(Y, F Z)X+ + g(X, Z)F Y − g(X, F Z)Y − g(Y, Z)F X

(3.20)

On the other hand, the assertion a) from Proposition 3.2.1 has a converse, namely we can state the following: Theorem 3.2.2. A contact Riemannian manifold M is Sasakian if and only if R(X, Y )ξ = η(Y )X − η(X)Y

(3.21)

for any X, Y ∈ X (M ). Proof. If (3.21) is satisfied then Ric(ξ, ξ) = 2n and from Proposition 3.1.1 we deduce (N (3) )2 = 0. Now, since N (3) is a symmetric operator (see Proposition 1.3.8, a)) we obtain N (3) = 0. Then from 3.3 it follows g(R(Z, Y )ξ, X) = (∇X Ω) (Y, Z) and taking into account (3.21), the definition of the fundamental 2-form Ω and the fact that the metric g is Riemannian, we obtain the equality (1.56) and then, by Theorem 1.5.3 M is Sasakian. D. E. Blair, T. Koufogiorgos and B. J. Papantoniou have been studied the contact Riemannian manifolds whose curvature vector satisfies a more general condition than (3.21), obtaining the following result: Proposition 3.2.3. [BKP95] Let M be a contact Riemannian manifold whose curvature vector satisfies the condition (3.14). Then k ≤ 1. Moreover, for k = 1 the manifold M is Sasakian and for k < 1 the scalars k, µ determine completely the sectional curvature of the manifold. Proof. Taking into account Proposition 1.3.8, c) and since on a contact Riemannian manifold we have N (3) ξ = 0, the formula c) from Proposition 3.1.1 yields (1 − k)[X − η(X)ξ] − (N (3) )2 X = 0

(3.22)


99

But N (3) is a symmetric operator (see Proposition 1.3.8, a)), hence all its eigenvalues are real. Or, if λ is an eigenvalue and X is a corresponding eigenvector then (3.22) becomes (1 − k)[X − η(X)ξ] − λ2 X = 0

(3.23)

and we distinguish the following two cases: case A. If all eigenvalues of N (3) are equal to 0 (that is N (3) ≡ 0) then from (3.22) we deduce k = 1 and the equality from our statement reduces to (3.21), which characterizes Sasaki manifolds. case B. If there exists λ 6= 0 then any corresponding eigenvector is orthogonal to ξ and the equality (3.23) becomes (1 − k) − λ2 =√0, which proves that k < 1 and the unique nonzero eigenvalues of N (3) are ± 1 − k, each √ of them with the multiplicity equal to n. For instance, if we take λ = 1 − k and if X is an eigenvector corresponding to λ then we have K(X, ξ) = k + λµ. But F X is an eigenvector corresponding to the eigenvalue −λ and then K(X, ξ) = k − λµ. Moreover, we have K(X, F X) = −k − µ and we will see in Section 3.4, Proposition 3.4.1 that these 3 values determine completely the sectional curvature of the manifold. This result has an important particular case, namely Theorem 3.2.4. [Ol79] Let M be a contact Riemannian manifold of dimension 2n + 1 ≥ 5. If M has constant sectional curvature K then K = 1 and M is Sasakian. Proof. We can apply Proposition 3.2.3 for k = K and µ = 0. Then for K = 1 the manifold M is Sasakian and for K < 1 from the argument used in the case B we deduce K = 0 and we finish the proof taking into account Theorem 3.1.4. Remark that in the 3-dimensional case we have a weaker result stated in Theorem 3.1.6. Proposition 3.2.5. [Ta77] Let M be a Sasaki manifold. a) For all X, Y, Z ∈ X (M ) and U, V, W ∈ D the following equalities hold [(∇Z R) (X, Y )] ξ = F R(X, Y )Z + g(X, Z)F Y − g(Y, Z)F X, [(∇Z R) (X, ξ)] Y = R(X, F Z)Y + g(F Y, Z)X + g(X, Y )F Z,

(3.24)

[(∇ξ R) (U, V )] W = 0

(3.25)

F 2 [(∇Z R) (X, Y )ξ] = 0

(3.26)

b) If

for any X, Y, Z ∈ D then M has constant sectional curvature equal to 1.

100 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Proof. (3.24) follows by a straightforward computation using Theorem 3.2.2 and the formulas (1.54), (3.20) for Z = ξ. (3.25) results by computing ∇ξ R. b) The right hand-side of the first equality (3.24) belongs to D, hence from (3.26) it follows R(X, Y )Z = −g(X, Z)Y +g(Y, Z)X for X, Y, Z ∈ D. Then, taking into account Proposition 3.2.1, f) we deduce that M has constant sectional curvature equal to 1. The linear connection ∇ on the manifold M is called locally symmetric at the point x if there exists an involutive affine transformation of an open neighborhood of x such that x is an isolated fixed point. If ∇ is locally symmetric at each point of the manifold M then ∇ is said a locally symmetric connection. Let M be a manifold with the connection ∇ and let x ∈ M . A diffeomorphism sx : U −→ U of a neighborhood U of x with the property sx (exp X) = exp(−X)

for X ∈ Tx M

is called a symmetry at x. Of course, two symmetries at x defined on U and V , respectively, coincide on U ∩ V , so we say the symmetry at x. From the definition we deduce s2x = 1Tx M , hence the symmetry at x is involutive. M is a locally symmetric manifold if for any x ∈ M the symmetry sx is an affine transformation, that is if for any γ, the symmetry sx maps each parallel vector field along the curve γ in U into a parallel vector field along the image curve sx ◦ γ. We have the following classical characterization of such manifolds. Proposition 3.2.6. The manifold M equipped with the linear connection ∇ is locally symmetric if and only if ∇R = 0 and T = 0. For the proof see for instance [KN69], vol. II, pg. 222. The Riemannian manifold M is locally symmetric if the Levi-Civita connection is locally symmetric. If for each x ∈ M the symmetry sx can be extended to an affine transformation of M then M is a (globally) symmetric manifold. A notion generalizing in a certain sense the notion of globally symmetric space is the following: The connected Riemannian manifold M is a weakly symmetric space if any two distinct points can be interchanged by an isometry. Such a space has the following characterization Theorem 3.2.7. [BeTV97] The connected simply connected Riemannian manifold M is a weakly symmetric space if and only if the following conditions are fulfilled: a) there exists a complete Riemannian connection ∇ with parallel curvature and torsion tensors R and T (i. e. R = ∇, T = 0, where denote of ∇); b) there exists a point x ∈ M such that for any unitary vector X ∈ Tx M there exists an isometry IX : Tx M −→ Tx M preserving Rx , T x and satisfying the condition IX (X) = −X.


101

Proof. if M is a weakly symmetric space then for any x ∈ M and for any unitary vector X ∈ T P xM there exists an isometry µX : M −→ M so that µX∗,x X = −X. Then IX = µX∗,x is an isometry of Tx M . On the other hand, we can choose a reductive decomposition of the Lie algebra associated to the isometries group of M . The curvature and the torsion of the induced canonical connection ∇ are parallel and since ∇ is invariant, it follows that IX preserves Rx and T x . Conversely, let ∇ be a complete Riemannian connection satisfying a), b) and let X ∈ Tx M be an unitary vector. Then there exists an unique affine transformation µ of M such that µ(x) = x and µ∗,x = IX (see for instance [KN69], vol. I, Corollary 7.9, pg. 265) and since ∇ is Riemannian, µ is an isometry. Then for any maximal geodesic γ through x there exists an isometry I of M such that I|γ is an involution and I(x) = x. Denote by ∇ the Levi-Civita connection of the Riemannian metric of M . Since ∇ − ∇ is a homogeneous structure on the complete, connected and simply connected manifold M , it follows that M is a Riemannian homogeneous space and then it is weakly symmetric. Theorem 3.2.8. Any locally symmetric K-contact manifold is Sasakian and has constant sectional curvature equal to 1. Proof. From [(∇X R) (ξ, Y )] ξ = 0 and taking into account (1.54) and Proposition 3.1.1, a) we obtain R(F Y, X)ξ + R(ξ, X)(F Y ) = g(X, F Y )ξ − η(X)F Y Now replacing Y by F Y and then, using the linearity properties of the curvature tensor we get R(ξ, X)Y − R(X, Y )ξ = g(X, Y )ξ − 2η(Y )X + η(X)Y

(3.27)

By computing again the left hand side of the equality [(∇X R) (Y, Z)] ξ = 0, taking into account the formula (3.27) and then by proceeding as above we deduce R(X, Y )Z + R(X, Z)Y = 2g(Y, Z)X − g(X, Y )Z − g(X, Z)Y

(3.28)

From the scalar product of the equality 3.8.9 with U we get 2

2

g (R(Y, X)X, Y ) = kXk kY k − g 2 (X, Y ) hence the sectional curvature of the manifold M is equal to 1. But in this case the curvature tensor has the expression R(X, Y )Z = g(Y, Z)X−g(X, Z)Y . Now, by putting Z = ξ and by using Theorem 3.2.2 it follows that M is Sasakian. For Sasaki manifolds a result similar to Theorem 3.1.13 is valid.

102 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Theorem 3.2.9. [HOT63] If the odd dimensional Riemannian manifold M has an unitary Killing vector field ξ with the property R(X, Y )ξ = g(ξ, Y )X − g(ξ, X)Y then M is Sasakian. Proof. As in the proof of Theorem 3.1.13 we define the tensor fields η and F by the formulas (3.18) and then we obtain the almost contact Riemannian structure (F, ξ, η, g). From the second equality (3.19) it results R(X, ξ)Y = − (∇X F ) Y and then from the hypothesis it follows g ((∇X F ) Y, Z) = g(R(ξ, X)Y, Z) = g(R(Y, Z)ξ, X) = g(η(Z)Y − η(Y )Z, X) = g(X, Y )η(Z) − η(Y )g(X, Z) hence by Theorem 1.5.3 the manifold M is Sasakian. The converse of Theorem 3.2.9 is also true by Theorem 3.2.2. Theorem 3.2.10. [HS81] Let M be a complete Sasaki manifold. If there exists δ ∈ R such that Ric ≥ δ > −2 then M is compact and its fundamental group is finite. Proof. If δ > 0 then the assertion holds by the classical Myers Theorem2 . Hence we can assume that 0 ≥ δ > −2 and remark that there exists α with the properties δ+2 ≤ 1; 2nα2 + α − δ − 2 ≥ 0 (3.29) 2 where dim M = 2n + 1. For such a real number α, let g˜ be the metric of the ˜ η˜, g˜) with constant α of the Sasaki structure D-homothetic deformation (F˜ , ξ, (F, ξ, η, g) (see the formulas (1.68)). But M is complete with respect to the metric g and then M is also complete for the metric g˜. Moreover, the Ricci g of the metrics g and g˜ respectively, satisfy the following equality tensors Ric, Ric g = Ric − 2(α − 1)g + 2(α − 1)(nα + n + 1)η ⊗ η Ric (3.30) 0 0 [or more generally if Ric(X, X) ≥ (dim M −1)k0 > 0 for any unitary vector field X ∈ X (M )] then M is compact, its diameter is ≤ √πk and has 0 finite fundamental group. For a proof of this result see the cited work.


103

because δ + 2 > 2α. Now, we can apply Myers Theorem. D. E. Blair and R. Sharma generalize this result to a more general class of contact Riemannian structures and they obtain the following Theorem 3.2.11. [BS90b] Let M be a complete and connected contact Riemannian manifold of dimension 2n+1, with the property that there exists f ∈ F(M ) such that div(N (3) F ) = f η. M is compact if Ric ≥ δ > −2 and if one of the following conditions is fulfilled: a) δ > 0; b) −2 < δ ≤ 0 and the sectional curvatures the qof the planes containing √ 0 0 Reeb vector field ξ are ≥ > δ > 0 with δ = 2 n(n + 2 − δ − 2 −2δ) + δ + √ 2 −2δ − 2n − 1. But δ 0 < 1 and on a K-contact manifold we have N (3) ≡ 0. Then taking into account Theorem 3.1.12 it results that the above theorem is valid in the case of K-contact manifolds, namely Theorem 3.2.12. Let M be a connected complete K-contact manifold. If Ric ≥ δ > −2 then M is compact. Let M be a manifold with the Sasaki structure (F, ξ, η, g) and denote by ∇ o

the Levi-Civita connection of the metric g. Then ∇, defined by o

∇X Y = ∇X Y − η(X)F Y + η(Y )F X + g(X, F Y )ξ

(3.31)

is a linear connection, called the Okumura connection. By a straightforward computation we get Proposition 3.2.13. [Ok62] The Okumura connection of a Sasaki manifold has the following properties o

∇ F = 0,

o

∇ ξ = 0,

o

∇ η = 0,

o

∇g=0

(3.32)

Conversely, there exists an unique linear connection satisfying (3.32). Proposition 3.2.14. [Ok62] The curvature and torsion tensors of the Okumura connection of a Sasaki manifold M are given by the formulas o

R (X, Y )Z = R(X, Y )Z + [η(Y )g(X, Z) − η(X)g(Y, Z)] ξ + η(Z) [η(X)Y − −η(Y )X + g(Y, F Z)F X + g(Z, F X)F Y − 2g(X, F Y )F Z] , o

T (X, Y ) =2 [g(X, F Y )ξ + η(Y )F X − η(X)F Y ] o o

for any X, Y, Z ∈ X (M ). Moreover, we have ∇T = 0. From Propositions 3.2.13, 3.2.14 we obtain the following

104 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Proposition 3.2.15. [In92] The curvature tensor of the Okumura connection of a Sasaki manifold M satisfies the equalities o o o R (X, Y )ξ = 0, R (ξ, X)Y = 0, η R (X, Y )Z = 0 for any X, Y, Z ∈ X (M ).

3.3

Curvature tensor of a Kenmotsu manifold

Proposition 3.3.1. Let M be a Kenmotsu manifold of dimension 2n + 1. The following assertions hold for any X, Y, Z ∈ X (M ): a) R(X, Y )ξ = η(X)Y − η(Y )X; b) the sectional curvature of any plane containing the Reeb vector field ξ is equal to −1; c) [(∇Z R)(X, Y )]ξ = g(X, Z)Y − g(Y, Z)X − R(X, Y )Z; d) R(X, Y )(F Z) − F R(X, Y )Z = = g(Y, Z)F X − g(X, Z)F Y + g(X, F Z)Y − g(Y, F Z)X; e) R(F X, F Y )Z − R(X, Y )Z = = g(Y, Z)X − g(X, Z)Y + g(Y, F Z)F X − g(X, F Z)F Y ; f ) Ric(X, ξ) = −2nη(X). Proof. a) follows easily from the formula (1.86) and from the first formula of Proposition 1.6.4. b) is a consequence of a). c) follows from the definition of ∇R [(∇Z R) (X, Y )] ξ = ∇Z (R(X, Y )ξ) − R (∇Z X, Y ) ξ− − R (X, ∇Z Y ) ξ − R(X, Y )∇Z ξ by using the same formulas like in the proof of the assertion a). d) follows by computation taking into account the formulas (1.84), (1.86). e) By using a) we have η(R(X, Y )Z) = η(Y )g(X, Z) + η(X)g(Y, Z) On the other hand from d) it results g(R(X, Y )F Z, F U ) − g(F R(X, Y )Z, F U ) = g(Y, Z)g(F X, F U )− − g(X, Z)g(F Y, F U ) + g(X, F Z)g(Y, F U ) − g(Y, F Z)g(X, F U ) Now, by applying (3.33) the above equality becomes g(R(F Z, F U )X, Y ) = g(R(Z, U )X, Y ) + g(Y, Z)g(X, U )− − g(X, Z)g(Y, U ) + g(X, F Z)g(Y, F U ) − g(Y, F Z)g(X, F U ) and then e) follows easily. f) is a corollary of a).

(3.33)

3.3. CURVATURE TENSOR OF A KENMOTSU MANIFOLD

105

The analogous of Proposition 3.3.1 concerning the semi-symmetric connection ∇∗ from Proposition 3.1.8 is the following Proposition 3.3.2. [TK01] Let M be a 2n+1-dimensional Kenmotsu manifold. Then: P ∗ a) cycl R (X, Y )Z = 0; ∗ b) R (X, Y, Z, U ) = −R∗ (Y, X, Z, U ), R∗ (X, Y, Z, U ) = R∗ (Z, U, X, Y ); c) Ric∗ (X, Y ) = Ric(X, Y ) + 2ng(X, Y ), ρ∗ = ρ + 2n(2n − 1). Proof. Indeed, taking into account the formulas (∇∗X F ) Y = −Ω(X, Y )ξ,

∇∗X ξ = η(X)ξ,

α(X, Y ) = −g(X, Y )

(3.34)

these equalities result from Proposition 3.1.8. We can state a result similar to the Theorem 3.2.8 concerning K-contact manifolds, namely Theorem 3.3.3. Any locally symmetric Kenmotsu manifold has constant sectional curvature equal to −1. Proof. Taking into account ∇R=0 in Proposition 3.3.1, c) we obtain R(X, Y )Z = g(X, Z)Y − g(Y, Z)X, hence the sectional curvature is equal to −1. Remark 3.3.4. K. Kenmotsu [Ke72] notices that this result remains valid under more general conditions, namely if we replace the hypothesis ”locally symmetric” by ”semi-symmetric3 ” or under the supposition ”the manifold is conformally flat and dim M ≥ 5”. In [TR99] are also studied the conformal curvature tensor, the projective curvature tensor, the concircular curvature tensor and the coharmonic curvature tensor of a Kenmotsu manifold. Also, the authors find a characterization of the deformation algebra of a Kenmotsu manifold M . Before stating this result we remark that the module X (M ) endowed with the product ˜ X Y − ∇X Y X ·Y =∇ becomes an algebra over the ring F(M ). It is called the deformation algebra of ˜ and is denoted by Π(M, ∇, ∇). ˜ Then we have the the pair of connections ∇, ∇ following Proposition 3.3.5. [TK01] Let M be a Kenmotsu manifold. For any element ˜ of M there exists f ∈ F(M ) such that X of the deformation algebra Π(M, ∇, ∇) X · X = f X. 3 The manifold M is called semi-symmetric if there exists a linear connection satisfying the condition R(X, Y )R = 0 for any X, Y ∈ X (M ). This notion was introduced by K. Nomizu, [No68], and the expression Nomizu’ condition is also used.

106 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS

3.4

F -sectional curvature

Let x be a point of the almost contact Riemannian manifold M and let π ⊂ Tx M be a plane in the tangent space at x to M . To the plane π is associated the sectional curvature K(π) and because this does not depend on the orthonormal basis {X, Y } ⊂ π, we write K(π) = K(X, Y ). Then it is natural to study the restriction of K to the planes π invariant by F , that is F π = π (called F -planes). But if π is a F -plane then for any unitary vector X ∈ π, the pair {X, F X} is an orthonormal basis of π and so the sectional curvature of such a plane is K(X, F X). Therefore it is natural to consider the restriction of the sectional curvature to F -planes and we call it the F -sectional curvature of the almost contact Riemannian manifold M . It is denoted by H and its value relative to the F -plane π = hX, F Xi is H(X). Similarly with Riemannian or Kählerian geometries, the following problems concerning the F -sectional curvature H of Sasaki manifolds must be solved: a) to decide if H determines completely the sectional curvature K and consequently if the Riemann-Christoffel curvature tensor is determined by H; b) the validity of a Schur type theorem relative to the F -sectional curvature; c) to find an explicit formula for the curvature vector, at least in the case of Sasaki manifolds of constant F -sectional curvature; d) to find examples of Sasaki manifolds with constant F -sectional curvature. Afterward we will give answers to these questions. Proposition 3.4.1. The F -sectional curvature of a Sasaki manifold M determines completely the sectional curvature of M . Proof. For Z ∈ Tx M orthogonal to ξ (but not necessary unitary), the F sectional curvature of the F -plane determined by Z is given by the formula H(Z) = −

g(R(Z, F Z)Z, F Z) g(Z, Z)2

(3.35)

Now, by using Proposition 3.2.1, c) and d), a direct computation shows that if {X, Y } ⊂ Tx M is an orthonormal system with X and Y orthogonal to ξ then 2

8K(X, Y ) = 3 [1 + Ω(X, Y )] H(X + F Y )+ 2

+ 3 [1 − Ω(X, Y )] H(X − F Y ) − H(X + Y )− − H(X − Y ) + H(X) + H(Y ) − 6 1 + Ω(X, Y )2

(3.36)

If X or Y is equal to ξ then, taking into account Proposition 3.2.1, a), we have K(X, Y ) = 1. It remains to study the case when X = η(X)ξ + aZ, Y = η(Y )ξ + bU , with ∗ U, V unitary and orthogonal p to ξ and a, b ∈ Rp. Since X and Y are orthogonal 2 and unitary, we have a = 1 − η(X) , b = 1 − η(Y )2 . Taking into account Proposition 3.2.1, a), b) and the linearity properties of the curvature tensor, we have K(X, Y ) = b2 η(X)2 − 2abη(X)η(Y )g(Z, U ) + a2 η(Y )2 + a2 b2 R(Z, U, Z, U )

3.4. F -SECTIONAL CURVATURE But g(Z, U ) +

107

1 1 g(X − η(X)ξ, Y − η(Y )ξ) = − η(X)η(Y ) ab ab

and then 2 R(Z, U, Z, U ) = 1 − g(Z, U ) K(Z, U ) = 1 −

1 2 2 η(X) η(Y ) K(Z, U ) a2 b2

so that the equality (3.36) becomes K(X, Y ) = η(X)2 1 − η(Y )2 + 2η(X)2 η(Y )2 + η(Y )2 1 − η(X)2 + (3.37) + 1 − η(X)2 1 − η(Y )2 − η(X)2 η(Y )2 K(Z, U ) The proof ends with the remark that in the above expression K(Z, U ) is given like in formula (3.36), but depending only on the F -sectional curvature H. From the formula (3.36) and from Proposition 3.2.1 it follows immediately the following Theorem 3.4.2. Let M be a Sasaki manifold and x ∈ M . If at the point x the F -sectional curvatures corresponding to vectors orthogonal to ξ are equal to 1 then all the sectional curvatures at x are equal to 1. Let x be a point of the Sasaki manifold M and let S(x) = {X ∈ Tx M ; kXk = 1, X ⊥ ξ} be the 2n − 1-dimensional unit sphere in the hyperplane orthogonal to ξ in Tx M , that is S(x) ⊂ Dx . Denote by V (S(x)) its volume and by dvX the volume element of S(x) at the end of the vector X ∈ S(x). Then V (S(x)) can be expressed depending on the F -sectional curvature only, namely we have Proposition 3.4.3. Let M be a Sasaki manifold of dimension 2n + 1. The volume V (S(x)) of the sphere S(x) at x ∈ M is given by the formula Z n(n + 1) V (S(x)) = H(X)dvX (3.38) ρ − n(3n + 1) S(x) where ρ is the scalar curvature of M at x. For a proof of this result see [Sa68], vol. 3, pg. 43.5–43.10. The explicit formula for the curvature vector of a Sasaki manifold with constant F -sectional curvature, given by K. Ogiue, is the following Theorem 3.4.4. [Og64] Let M be a connected Sasaki manifold of dimension ≥ 5. If at each point x ∈ M the F -sectional curvature H does not depend on the F -plane then H = c = constant on M and the curvature vector is given by R(X, Y )Z =

c+3 [g(Y, Z)X − g(X, Z)Y ]+ 4

c−1 [η(X)η(Z)Y − η(Y )η(Z)X + g(X, Z)η(Y )ξ− 4 − g(Y, Z)η(X)ξ + Ω(Z, Y )F X + Ω(X, Z)F Y + 2Ω(X, Y )F Z] .

+

(3.39)

108 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Proof. Denote by H(x) the F -sectional curvature at the point x, assumed to be independent on the F -plane and we search for a vector field R1 (X, Y )Z which coincides with R(X, Y )Z for some vectors X, Y, Z. According to the formula for any plane hX, Y i ⊂ Tx M orthogonal to (3.36) we have K(X, Y ) = H(x)+3 4 ξ, hence neither the sectional curvature of such a plane does not depend on the plane. By analogy with the proof of Schur theorem (see for instance [KN69], vol. I, pg. 202–203), it follows that for Z ⊥ ξ the curvature vector R(X, Y )Z 2 has one component of the form g(Y, Z)X − g(X, Z)Y . But F|D = −IDx , hence x 1 for any Z ⊥ ξ the vector R (X, Y )Z also contains a component of the form Ω(Z, Y )F X − Ω(Z, X)F Y + 2Ω(X, Y )F Z, like in the case of Kähler manifolds (see for instance [KN69], vol. II, pg. 166). By Proposition 3.2.1, b) the ξ component of R1 is g(R(X, Y )Z, ξ) = −g(R(X, Y )ξ, Z) = −η(Y )g(X, Z) + η(X)g(Y, Z) Now, for any Z we have Z = Z 0 + η(Z)ξ with Z 0 ⊥ ξ, hence for X, Y ⊥ ξ the vector R1 (X, Y )Z has another component R(X, Y )[η(Z)ξ] = η(Z)[η(Y )X − η(X)Y ] Therefore for arbitrary X, Y, Z we can search for R1 (X, Y )Z on the form R1 (X, Y )Z = A[g(Y, Z)X − g(X, Z)Y ]+ + B[Ω(Z, Y )F X − Ω(Z, X)F Y + 2Ω(X, Y )F Z]+ +C[η(X)g(Y, Z) − η(Y )g(X, Z)]ξ + Dη(Z)[η(Y )X − η(X)Y ]

(3.40)

By applying the formula (3.36) to the orthonormal system {X, Y }, assumed to be orthogonal to ξ, we have K(X, Y ) =

H(x) + 3 + 3 [H(x) − 1] g(X, F Y )2 4

(3.41)

If we compute K(X, Y ) with R1 (X, Y )Z given by the formula (3.40) then we obtain K(X, Y ) = A + 3Bg(X, F Y )2

(3.42)

By identification of the equalities (3.41) and (3.42) and taking into account H−1 that dim M ≥ 5, we deduce A = H+3 4 , B = 4 . On the other hand, if we put Y = ξ, Z = X with X unitary then, by comparing with Proposition 3.2.1, a) we obtain C = 1 − A = 1−H 4 . By an analogous argument, but using the formula b) from the same Proposition, we obtain D = 1−H and thus 4 R1 (X, Y )Z has the same expression like in formula (3.39). We check easily that R1 (X, Y, Z, U ) = g(R1 (Z, U )Y, X) has the same symmetry properties with R(X, Y, Z, U ), it verifies the conditions c) and d) from Proposition 3.2.1 and R1 (X, F X, X, F X) = R(X, F X, X, F X),

3.5. SASAKI SPACE FORMS

109

hence R1 ≡ R and then R1 ≡ R. Finally, from the second Bianchi identity and from the formula (3.39), by a hard computation we obtain (n − 1)dH + (ξH)η = 0

(3.43)

and applying this equality to ξ it follows ξH = 0. Hence we have dH = 0, that is H is constant.

3.5

Sasaki space forms

A connected Sasaki manifold M with constant F -sectional curvature equal to c is called a Sasaki space form and we denote it by M (c). The Sasaki space form M (c) is elliptic, hyperbolic or parabolic according to c > −3, c < −3 or c = −3. A simple characterization of Sasaki space form is the following Proposition 3.5.1. Let M be a connected Sasaki manifold of dimension ≥ 5. M is a Sasaki space form if and only if there exists f ∈ F(M ) such that R(X, F X)X = f F X

(3.44)

for any unitary vector field X ⊥ ξ. In this case f = constant. Proof. If M is a Sasaki space form with F -sectional curvature c then from (3.39) it follows R(X, F X)X = −cF X for any unitary X orthogonal to ξ. Conversely, by using the condition (3.44) in (3.35) we obtain H(X) = −f and then by Theorem 3.4.4 we get f = constant, hence M is a Sasaki space form of F -sectional curvature equal to −f . Proposition 3.5.2. Let M (c) be a Sasaki space form of dimension 2n + 1. a) The Ricci tensor and the scalar curvature are given by the formulas 2Ric = {n(c + 3) + c − 1} g + {(n + 1)(c − 1)} η ⊗ η, 2ρ = n(2n + 1)(c + 3) + n(c − 1);

(3.45)

b) If c < 1 then c ≤ K(X, Y ) ≤

c+3 4

(3.46)

and if c > 1 then in (3.46) the inequalities have opposite sense. Proof. Follows from (3.39) a by straightforward computation. Consider the horizontal vector fields X H , Y H , Z H associated to the vector ˜ Y˜ , Z˜ ∈ X (Mξ ), as in the proof of Theorem 2.6.4. Denoting by ∇ ˜ the fields X,

110 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS connection induced on the orbit space Mξ by the Levi-Civita connection ∇ of the Sasaki manifold M we have ˜ ˜ Y˜ = −π∗ F 2 ∇X H Y H ∇ X

(3.47)

˜ R of the orbit space and the Sasaki manifold and then the curvature tensors R, M , respectively, satisfy the relation ˜ R(X, Y )Z = π∗ R(X H , Y H )Z H − g(F X H , Z H )π∗ F Y H + + g(F Y H , Z H )π∗ F X H − 2g(F X H , Y H )π∗ F Z H

(3.48)

Moreover, from (3.47), (3.48), from Proposition 3.2.1, b) and Theorem 3.2.2 we obtain ˜ (X, ˜ Y˜ )Z˜ = −π∗ F 2 (∇W H R) (X H , Y H )Z H ˜ ˜R (3.49) ∇ W ˜ Y˜ , Z, ˜ W ˜ ∈ X (Mξ ). Now, from the formula (3.48) and from Theorem for any X, 3.4.4 it results Proposition 3.5.3. [Og65] The orbit space of the compact regular Sasaki space form M (c) is a complex space form with constant holomorphic curvature c+3 4 . In the case of contact hypersurfaces of a complex space form, Theorem 3.1.7 has the following expression Theorem 3.5.4. [Sh03] Let S be a contact hypersurface of the complex space form M (c). Then either a) S is a Sasaki space form S = S(c + 1) and it is C-umbilical (i. e. h = αg + βη ⊗ η) or b) S is locally isometric (up to a D-homothetic deformation) with the sphere tangent bundle of a real space form p of sectional curvature 6= 1, c > −4 and the principal curvatures are − 2c , 1± 1 + 4c . For c = 0 the hypersurface S is locally isometric with Rn+1 × S n (4) if n > 1 and S is flat for n = 1. Proof. The argument is similar to that one used in the proof of Theorem 3.1.7, taking into account that the expressions of the curvature vectors of complex and Sasaki space forms (see the formula (1.50) and Theorem 3.4.4) and the Gauss equation. On a Sasaki manifold of dimension 2n + 1, M. Matsumoto and G. Chuman [MC69] defined the following tensor field, called the C-Bochner curvature tensor B(X,Y, Z, U ) = R(X, Y, Z, U )+ 1 + [Ric(Y, Z)g(X, U ) − Ric(Y, U )g(X, Z) + Ric(X, U )g(Y, Z)− 2(n + 2)


111

− Ric(X, Z)g(Y, U ) − Ric(Y, Z)η(X)η(U ) + Ric(Y, U )η(X)η(Z)− − Ric(X, U )η(Y )η(Z) + Ric(X, Z)η(Y )η(U ) − Ric(F Y, Z)Ω(X, U )+ + Ric(F Y, U )Ω(X, Z) − Ric(F X, U )Ω(Y, Z) + Ric(F X, Z)Ω(Y, U )+ + 2Ric(F X, Y )Ω(Z, U ) + 2Ric(F Z, U )Ω(X, Y )]− ρ − 6n − 8 − [g(Y, Z)g(X, U ) − g(Y, U )g(X, Z)] − (3.50) 4(n + 1)(n + 2) 2 ρ + 4n + 6n [Ω(Y, Z)Ω(X, U ) − Ω(Y, U )Ω(X, Z)− − 4(n + 1)(n + 2) ρ + 2n −2Ω(X, Y )Ω(Z, U )] + [g(Y, Z)η(X)η(U )− 4(n + 1)(n + 2) − g(Y, U )η(X)η(Z) + g(X, U )η(Y )η(Z) − g(X, Z)η(Y )η(U )] The vector field B(X, Y )Z defined by g(B(Z, U )Y, X) = B(X, Y, Z, U ) is called the C-Bochner curvature vector. Theorem 3.5.5. [MC69] Let M be a compact and connected Sasaki manifold of dimension ≥ 5 with vanishing C-Bochner curvature vector and Ric > −2. Then M is a Sasaki space form. The above result (for its proof see the cited work) was improved by J. S. Pak [Pa76], who proved that it remains valid under the stronger hypothesis Ric ≥ 0, but at the same time giving up the compactness condition. The tensor field S η , defined by ρ ρ S η = Ric − −1 g+ − 2n − 1 η ⊗ η (3.51) 2n 2n is called the η-Einstein tensor of the Sasaki manifold M and from the formulas (3.50) and (3.51), by a straightforward computation it follows Theorem 3.5.6. Let M be a 2n + 1-dimensional connected Sasaki manifold whose C-Bochner curvature tensor vanishes. a) If M has constant scalar curvature then its Ricci tensor is η-parallel, i. e. (∇X Ric) (F Y, F Z) = 0 for any X, Y, Z ∈ X (M ). b) If the η-Einstein tensor S η of M vanishes then ρ = constant and M is a 2 −n Sasaki space form with F -sectional curvature equal to ρ−3n n(n+1) . The assertion b) from the previous theorem was obtained by I. Hasegawa and T. Nakane, who also proved the following Theorem 3.5.7. [HT80] Let M be a 2n+1-dimensional connected Sasaki manifold of constant scalar curvature and with vanishing C-Bochner curvature tensor. M is a Sasaki space form if one of the following conditions is satisfied: 2 2 (ρ+2n)2 a) n ≥ 3 and kS η k < (n−1)(n+2) 2n(n+1)2 (n−2)2 ; b) n = 2 and ρ 6= −4.

112 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Examples of Sasaki space forms 1. R2n+1 The space R2n+1 equipped with the Sasaki structure defined in Example 1, Section 1.5 is a Sasaki space form with F -sectional curvature c = −3. 2. S 2n+1 Let S 2n+1 be the unit sphere with the Sasaki structure (F, ξ, η, g) defined in Example 2, Section 1.5. Since g is the restriction to S 2n+1 of the standard metric in R2n+2 , it results that it has sectional curvature equal to K = 1. ˜ η˜,˜) of the Sasaki strucNow, we consider the D-homothetic deformation (F˜ , ξ, ture (F, ξ, η, g) (see the formulas (1.68)). Taking into account that the structure ˜ η˜,˜) has (F, ξ, η, g) has K = 1, a straightforward computation shows that (F˜ , ξ, constant F -sectional curvature c = α4 − 3. 3. B n × R, where B n is a simply connected bounded domain from Cn Consider on B n a K¨ ahler structure (J, G) with constant holomorphic curvature K < 0 and denote by Ω its fundamental 2-form and by π : B n × R −→ B n , π(x, t) = x the projection on the first factor. Ω is closed and B n is a simply connected bounded domain from Cn , hence by Poincaré lemma there exists a real 1-form ω ∈ F 1 (B n ) such that Ω = dω. Now, we can define on B n × R the d 1-form η = π ∗ ω + (0, dt), the vector field ξ = 0, dt , the Riemannian metric ∗ n g = π G+η ⊗η and the endomorphism F : X (B ×R) −→ X (B n ×R), given by F X = −∇X ξ, where ∇ is the Levi-Civita connection of the metric g on B n × R. By direct verification it follows that (F, ξ, η, g) is a Sasaki structure on B n × R. Denote by K and K ∗ the sectional curvatures of B n ×R and B n , respectively. With the same notations as in Example 3 of Section 1.3, but replacing T M by B n × R and M by B n , a direct computation shows that for any X, Y ∈ X (B n ) we have 2 K(X H , Y H ) = K ∗ − 3η ∇X H Y H (3.52) where {X, Y } is an orthonormal system in X (B n ). On the other hand we have (3.53) g ∇X H Y H , ξ = −g Y H , ∇X H ξ = g(Y H , F X H ) By setting Y H = F X H with X H ⊥ ξ, from the formulas (3.52) and (3.53) we deduce that B n × R has constant F -sectional curvature equal to K − 3, hence it is a hyperbolic Sasaki space form. 4. H 3 (R) Let H3 (R) be the Heisenberg group with Sasaki structure given in Example 6, Section 1.5. From the expression of its metric we deduce the values of the sectional curvature, namely K(X1 , X2 ) = −3, K(X1 , X3 ) = K(X2 , X3 ) = 1. Hence H3 (R) is a hyperbolic Sasaki space form with F -sectional curvature equal to −3. 5. P + × S 1 The upper semi plane P + = (x, y) ∈ R2 : y > 0 with the Riemannian metric g + = 4y12 (dx2 + dy 2 ) is a space of constant sectional curvature


113

equal to −1 and consider the product manifold P + × S 1 with the metric g given by 2 dx + , g = g + dθ + 2y where θ is the polar angle on the circle S 1 . But S 1 is parallelizable, hence we can consider on P + × S 1 the vector fields X1 = 2y

∂ ∂ − , ∂x ∂θ

X2 = 2y

∂ , ∂y

X3 =

∂ ∂θ

which form an orthonormal frame with respect to the metric g and its dual frame 1 1 1 dx, ω 2 = 2y dy, ω 3 = dθ + 2y dx. It is easy to check that η = −ω 3 is ω 1 = 2y is a contact form on P + × S 1 . We set ξ = −X3 and define the endomorphism F : X (P + × S 1 ) −→ X (P + × S 1 ) by F X1 = X2 , F X2 = −X1 , F X3 = 0. Thus we obtain an almost contact Riemannian structure on P + × S 1 . On the other hand, an elementary computation gives the expression of the Levi-Civita connection ∇X1 X1 = 2X2 , ∇X1 X2 = −2X1 − X3 , ∇X1 X3 = X2 , ∇X2 X1 = X3 , ∇X2 X2 = 0, ∇X2 X3 = −X1 , ∇X3 X1 = X2 , ∇X3 X2 = −X1 , ∇X3 X3 = 0 and these relations show that the formula (1.56) from Theorem 1.5.3 is satisfied, hence this structure is Sasakian. Moreover, its sectional curvatures are K(X1 , X2 = −7, K(X1 , X3 ) = K(X2 , X3 ) = 1 and then P + ×S 1 is a hyperbolic Sasaki space form with F -sectional curvature c = −7. 3∗ 6. M3∗ λµ Let Mλµ be a Bianchi-Cartan-Vranceanu space with the Sasaki struc∗ ∗ ∗ ∗ ture (F , ξ , η , g ) defined in Example 4, Section 1.5 and assume that λ 6= 0. By using the formulas (1.8) it follows that M3∗ λµ is Sasaki space form with F 16µ sectional curvature c = −3 + λ2 .

Classification of Sasaki space forms M. Belkhelfa, F. Dillen and J-I. Inoguchi, [BDI02], give a complete classification of the Bianchi-Cartan-Vranceanu spaces M3∗ λµ equipped with this Sasakian structure. By their classification such a space is one of the following: 3 3 i) M3∗ 00 ≡ R (where R is endowed with the standard Sasaki structure from Example 1); ii) for λ 6= 0 and µ = 0 the Bianchi-Cartan-Vranceanu space M3∗ λ0 is homothetic to the Heisenberg group H3 (R) endowed with the Sasaki structure from the Example 4; iii) for λ 6= 0 and µ < 0 the space M3∗ λµ is identified with the hyperbolic Sasaki space form P + × S 1 (−7) given in Example 5; iv) for λ 6= 0 and µ > 0 the Bianchi-Cartan-Vranceanu space M3∗ λµ is identified to a dense open subset of an elliptic space form;

114 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS v) for λ = 0 and µ < 0 the space M3∗ 0µ is identified to the Riemannian √ product D2 (1/ −µ) × R (see Example 3); vi) for λ 6= 0 and µ > 0 the space M3∗ λµ is the Riemannian product of the plane R2 with the real line R, the plane being endowed with the metric dx2 +dy 2 [1+µ(x2 +y 2 )]2 of positive constant sectional curvature. While the class of Sasaki manifolds is, as we have already seen, extremely rich, that one of Sasaki manifolds with constant F -sectional curvature seems to be much more poor, at least under certain topological restrictions. So, S. Tanno proved that a complete and simply connected Sasaki manifold with constant F sectional curvature is diffeomorphic to one of the following spaces: Theorem 3.5.8. [Ta69b] Let M (c) be a complete and simply connected Sasaki space form. M (c) is diffeomorphic (by a contact diffeomorphism) to one of the following Sasaki space forms: a) a D-homothetic deformation of the Sasaki structure on the unit sphere S 2n+1 if c > −3; b) R2n+1 (−3) if c = −3; c) B n × R(k − 3) with k < 0 if c < −3. For a proof of Theorem 3.5.8 see [Ta69b]. In fact this result is a stronger version of the following classification theorem, given also by S. Tanno a few time ago, and it completes Theorem 2.1.11. Theorem 3.5.9. [Ta69a] Let M be a connected almost contact manifold of dimension 2n+1. dimAci (M ) = (n+1)2 if and only if M has constant sectional curvature K for all the planes containing the Reeb vector field ξ. In this case: a) for K > 0 M is diffeomorphic to a homogeneous Sasaki manifold M 0 with constant F -sectional curvature c (or with a -deformation of its structure) and a1 ) for c > −3 M 0 is S 2n+1 with a D-homothetic deformation of the usual structure or M 0 is S 2n+1 /G(t), where G(t) is the group finitely generated by exp tξ, with 2π/t ∈ Z; a2 ) for c = −3 M 0 is R2n+1 or R2n+1 /G(t), where G(t) is the cyclic group generated by exp tξ, with t ∈ R; a3 ) for c < −3 M 0 is B n × R or B n × R/G(t), where G(t) is defined as in a2 ); b) for K = 0 M is diffeomorphic to one of the following Riemannian products Pn (C) × R, Cn × R, B n × R, Pn (C) × S 1 , Cn × S 1 , B n × S 1 ; c) for K < 0 M is diffeomorphic to the warped product R ×f Cn , with f (t) = et and the metric given by the formula (1.88). Finally we remark that Theorem 3.5.8 is a corollary of Theorem 3.5.9. On the other hand, both these results are remarkable classification theorems. Indeed, we see that the cases a), b), c) in Theorem 3.5.9 correspond to Sasakian, cosymplectic and Kenmotsu structures, respectively. So, the definition of Kenmotsu manifolds appears to be necessary in order to complete this classification. Therefore by Theorem 3.5.8 the classification of simply connected Sasaki space forms is complete.


115

Another classification of Sasaki space forms was obtained by J. Berndt, [Ber89], [Ber90], and N. Ejiri, [Ej83] starting from the remark that any Sasaki space form M (c) can be viewed as a real hypersurface of a complex space form with holomorphic sectional curvature equal to c−1. Their result is the following Theorem 3.5.10. Let M (c) be a Sasaki space form of dimension 2n + 1.√ Then: in a) for c ∈ (1, ∞) M (c) is a geodesic sphere with radius r = arctan c−1 2 the complex projective space Pn+1 (C) with holomorphic curvature c − 1; b) for c = 1 M (c) is the unit sphere in the standard complex manifold√ Cn+1 ; in c) for c ∈ (−3, 1) M (c) is the geodesic sphere with radius r = arcctg 1−c 2 the open unit disc Dn+1 with holomorphic curvature equal to c − 1; d) for c = −3 M (c) is the horosphere in Dn+1 with holomorphic curvature equal to −4; e) for c√ ∈ (−∞, −3) M (c) is the universal covering of a tube4 of radius r = arcctg 1−c around of a totally geodesic complex hyperbolic hyperplane with 2 holomorphic curvature c − 1 in Dn+1 . In the case when the total space of a Sasaki fibered space is a Sasaki space form, we can get deeply into the results from Section 2.5 and we notice some of them. Thus the statement of Theorem 2.5.6 can be completed by the following result: Proposition 3.5.11. [Ki88] Let (M, M 0 , π, g) be a Riemannian fibered space with the fibers almost contact Riemannian manifolds and the base M 0 a manifold with the almost Hermitian structure (J, g 0 ). If M with the induced almost contact Riemannian structure is a Sasaki space form M = M (c) then M 0 is a complex space form with holomorphic curvature c + 3. If in addition (M, M 0 , π, g) is a fibered space with conformal fibers then it is a fibered space with isometric fibers and its fibers are totally geodesic submanifolds. In the case of Sasaki fibered spaces the last assertion from Proposition 3.5.11 can be completed by the following Theorem 3.5.12. [Ki88] Let (M, M 0 , π, g) be a Sasaki fibered space with conformal fibers. If M is Sasaki space form M = M (c) then: a) c = −3; b) the base space M 0 is locally Euclidean; c) the fibers are Sasaki space forms with F -sectional curvature equal to −3. Conversely, if the base space M 0 is locally Euclidean and if all the fibers are Sasaki space forms with F -sectional curvature equal to −3 then the total space M is a Sasaki space form M = M (−3). 4 Let µ : M −→ E be an immersion of the manifold M into the Euclidean space E. For x ∈ M we consider the subspace Nx M = (µ∗,x Tx M )⊥ of Tµ(x) E. We also denote by N M the normal bundle of the immersion µ and notice that its total space is N M = ∪x∈M Nx M = {(x, v) ∈ M × E; x ∈ M, v ∈ Nx M }. Consider the map T : N M −→ E, defined by T (x, v) = µ(x) + v. Then the set {T (x, v) ∈ N M : kvk < } is called the tube of radius around (M, µ). If µ is the inclusion map then we simply say: the tube of ray around the manifold M .

116 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS By giving up the condition ”the fibers of the Sasaki fibered space are conformal ”, as it is imposed in the last part of Theorem 3.5.12, K. Takano proves the following Theorem 3.5.13. [Tak93] Let (M, M 0 , π, g) be a Sasaki fibered space such that the total space M is a Sasaki space form M = M (c). If n − n0 ≥ 3 then: a) c ≤ −3; b) the base space M 0 is a complex space form with holomorphic curvature c + 3. In [Tak93] it is proved that under the hypotheses of Theorem 3.5.13 the fibers are η-Einstein manifolds (these manifolds will be defined and studied in Section 3.8). If the difference of dimensions n − n0 is ≥ 4 then the following characterization theorem for the base space of a Sasaki fibered space can be stated: Theorem 3.5.14. [Tak93] Let (M, M 0 , π, g) be a Sasaki fibered space and m = n − n0 ≥ 3. If the C-Bochner curvature tensor of the total space M vanish then: a) the scalar curvatures ρ0 and ρ of the base space M 0 and of the fibers respectively, are constant and ρ0 ≤ −n0 (n0 + 2), ρ ≤ m(m − 1); b) M 0 is either a complex space form with holomorphic curvature c0 ≤ −4 or locally, it is the product between two complex space forms with holomorphic curvatures c and −c, with |c| > 4. At the end of this section we notice that Theorem 3.5.9 allowed to A. Bejancu and H. R. Farran to find an interesting relation between Sasaki space forms and Finsler spaces. But first we recall some basic notions concerning these spaces. Let M be a m-dimensional manifold and denote by (xi , y i ) the local coordinates of (x, y) ∈ T M , induced by the local coordinates (xi ) of the point x ∈ M . The function F : T M −→ R is called Finsler metric if the following conditions are fulfilled: 1. F (x, y) ≥ 0 and F (x, y) = 0 if and only if y = 0; 2. F (x, y) is smooth on T M − 0; 3. F (x, ky) = kF (x, y) for any k > 0; 2

F 4. the matrix gij (x, y) = 12 ∂y∂i ∂y is positively defined at any point (x, y) ∈ j T M − 0. The pair (M, F ), where M is a m-dimensional manifold and F is a Finsler metric, is denoted by F n and we call it a m-dimensional Finsler space. The Finsler metric F is projectively flat if any point of M has a neighborhood U which can be embedded in Rm+1 so that the geodesics of U with respect to the metric F are carried in straight segments from Rm+1 . We suppose that M is a Riemannian manifold with the metric a = (aij (x)) and b = (bi (x)) is a 1-form p nowhere zero on M . Moreover, a and b have the property that F (x, y) = aij (x)y i y j + bi (x)y i is a Finsler metric on M . In this case F is called Randers metric and we say that (M, F, aij , bi ) is a Randers space.


117

∂ 1 The vector field L = y i ∂y i is globally defined on T M − 0 and ` = F L is unitary with respect to the metric (gij (x, y)). We denote by `∧V flag determined ∂ by ` and by V = V i ∂x i . Then the number

K(` ∧ V ) =

Rij V i V j gij V i V j − (gij ì V j )2

is called the flag curvature of the flag ` ∧ V . In this formula the notations have the following meaning " !# k k i G G y δ δ j h ì = , Rij = gik `h − h F δxj F δx F j ∂ ∂ where the functions Gji have the property that δxδ i = ∂x i − Gi ∂y j is a local frame in the vertical subbundle of T M − 0. For a deeply study of these spaces see the monographs written by R. Miron and M. Anastasiei, [MA94], by R. Miron, D. Hrimiuc, H. Shimada, V. S. Sabau, [MHSS01], or by D. Bao, S. S. Chen and Z. Shen, [BCS00].

Theorem 3.5.15. [BF02] Let M (c) be a Sasaki space form with constant F sectional curvature c ∈ (−3, 1). For any K > 0 there exists a Randers metric F on M (c) such that (M (c), F ) has constant flag curvature K and it is not projectively flat. √

Proof. It is enough to take b = αη, α =

1−c 2 .

In [BF03], the same authors prove also a converse of this result. Theorem 3.5.16. [BF03] Let (M, F, aij , bi ) be a Randers space with positive flag curvature and (3.54) ajk bk bj|i − bi|j = 0 Then M has a Sasaki space form structure with F -sectional curvature c ∈ (−3, 1). Proof. We set η=

1 b, kbk

ξ : (ξ i = ηj aij ),

i Fji = −ξ|j

where kbk = aij (x)bi (x)bj (x). Then one verifies that (F, ξ, η, a) is a Sasaki structure on M . Taking into account Theorem 3.5.8, the above result can be expressed under the following form: Theorem 3.5.17. [BF03] Let (M, F, aij , bi ) be a Randers space with positive flag curvature If the condition (3.54) is satisfied then the following assertions hold: a) dim M = 2n + 1; b) if the manifold M is simply connected and complete relative to the metric (aij ) then it is diffeomorphic to the sphere S 2n+1 .

118 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Let M be a (n + 1)-dimensional manifold and denote by π : T M −→ M the natural projection of its tangent bundle. The relation between contact manifolds and Finsler spaces is considered by R. Bryant, [Br01] in a more general background by defining the generalized Finsler structure on M as a pair (Σ, ı), where Σ is a connected (2n + 1)-dimensional manifold and ı : Σ −→ T M is a radial transverse immersion (that is transverse to the orbits of the multiplication by scalars on T M ) such that the following conditions are fulfilled: 1. π ◦ ı : Σ −→ M is a submersion with connected fibers; 2. for any x ∈ M the fiber Σx = ı−1 (Tx M ) is a hypersurface in Tx M by the immersion ıx = ı|Σx and it is locally strictly convex in a neighborhood of the origin 0x ∈ Tx M . For each u ∈ Σ we have ı(u) ∈ Tx M , where x = π(ı(u)) and the hyperplane (π ◦ ı)∗ (Tu Σx ) ⊂ Tx M does not contain ı(u). Then there is a unique u∗ ∈ Tx∗ M with the properties ker u∗ = (π ◦ ı)∗ (Tu Σx ), u∗ (ı(u)) = 1 and then we can define a 1-form η0 on Σ by (η0 )u = (π ◦ ı)∗ (u∗ ) But ı is radially transverse and then η0 ∧ (dη0 )n 6= 0, i. e. η0 is a contact form. R. Bryant also proves that there exists a foliation M whose leaves are the fibers Σx and these are Legendre submanifolds of the manifold Σ (we will study the Legendre submanifolds in Section 5.4).

3.6

Kenmotsu space forms

Let M be a Kenmotsu manifold and denote by H its F -sectional curvature. With few minor modifications we obtain a result similar to Theorem 3.4.4, namely Theorem 3.6.1. [Ke72] Let M be a Kenmotsu manifold of dimension ≥ 5. If at the point x ∈ M the F -sectional curvature H is independent on the F -plane then there is a constant c so that the curvature vector of the manifold M at x is given by c−3 [g(Y, Z)X − g(X, Z)Y ] + 4 c+1 (3.55) + [η(X)η(Z)Y − η(Y )η(Z)X + η(Y )g(X, Z)ξ− 4 − η(X)g(Y, Z)ξ + Ω(X, Z)F Y − Ω(Y, Z)F X + 2Ω(X, Y )F Z]

R(X, Y )Z =

A technical proof of this result was initially given by K. Kenmotsu [Ke72]. A Kenmotsu manifold M with the property that its F -sectional curvature c is depending only on the point x ∈ M , but not on the F -planes at x is called pointwise Kenmotsu space form and we denote it by M (c(x)). A connected pointwise Kenmotsu space form whose F -sectional curvature does not depend on the point is called Kenmotsu space form and we set M (c). From the formula (3.55), by an elementary computation we deduce the following

3.6. KENMOTSU SPACE FORMS

119

Proposition 3.6.2. The Ricci tensor and the scalar curvature of a pointwise Kenmotsu space form M (c) of dimension 2n + 1 ≥ 5 have the following expressions 2Ric = [n(c − 3) + c + 1] g − (n + 1)(c + 1)η ⊗ η; 2ρ = (2n + 1) [n(c − 3) + c + 1] − (n + 1)(c + 1) From Theorem 3.6.1 and Proposition 3.6.2 we obtain the following characterization of Kenmotsu space forms: Theorem 3.6.3. [Ke72] Any Kenmotsu space form M (c) has constant sectional curvature equal to c = −1. Proof. From Proposition 3.6.2 and from the Ricci identity, by a straightforward computation we obtain n(n + 1)(c + 1)η(Y ) = 0 for any Y ∈ X (M ) Hence c = −1 and then from (3.55) it follows that M (c) has sectional curvature equal to −1. Example. Let M be a K¨ ahler and consider the warped product R ×f M , f (t) = cet with the Kenmotsu structure given in Example 1, Section 1.6. If we consider on M = Cn the usual Kähler structure then a simple computation shows that R ×f Cn has constant F -sectional curvature equal to −1, hence it is a Kenmotsu space form. We can ask if there are other K¨ ahler manifolds whose warped products with R are Kenmotsu space forms. We limit ours considerations to the case when M is a Kähler manifold with constant holomorphic curvature equal to k. By using the results from the Example 3, Section 1.6, by a straightforward computation we obtain the following expression for the curvature tensor of the manifold R ×f M R(X, Y )Z = H1 (t) [g(Y, Z)X − g(X, Z)Y ] + + (H1 (t) + 1)[η(X)η(Z)Y − η(Y )η(Z)X + η(Y )g(X, Z)ξ− (3.56) − η(X)g(Y, Z)ξ + Ω(X, Z)F Y − Ω(Y, Z)F X + 2Ω(X, Y )F Z] for any X, Y, Z ∈ X (R ×f M ), where H1 : R −→ R is a function depending on k and c. But from Theorems 3.6.1 and 3.6.3 it follows that R ×f M is a Kenmotsu space form if and only if H1 = H−3 = −1. Then from the expression of H1 it 4 follows k = 0, so that assuming that M is simply connected and complete it must be isometric to Cn . We remark that the previous argument also proves that in this manner we cannot construct strictly pointwise Kenmotsu space forms. In fact we do not know examples of pointwise Kenmotsu space forms which are not Kenmotsu space forms. From the formulas (3.17) and (3.34) it follows Theorem 3.6.4. a) A Kenmotsu manifold has constant sectional curvature equal to −1 if and only if the semi-symmetric connection ∇∗ is flat.

120 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS b) [TR99] If M is a Kenmotsu space form then the semi-symmetric Riemannian connection ∇∗∗ is flat. As in the case of Sasaki space forms (see Theorems 3.5.15 and 3.5.16), I. Hasegawa, V. S. Sabau and H. Shimada established the following correspondence between Kenmotsu space forms and Randers spaces: Theorem 3.6.5. [HSS04] Let M (c) be a Kenmotsu space form. On M (c) there exists a Randers metric with negative constant flag curvature equal to − 14 and it is projectively flat. In fact, in [HSS04] the authors close this study by proving that a similar result is valid in the cosymplectic case too, namely: On any cosymplectic space form with F -sectional curvature equal to 0 there exists a Randers metric of constant flag curvature equal to 0.

3.7 3.7.1

Bisectional curvature of Sasaki manifolds F -bisectional curvature

Let M be a Riemannian manifold of dimension ≥ 3 and denote by π1 , π2 two oriented planes in Tx M . This means that in π1 and in π2 two positively oriented bases {X, Y }, {Z, V } are fixed. The bisectional curvature ℵπ1 π2 of the manifold M at the point x, relative to these two oriented planes π1 , π2 is defined by the formula ℵ π1 π2 =

R(X, Y, Z, U ) 2

2

2

2

[kXk kY k − g(X, Y )2 ]1/2 [kZk kU k − g(Z, U )2 ]1/2

(3.57)

The notion of bisectional curvature was introduced by E. Bompiani, [Bom24]. It is easy to prove that: a) the definition (3.57) is not depending on the bases of the planes π1 , π2 ; b) ℵπ1 π2 = ℵπ2 π1 ; c) if we denote by π10 , π20 the planes π1 , π2 with the opposite orientation then ℵπ10 π20 = −ℵπ10 π2 = −ℵπ1 π20 = ℵπ1 π2 d) if π1 = π2 then ℵπ1 π2 is the sectional curvature of the plane π1 . G. B. Rizza obtained a useful formula for the bisectional curvature, namely Proposition 3.7.1. [Ri87] Let us consider two oriented planes π1 , π2 and denote by {X1 , X2 }, {X3 , X4 } two orthonormal bases for π1 and π2 , respectively.

3.7. BISECTIONAL CURVATURE OF SASAKI MANIFOLDS

121

Then X 3 1\ 21 \ ℵπ1 π2 = sgn(i, j) K(Si3 , Sj4 )cos2 X Xj , X4 sin2 S\ i , X3 cos i3 , Sj4 − 2 2 2 σ 2

1\ 21 \ − K(Si3 , Dj4 )cos2 X Xj , X4 sin2 S\ i , X3 sin i3 , Dj4 − 2 2 1\ 21 \ Xj , X4 sin2 D\ − K(Di3 , Sj4 )sin2 X i , X3 cos i3 , Sj4 + 2 2 1\ 21 \ 2 \ , X sin X , X sin D , D + K(Di3 , Dj4 )sin2 X i 3 j 4 i3 j4 2 2 where Sij = Xi + Xj , Dij = Xi − Xj , σ2 is the permutations group of the set [ {1, 2} and X, Y is the angle between the vectors X and Y . From Proposition 3.7.1 it results that if |K(π)| ≤ C for any π ⊂ Tx M then |ℵπ1 π2 | ≤ 34 C for all the oriented planes π1 , π2 ⊂ Tx M . This result is improved in the following Proposition 3.7.2. Let C such that |K(π)| ≤ C for all the planes π ⊂ Tx M . If α ∈ −1, 12 and K(π) ∈ [αC, C] or K(π) ∈ [−C, αC] then |ℵπ1 π2 | ≤ C for all oriented planes π1 , π2 ⊂ Tx M . Proof. The orthonormal bases {X1 , X2 }, {X3 , X4 } of the planes π1 , π2 can be chosen so that g(X1 , X3 ) = 0, g(X2 , X4 ) = 0 and then from Proposition 3.7.1 and by using an elementary argument we obtain the result. Generally we cannot assert that if |K(π)| ≤ C then |ℵπ1 π2 | ≤ C. Indeed, we have Proposition 3.7.3. Let M be a Sasaki manifold and α ∈ ( 12 , 1]. There are two oriented planes π1 , π2 ⊂ Tx M so that |ℵπ1 π2 | > C provided that for any plane π ∈ Tx M one of the following conditions is fulfilled: 2 a) K(π) ∈ [−αC, C] and C ∈ [1, 2−α ); 1 2 b) K(π) ∈ [−C, αC] and C ∈ [ α , α ). Proof. Consider the unitary vectors X, Y ∈ Tx M . Then X = aξ + bT1 , Y = cξ + dT2 , where a2 + b2 = 1, c2 + d2 = 1, T1 ⊥ ξ, T2 ⊥ ξ and kT1 k = 1, kT2 k = 1. Under the supposition X ⊥ Y , from Proposition 3.2.1, a) it follows K(X, Y ) = 2 − b2 − d2 + b2 d2 1 − g(T1 , T2 )2 K(T1 , T2 ) and then K(T1 , T2 ) = −αC. But if {ei , ei∗ = F ei , ξ}i∈1,n is a F -basis then we have R(ei , ei∗ , ej , ej ∗ ) = 2 − K(ei , ej ) − K(ei∗ , ej ∗ ) for i 6= j, therefore in the case a) we deduce R(T1 , F T1 , T2 , F T2 ) ≥ 2 + αC − C > C

(3.58)

122 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Under the hypothesis from b), the same argument shows that R(T1 , F T1 , T2 , F T2 ) ≥ 2 − αC + C > C

(3.59)

Now, consider the oriented planes π1 , π2 spanned by {T1 , F T1 } and {T2 , F T2 } respectively. From the formula (3.57) we deduce ℵπ1 π2 = R(T1 , F T1 , T2 , F T2 ) and then the assertion follows from (3.58) and (3.59). Let M be a manifold with the Sasaki structure (F, ξ, η, g) and x a point in M . We assume here that any F -plane of M is canonically oriented, i.e. it is oriented by a basis {X, F X}. The restriction of the bisectional curvature ℵπ1 π2 to F -planes is called the F -bisectional curvature relative to the F -planes π1 and π2 . For the nonzero vectors X ∈ π1 , Y ∈ π2 we have π1 = hX, F Xi, π2 = hY, F Y i and because ℵπ1 π2 does not depend on the choice of the bases in the F -planes π1 , π2 , we set H(X, Y ) = ℵπ1 π2 . From (3.57) it follows that the F -bisectional curvature relative to the F -planes π1 , π2 is given by H(X, Y ) =

R(X, F X, Y, F Y ) 2

2

kXk kY k − g(X, Y )2

(3.60)

Proposition 3.7.4. For any nonzero vectors X, Y ∈ Tx M , orthogonal to the Reeb vector field of the Sasaki manifold M we have 1

{R(X, Y, X, Y ) + R(X, F Y, X, F Y ) kXk kY k − g(X, Y )2 h io 2 2 + 2 g(X, Y )2 − kXk kY k + g(X, F Y )2

H(X, Y ) =

2

2

Proof. It results by a straightforward computation from the formula (3.60) and from Proposition 3.2.1. Proposition 3.7.5. Let π1 , π2 be two canonically oriented F -planes of Tx M in the Sasaki manifold M . If X ∈ π1 , Y ∈ π2 are non zero vectors then [ \ H(X, Y ) = ℵπF π sin2 X, Y + ℵπ∗ F π∗ sin2 X, FY where π, π ∗ are the planes spanned by {X, Y } and {X, F Y }, respectively. Proof. From (3.60) and from the first Bianchi identity we obtain R(X, F X, Y, F Y ) = R(X, Y, F X, F Y ) − R(X, F Y, F X, Y ) and by simple computation the desired formula follows. From Proposition 3.7.5 it follows that the F -bisectional curvature is completely determined by the bisectional curvatures of the type ℵπF π . Moreover, the bisectional curvatures determine the F -bisectional curvature of the Sasaki manifold in the following sense

3.7. BISECTIONAL CURVATURE OF SASAKI MANIFOLDS

123

Proposition 3.7.6. Let M be a Sasaki manifold. If ℵπF π = c = constant for any oriented plane π ⊂ Tx M which is not a F -plane, then for all non zero vectors X, Y ∈ Tx M we have a) H(X, Y ) = c 2 − cos π\ 1 , π2 ,where π1 = hX, F Xi, π2 = hY, F Y i; b) M has constant F -sectional curvature at the point x. Proof. The angle π\ 1 , π2 between the planes π1 = hX1 , X2 i, π2 = hY1 , Y2 i is given by the formula g(X1 , Y1 )g(X2 , Y2 ) − g(X2 , Y1 )g(X1 , Y2 ) q cos π\ 1 , π2 = q 2 2 2 2 kX1 k kX2 k − g(X1 , X2 )2 kY1 k kY2 k − g(Y1 , Y2 )2 and then the assertions a) and b) follow from (3.60). From Proposition 3.7.6 we deduce that if the F -bisectional curvature of the Sasaki manifold M relative to the planes pi, F π does not depend on the oriented plane π ∈ Tx M then M is a Sasaki space form.

3.7.2

F -bisectional curvature of order q

Consider the Sasaki manifold M of dimension 2n+1 and let S = {X1 , . . . .Xq } be an orthonormal system of vectors tangent to M at the point x, whose vectors are orthogonal to the Reeb vector field of M . Evidently, the system F S = {F X1 , . . . , F Xq } has the same properties, that is it is orthonormal and its vectors are orthogonal to ξ. S is called a F -orthonormal q-system at x ∈ M if S ∪ F S is orthonormal. Such kind of systems can exist only for q ≤ n and a F -orthonormal n-system generates in a obvious manner a F -basis. In Section 6.2 we will indicate a construction for these systems on a class of Sasaki manifolds (see Proposition 6.2.4). For any X ∈ Tx M we consider the scalar Hq (X, S) =

q X

H(X, Xi )

i=1

From Proposition 3.7.4 it follows easily the elementary properties of Hq , namely Proposition 3.7.7. Let X ∈ Tx M be an unitary vector and let S be a F orthonormal q-system at the point x ∈ M . If S 0 ⊂ Tx M is an orthonormal system such that hS 0 i = hSi then: a) S 0 is a F -orthonormal q-system; b) H(X, S 0 ) = H(X, S). From the above Proposition it results that Hq (X, S) does not depend on the F -orthonormal q-system S, but only on the subspace spanned by S in Tx M . Hq (X, S) is called the F -bisectional curvature of order q at the point x ∈ M and we remark that for q = 1 it is just the F -bisectional curvature. We say that the manifold M has lower bounded F -bisectional curvature of order q at x ∈ M if there exists k0 ∈ R such that Hq (X, S) ≥ k0 for any unitary

124 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS tangent vector X ∈ Tx M and for any F -orthonormal q-system S at the point x. When k0 > 0 we say that M has positive F -bisectional curvature of order q. Theorem 3.7.8. Let M be a complete and connected Sasaki manifold with lower bounded F -bisectional curvature of order q, Hq ≥ k0 > 0. Then: a) M is compact; q 2q b) the diameter of the manifold M is ≤ π 2q+k ; 0 c) M has finite fundamental group. Proof. For two arbitrary points x, y in M we consider the geodesic γ for which the minimum of the distance between x and y is attained and assume that γ is parametrized by its arc length. For a, b such that γ(a) = x, γ(b) = y, the arc γ|(a,b) has no conjugate points5 . Let S = {e1 , . . . , eq } ⊂ Tx M be a F -orthonormal q-system, orthogonal to the geodesic γ at the point γ(a). If Ei is obtained by the parallel translation of ei along the curve γ then we have ∇γ 0 Ei = 0, Ei (γ(t)) = ei for i ∈ 1, q. Hence Ei is orthogonal to γ at the point γ(a). Analogous equalities hold for the vector ˜i , obtained from F ei by parallel translation along γ. From Theorem fields E ˜i = F ei , 1.5.3 it follows ∇γ 0 (F Ei ) = 0 and because F Ei (γ(t)) = F ei we have E ˜ which proves that S = {E1 , . . . , Eq } is a F -orthonormal q-system at any point of the geodesic γ and it is orthogonal to γ at γ(a). Let Φ : [a, b] −→ R be a non zero differentiable function such that Φ(a) = Φ(c) = 0 for some c ∈ (a, b). Then by the formulas Xi (t) = Φ(t)Ei (γ(t)), ˜i (γ(t)) we define the vector fields Xi , Yi along γ|[a,b] . These are Yi (t) = Φ(t)E orthogonal to the curve γ at the point γ(a) and Xi (a) = Xi (c) = 0, Xi0 = Φ0 Ei ; ˜i . The index form I of the geodesic γ along the arc Yi (a) = Yi (c) = 0, Yi0 = Φ0 E γ|[a,b] is given by (see for instance [KN69], vol. II, Theorems 5.4 and 5.5, pg. 81 and Corollary 3.2, pg. 74) Iab (X, Y

Z )=

b

[g(X 0 , Y 0 ) − R(X, γ 0 , Y, γ 0 )] dt =

a 0

0

Z

= g(X , Y )(b) − g(X , Y )(a) −

(3.61)

b 00

0

0

[g(X , Y ) + R(X, γ , Y, γ )] dt a

5 The points x, y of the geodesic γ are conjugate each other or simply conjugate if there exists a vector field X along γ such that X(x) = X(y) = 0 and

∇2γ 0 X + ∇γ 0 (T (X, γ 0 )) + R(X, γ 0 )γ 0 = 0 The vector field X along the geodesic γ satisfying the above condition is called a Jacobi field and the condition itself is said the Jacobi equation. We have the following Theorem. Let γ : [a, b] −→ M be a geodesic. If for some c ∈ (a, b) the points γ(a) and γ(b) are conjugate then the length of the geodesic γ is greater than the distance between γ(a) and γ(b), i. e. γ is a not minimizing geodesic joining γ(a) and γ(b). This result is proved in [KN69], vol. II, pg. 87.

3.8. EINSTEIN AND η-EINSTEIN MANIFOLDS

125

and then we obtain q X

c

Z

= a

02

2qΦ − Φ

2

i=1 q X

[Iac (Xi , Xi ) + Iac (Yi , Yi )] = ) 0

0

0

(3.62)

0

[R(Ei , γ , Ei , γ ) + (F Ei , γ , F Ei , γ )] ds

i=1

But γ|[a,c] has no conjugate points and taking into account Proposition 3.7.4, from (3.62) we deduce Z cn h io ˜ + 2q ds ≤ 0< 2qΦ02 − Φ2 Hq (γ 0 , S) (3.63) Zac 02 2 2qΦ − (k0 + 2q)Φ ds ≤ a

The function Φ0 (s) = sin π s−a c−a satisfies the conditions imposed on Φ and by 2q using it in the inequalities (3.63) we get (c − a)2 < 2q+k π 2 . Since c is arbitrary 0 q 2q in the interval (a, b) we have b − a < π 2q+k . But γ is parametrized by its 0 arc length and then r 2q distance(x, y) = distance(γ(a), γ(b)) ≤ π 2q + k0 hence b) is proved and in addition M is bounded. The manifold M is also complete and therefore M is compact according to a classical theorem which connects the completeness of a manifold with its topology6 . c) By Proposition 1.5.11 the universal covering space M ∗ is complete. On the other hand, for an unitary vector field X ∗ ∈ X (M ∗ ) and for any F -orthonormal q-system S of the manifold M , the F -bisectional curvatures of order q of M and M ∗ are related by Hq∗ (X ∗ , S ∗ ) < Hq (π∗ X ∗ , S) hence M ∗ has lower bounded q-bisectional curvature Hq∗ ≥ k0 > 0. Therefore M ∗ satisfies the hypothesis of the Theorem and by a) it follows that M ∗ is compact and then the fundamental group of M is finite.

3.8 3.8.1

Einstein and η-Einstein manifolds General properties and examples

The 2n + 1-dimensional manifold M with the almost contact Riemannian structure (F, ξ, η, g) is called η-Einstein if there are a, b ∈ F(M ) such that the 6 Theorem. For a Riemannian manifold M the following assertions are equivalent: i) M is a complete Riemannian manifold; ii) M is a complete metric space with respect to the function distance d (defined as the infimum of the length of arcs which join two given points); iii) any bounded subset (relative to d) of M is relative compact. For the proof of this theorem see for instance [KN69], vol. I, pg. 172–175.

126 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Ricci tensor satisfies the condition Ric = ag + bη ⊗ η

(3.64)

We say that a and b are the Einstein functions of M . In the case when in (3.64) we have b = 0 the almost contact Riemannian manifold M is called an Einstein manifold. For η-Einstein Sasaki and Kenmotsu manifolds the Einstein functions a and b have the following properties: Proposition 3.8.1. a) If M is a connected η-Einstein Sasaki manifold of dimension 2n + 1 ≥ 5 then the Einstein functions a and b are constant, a + b = 2n and M has constant scalar curvature ρ = 2n(a + 1). b) If M is a η-Einstein Kenmotsu manifold then a + b = −2n, ρ = 2n(a − 1). Proof. a) By using the equality (3.64) and Proposition 3.2.1, e) we obtain a+b = 2n and the desired expression for the scalar curvature ρ. On the other hand, by putting i = l in the second Bianchi identity and summing up on i we obtain i ∇m Ricjk − ∇k Ricjm + ∇i Rjmk =0

Multiplying the last equality by g jk and then summing up by j and k and taking into account the formula (3.64) we deduce Xρ = 2[Xa − (ξb)η(X)]

(3.65)

for any X ∈ X (M ). But ξb = −ξa and from (3.65) it follows Xa+(ξa)η(X) = 0. By setting X = ξ we have ξa = 0 and then Xa = 0 for any X ∈ X (M ), hence a = constant. b) The first equality b) and the expression of ρ follow from Proposition 3.3.1, f) and from (3.64). The notion of η-Einstein metric (i. e. a metric satisfying 3.64)) was introduced by M. Okumura, [Ok62]. Remark 3.8.2. From the proof of the assertion a) in Proposition 3.8.1 we deduce that it is also valid for K-contact manifolds. Since on Sasaki manifolds the functions a and b are constant, sometimes we call them the Einstein constants of such a manifold. But we have to remark that it is no more true for manifolds of dimension 3. In this case we have the following Proposition 3.8.3. Any Sasaki manifold M of dimension 3 is η-Einstein and a + b = 2. The Einstein functions a and b are constant if and only if M is a Sasaki space form with F -sectional curvature equal to a − 1.


127

Proof. Let {e, F e, ξ} be a F -basis and consider the vectors X = a0 e + b0 F e + c0 ξ, Y = a00 e + b00 F e + c00 ξ. A simple computation using Proposition 3.2.1 shows that Ric(X, Y ) = R(X, e, Y, e) + R(X, F e, Y, F e) + R(X, ξ, Y, ξ) = = (a0 a00 + b0 b00 )(α + 1) + 2c0 c00 = (1 + α)g(X, Y ) + (1 − α)η(X)η(Y ) where α = R(e, F e, e, F e). Now, the assertions from the Proposition are obvious. Examples. 1. The Sasaki manifold R2n+1 with the standard structure defined in Example 1, Section 1.5 is η-Einstein. Indeed, simple computation shows that the Christoffel coefficients are ∗ ∗ 1 1 1 δkj y i , Γkij = − δik y j + δkj y i , Γki∗ ∆ = −Γki∆ = − δik , 2 2 2 1 i j 1 i α = y y − δij , Γ∆ i∗ ∆ = − y , Γβγ = 0 otherwise 2 2

Γkij ∗ = Γ∆ ij ∗

where ∆ = 2n + 1. The components of the Ricci tensor are Ricij = Rici∗ j ∗

n 1 ny j y k − δjk , Ricij ∗ = 0, Rici∆ = − y i , 2 2 1 n = − δij , Rici∗ ∆ = 0, Ric∆∆ = 2 2

and we have Ric = −2g + 2(n + 1)η ⊗ η 2. From Propositions 3.5.2 and 3.6.2 it follows that the Sasaki and Kenmotsu space forms are η-Einstein manifolds with b 6= 0.

3.8.2

Einstein Sasaki manifolds

We observe that if the almost contact Riemannian manifold M is Einstein ρ then a = 2n+1 and taking into account Propositions 3.2.1 and 3.8.1, we have Proposition 3.8.4. The scalar curvature of an Einstein Sasaki manifold is positive equal to 2n(2n + 1) and its Ricci tensor is given by Ric = 2ng. From this result one deduces that for any unitary vector field X on the Sasaki manifold M we have Ric(X, X) = 2n and by applying Theorem 3.2.10 we obtain the following Theorem 3.8.5. Any complete and connected Einstein Sasaki manifold is compact, has diameter ≤ π and its fundamental group is finite. Proposition 3.8.6. [Ha91] There are no Einstein hypersurfaces in Sasaki space forms with F -sectional curvature 6= 1.

128 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS For a technical proof see [Ha91]. Theorem 3.8.7. The 2n + 1-dimensional compact Sasaki manifold M is Einstein if and only if the orbit space Mξ with the induced metric is an Einstein K¨ ahler manifold with scalar curvature equal to 4n(n + 1). Proof. Denoting by g˜ the Hermitian metric induced 0n Mξ by the metric g of the manifold M (see Section 2.6), a straightforward computation shows that Ricg (X H , Y H ) = Ricg˜ (X, Y ) − 2g(X H , Y H )

(3.66)

for any X, Y ∈ X (Mξ ). It follows that M is an Einstein manifold if and only if Mξ is Einstein. The scalar curvature of the orbit space can be easily computed taking into account the equality (3.66) and Proposition 3.8.4. From Theorem 3.2.8 it follows Proposition 3.8.8. Any locally symmetric Einstein K-contact manifold is Einstein Sasaki. Another characterization of Einstein Sasaki manifolds, depending on the Ricci tensor of the symplectic cone is the following Theorem 3.8.9. [BG00] The Sasaki manifold M is Einstein if and only if the metric G = dt2 + t2 g of its symplectic cone C(M ) is Ricci-flat, that is the Ricci tensor of C(M ) vanishes. Proof. In a local chart, the components of the metric G have the following expressions depending on the components of the metric g of the manifold M Gij = t2 gij ,

Gi∆ = 0,

G∆∆ = 1,

where i, j, k ∈ 1, 2n + 1 and ∆ = 2n + 2. Then from the formulas (1.59) we ¯ i of the Levi-Civita connection of the metric G deduce the coefficients Γ jk ¯ hij = Γhij , Γ

¯∆ Γ ij = −tgij ,

¯ h∆j = 1 δjh , Γ t

¯∆ Γ ∆j = 0

Now, by simple computation we obtain the components of the Riemann-Christo¯ of the symplectic cone ffel tensor R ¯ ijkl = t2 (Rijkl + gki glj − gkj gli ), R

¯ ∆ jkl = 0, R

¯ ∆j∆l = 0 R

We get ¯ Y¯ ) = t2 [Ric(X, Y ) − 2ng(X, Y )] RicC (X, and then our assertion follows easily. Remark 3.8.10. In the proof of Theorem 3.8.9 only the fact that the manifold M has the Einstein constant a = 2n is used and then from Proposition 3.1.10 it follows that this result is still valid for K-contact manifolds.


129

C. P. Boyer and K. Galicki proved a much stronger version of this result, namely Theorem 3.8.11. [BG01] Any compact Einstein K-contact manifold is Einstein Sasaki. Proof. Although the authors’ proof is elegant, we present here that one given by V. Apostolov, T. Draghici and A. Moroianu, [ADM04] which seems to be more elementary. Let M be a K-contact manifold of dimension 2n + 1. According to Theorem 1.4.4 its symplectic cone C(M ) is almost Kähler and then the following Weitzenb¨ ock type formula is valid 00 ∆C ρC∗ − ρC = −4δ Jδ C J RicC + 00 2

00 2 (3.67) + 8δ C rC∗ , ∇C ΩC + 2 RicC − 8 RC − 2

2 − ∇C∗ ∇C ΩC − |Ψ| + 4 hr, Ψi − 4 r, ∇C∗ ∇C ΩC where ρC , ρC∗ are the and the ∗-scalar curvature of the cone

scalar curvature 00 C(M ), i. e. ρC∗ = 2 RC (Ω), Ω , RicC is the J-anti-invariant component of the Ricci tensor RicC , r is the (1, 1)-form associate to the J-invariant component of 00 RicC , rC∗ = RC (ΩC ), RC is a component of the Riemann-Christoffel curvature tensor RC of the cone C(M ) given by 00

8RC (X, Y, Z, U ) = RC (X, Y, Z, U ) − RC (CX, CY, Z, U )− − RC (X, Y, CZ, CU ) + RC (CX, CY, JZ, JU ) + RC (CX, Y, CZ, U )+ + RC (CX, Y, Z, CU ) + RC (X, CY, CZ, U ) for any X, Y, Z, U ∈ X (C(M )) and Ψ is the semi-positive form of type (1,1) given by Ψ(X, Y ) = ∇CJX ΩC , ∇CY ΩC , where h, i is the classical scalar product on the space F p (C(M )) of p-forms. The formula (3.67) is valid for any almost Kähler manifold (see for instance [Ru99b], Proposition 1). The meaning of the notations is the same, with the difference that in the cited paper there is no index C, which, in our proof, tell that the operators are considered on the symplectic cone C(M ) of the manifold M . But M is an Einstein manifold, hence according to Theorem 3.8.9 we have RicC = 0, ρC = 0 and then the formula (3.67) becomes 00 2 2

2 ∆C ρC∗ = 8δ C rC∗ , ∇C ΩC − 8 RC − ∇C∗ ∇C ΩC − |Ψ| (3.68) By a simple computation, from the formulas (1.59) we obtain the following relations between the curvature tensors R and RC of the manifold M and of C(M ), respectively ∂ , X = 0, RC (X, Y )Z = R(X, Y )Z + g(X, Z)Y − g(Y, Z)X (3.69) RC ∂t

130 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS valid for any X, Y, Z ∈ X (M ). Also we have (see (1.60) and (1.61)) ∇C∂ ΩC = 0, ∂t

∇CX ΩC = t2 ω + tdt ∧ θX

(3.70)

where ω ∈ F 2 (M ), θX ∈ F 1 (M ). Then we obtain r C∗

∂ ,X ∂t

= 0,

rC∗ (X, Y ) =

n X

Xi Xi G RC ,J X, Y t t i=1

hence we have ρC∗ = t12 f forsome function f ∈ F(M ). But from the formula (3.70) we deduce rC∗ , ∇C ΩC = t12 α for some 1-form α ∈ F 1 (M ) and then the equality (3.68) becomes 00 2 1 1 C C C ∇C∗ ∇C ΩC 2 − |Ψ|2 f − 8δ f = −8 − (3.71) ∆ R t2 t2 It remains to compute the two terms form the left hand side of the equality (3.71). For this purpose we denote by {ei }i∈1,2n+1 a local orthonormal basis in X (M ). We have ∂ ∂ C k k C ∂ k δ (t α) = t α ∇ ∂ − t α + ∂t ∂t ∂t ∂t 2n+1 Xh ei ei k ei i − t α = + tk α ∇Cei t t t t i=1 2n+1 X ∂ = tk−2 α ∇ei ei − t − ei (α(ei )) = tk−2 δα, ∂t i=1 h ei ei k i ∂ ∂ k k C k C ei (t f ) − t f − (t f ) = ∆ (t f ) = ∇ ei t t t t ∂t ∂t 2n+1 X 1 ∂ k k−2 = ∇ei ei − t (t f ) − t ei (ei f ) − k(k − 1)tk−2 f = 2 t ∂t i=1 = −tk−2 (∆f − k(2n + k)f ) where δ, ∆ are the codifferential and the Laplace operator on M respectively. By using these relations, the equality (3.71) becomes 00 2 2 1 C 2 [∆f + 2(2n − 2)f − 8δα] = −8 R − ∇C∗ ∇C ΩC − |Ψ| 4 t

(3.72)

The manifold M is compact, hence for any t = constant the level surface Mt = M × t of the cone C(M ) is also compact and we can integrate the equality (3.72) over Mt . From the Green formula we deduce Z Z ∆f = 0, δα = 0 Mt

Mt


131

so that from (3.72) we obtain Z Z 2(2n − 2) C 00 2 C∗ C C 2 2 f= −8 R − ∇ ∇ Ω − |Ψ| (3.73) t4 Mt Mt 2 But ρ∗ = ρ∗ − ρ = ∇C ΩC , hence ρ∗ = t12 f ≥ 0 and then from the formula (3.73) we deduce f ≡ 0 and ∇C ΩC = 0. This last equality shows that C(M ) is a Kähler manifold, so that by Theorem 1.5.7 the manifold M is Sasakian. We remark that this result cannot be improved, that is the answer to the question ”is any compact Einstein contact Riemannian manifold an Einstein Sasaki manifold?” is generally negative. Thus, by modifying slowly the example of pg. 68 in [Bl02], we obtain the following construction: In R3 the 1-form η = cos t dx + sin t dy is invariant by translations with 2π on the direction of each coordinate, so that it is a 1-form on the torus T 3 . Moreover we have η ∧ dη = dx ∧ dy ∧ dz, hence η defines a contact structure on T 3 . With this structure we associate the almost contact Riemannian structure (F, ξ, η, g), where ∂ ∂ + sin t , ∂x ∂y  0 0 0 0 F :  − sin t cos t

ξ = cos t

g : gij = δij ,  sin t − cos t  0

Taking into account the expression of the metric g, it follows that the torus T 3 with the above structure is flat, hence it is an Einstein manifold. But ∇X ξ = 0 and then from Proposition 1.4.1 we deduce that this structure cannot be Kcontact, hence it cannot be Sasakian. Another characterization of Einstein Sasaki manifolds is the following Theorem 3.8.12. Let M be a compact connected simply connected Einstein Sasaki manifold of dimension 2n + 1. If M has strictly positive sectional curvature then M is isometric to the unit sphere S 2n+1 . Proof. The manifold M is compact and then the set {H(X) : x ∈ M, X ∈ Tx M, kXk = 1} has a maximum, that is there exist x0 ∈ M and an unitary vector X0 ∈ Tx0 M so that H(X0 ) is the maximum value of the F -sectional curvature. From Proposition 3.2.1, c) it results that the 2-form ω given by ω(Y, Z) = g(R(X0 , F X0 )Y, Z), satisfies the condition ω(F Y, Z) = −ω(Y, F Z). Then we can construct a F -basis {ei , ei∗ , ξ}i∈1,n with e1 = X0 and so that the non zero components of the form ω are ωii∗ and ωi∗ i = −ωii∗ . A hard computation proves that Ric(X0 , X0 ) = =

n X i=1 n X i=1

[R (X0 , ei , X0 , ei ) + R (X0 , ei∗ , X0 , ei∗ )] + 1 = [K(X0 , ei ) + K(X0 , ei∗ )] + 1

132 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS But M is Einstein and then taking into account Proposition 3.8.4 in the above equalities, we deduce 2n = H(X0 ) +

n X

[K(X0 , ei ) + K(X0 , ei∗ )] + 1

(3.74)

i=2

On the other hand we have R (X0 , F X0 , ei , ei∗ ) = − [K(X0 , ei ) + K(X0 , ei∗ ] + 2 and then from the equality (3.74) we obtain n X

R (X0 , F X0 , ei , ei∗ ) = H(X0 ) − 1

(3.75)

i=2

But 2R (X0 , F X0 , ei , ei∗ ) + H(X0 ) − 1 = 0 and then from (3.75) it follows (n + 1) [1 − H(X0 )] = 0, that is H(X0 ) = 1. In our case the formula (3.38) becomes Z V (S(X)) = H(X)dvX S(x)

and by applying the Lagrange formula in this integral we obtain H(X) = 1 for any X ⊥ ξ. Then from the formulas (3.36) and (3.37) it results that M has constant sectional curvature equal to 1 and the manifold M is isometric to S 2n+1 since any two connected, simply connected and complete Riemannian manifolds with the same constant sectional curvature are isometric (see for instance [KN69], vol. I, pg. 265). Remark 3.8.13. This result is still valid if we replace the hypothesis”M has strictly positive sectional curvature” by the weaker one, namely ”K(X, Y ) + K(X, F Y ) > 0 for any orthonormal system {X, Y } of vectors orthogonal to ξ”. By attending a method used by C. P. Boyer and K. Galicki, [BG00] we will associate to two compact, regular and simply connected Einstein Sasaki manifolds M1 , M2 of dimensions 2n1 + 1 and 2n2 + 1 respectively, another Einstein Sasaki manifold of dimension 2(n1 + n2 ) + 1. According to Theorem 3.8.7 the orbit spaces Mξ1 , Mξ2 of the manifolds M1 and M2 respectively, with the induced metrics g˜1 and g˜2 are Einstein Kähler with Einstein constants equal to 2n1 and 2n2 , respectively. The product Mξ1 × Mξ2 is a K¨ ahler manifold with the metric g˜1 + g˜2 , but it is generally not Einstein K¨ ahler. More precisely, we have the following Proposition 3.8.14. Let M1 and M2 be two compact regular simply connected Einstein Sasaki manifolds of dimensions 2n1 + 1 and 2n2 + 1, respectively. The product manifold Mξ1 × Mξ2 of theirs orbit spaces is Einstein K¨ ahler with the metric g˜1 + g˜2 if and only if n1 = n2 .


133

Proof. From the complex geometry we know that the product of two Einstein Kähler manifolds is Einstein K¨ ahler for the sum metric if and only if these manifolds have the same Einstein constant (see for instance [K87], Proposition 1.4, pg. 99–100). Now, it suffices to apply Proposition 3.8.4. However we can construct other metrics for which Mξ1 × Mξ2 is Einstein Sasaki, but without imposing conditions on the dimensions of the manifolds M1 , M2 . Indeed, it is easy to verify that for the metric g˜ =

n2 + 1 n1 + 1 g˜1 + g˜2 n1 + n2 + 1 n1 + n2 + 1

the product Mξ1 ×Mξ2 is a Einstein Kähler manifold with Einstein constant equal to 2(n1 + n2 + 1) and has scalar curvature 4(n1 + n2 )(n1 + n2 + 1). Moreover, Mξ1 × Mξ2 is a Hodge space. But the natural projections π1 : M1 −→ Mξ1 , π2 : M2 −→ Mξ2 define two principal bundles with the structure group S1 namely, the Boothby-Wang bundles of the Sasaki manifolds M1 and M2 , respectively (see Theorem 2.6.3). It follows that π1 × π2 : M1 × M2 −→ Mξ1 × M22 defines a principal bundle with the torus S1 × S1 as structure group. For a given groups homomorphism S1 −→ S1 × S1 , we can consider the principal bundle M1 ×M2 /S1 over Mξ1 ×Mξ2 and with the structure group S1 . The details of such a construction for the general case are given in [KN69], vol. I, pg. 57. Intuitively speaking, in this construction we identify the orbits of the Reeb vector fields ξ1 , ξ2 of the manifolds M1 and M2 , respectively. Following an argument similar to the one used in the proof of Theorem 2.6.4, but starting from the K¨ ahler structure on Mξ1 × Mξ2 , we can construct on the 1 manifold M1 × M2 /S a contact Riemannian structure with the property that Mξ1 × Mξ2 is the orbit space with the induced metric g˜. Moreover, the manifold M1 × M2 /S1 , denoted by M1 ∗ M2 and called the join of M1 and M2 , is regular and compact, so that by applying Theorem 3.8.7 we can state the following Theorem 3.8.15. [BG00] Let M1 and M2 be two compact simply connected regular Einstein Sasaki manifolds of dimensions 2n1 +1 and 2n2 +1, respectively. Then M1 ∗ M2 is a compact simply connected regular Einstein Sasaki manifold of dimension 2(n1 + n2 ) + 1. The operator ∗ defines an algebraic structure on the set S of compact, regular and simply connected Einstein Sasaki manifolds. Indeed, we can easily verify the following Theorem 3.8.16. [BG00] The set S endowed with the operation ∗ is an associative and commutative topological monoid (up to isomorphisms). By using these results, C. P. Boyer and K. Galicki, [BG00] obtain remarkable classes of Einstein Sasaki manifolds. Thus, they construct: a) an infinity of Einstein Sasaki manifolds M with the injectivity radius arbitrary small for any dim M ≥ 7;

134 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS b) Einstein Sasaki manifolds of dimension ≥ 7 with arbitrary second Betti number. In particular, the authors prove that in any dimension ≥ 7 there exist Einstein Sasaki manifolds M with the property that for any k < 0 there are no metrics with sectional curvature ≥ k; c) for any dimension ≥ 5 there are compact and simply connected manifolds admitting continuous families of Einstein Sasaki metrics. Even this simple enumeration shows that the problem of classification for Einstein Sasaki manifolds is very complicated, it depending, of course, on the classification of Einstein Kähler manifolds. In the case of compact, regular and simply connected Einstein Sasaki manifolds of dimensions 5 and 7, a classification is given in [BG00], Theorem 3.3 and Propositions 4.4, 4.5.

3.8.3

η-Einstein Sasaki manifolds

Taking into account the definition of the η-Einstein tensor, from Proposition 3.8.1 it follows that a Sasaki manifold is η-Einstein if and only if its η-Einstein tensor S η vanishes and then from Theorem 3.5.6 it follows Theorem 3.8.17. Any connected η-Einstein Sasaki manifold with vanishing C-Bochner curvature tensor is a Sasaki space form and conversely. Proposition 3.8.18. Let M be a 2n + 1-dimensional η-Einstein Sasaki mani˜ η˜, g˜) of fold with Einstein constants a, b. The D-homothetic deformation (F˜ , ξ, the Sasaki structure of M defines a η-Einstein Sasaki structure with Einstein , ˜b = 2n − a+2−2α , where α is the constant of the Dconstants a ˜ = a+2−2α α α ˜ η˜, g˜) homothetic deformation. For α = a+2 and a > −2 the structure (F˜ , ξ, 2(n+1)

is Einstein Sasaki. ˜ be the Levi-Civita connections of the metrics g and g˜, respecProof. Let ∇, ∇ tively. Taking into account the last relation (1.68), from (1.19) we obtain ˜ X Y, Z = αg(∇X Y, Z) + +α(α − 1) [η(Y )Ω(X, Z)+ g˜ ∇ (3.76) +η(X)Ω(Y, Z) + η(Z)g(∇X Y, ξ)] Now, if in (3.76) we replace the metric g by its expression depending on g˜ then we obtain ˜ X Y = ∇X Y − (α − 1)[η(Y )F Z + η(X)F Y ] ∇ (3.77) n o Xi X If {Xi , Xi∗ , ξ} is a F -basis then we observe that √ , √iα∗ , αξ is an orthonormal α basis for the metric g˜ and using (3.77) in the definition of the Ricci tensor, by a straightforward computation we obtain the following relation between Ricci tensors corresponding to these two metrics [Ta68] Ricg˜ = Ricg − 2(α − 1)g + (α − 1)(2n + 2 + 2nα)η ⊗ η Now, our assertion follows easily.


135

The following result, similar to the one given in Theorem 3.8.7, holds. Theorem 3.8.19. Let M compact regular η-Einstein Sasaki manifold with Einstein constants a and b. The orbit space Mξ with the induced metric is an Einstein K¨ ahler manifold with Einstein constant a + 2. Proof. This assertion follows from the formula (3.66), taking into account that if X ∈ X (Mξ ) then η(X H ) = 0. In [BGM04] the following more general result is proved: Theorem 3.8.20. If the η-Einstein Sasaki manifold M is quasi-regular then its orbit space is a Einstein Hodge orbifold. A result similar to that one from Theorem 3.8.11 holds in the case when the K-contact manifold is η-Einstein, namely Theorem 3.8.21. [BG01] Let M be a compact η-Einstein K-contact manifold. a+2 a) If a > −2 then M is Sasakian. Moreover, for α = 2(n+1) the deformed metric αg + α(α − 1)η ⊗ η defines on M an Einstein Sasaki structure. b) For a = −2 the manifold M is η-Einstein Sasaki. c) If a < −2 then M is a quasi-regular. Proof. a) For a > −2 we have α > 0 and g˜ = αg +α(α −1)η ⊗η is a Riemannian metric. A simple computation shows that the manifold M with the metric g˜ is Einstein, and (αη, g˜) is a K-contact structure on M . Hence by Theorem 3.8.11 M equipped with this structure is an Einstein Sasaki manifold. But the almost complex structure C on the symplectic cone is not influenced by the change of the metric on M and then it follows that the initial K-contact structure on M is Sasakian and thus M is a η-Einstein Sasaki manifold. b) Denoting by Ricg the Ricci tensor of the metric g, by direct computation we prove the equality RicgD = Ric|D×D + 2g|D×D

(3.78)

where gD is the metric on the contact distribution D (for the definition of this metric see Section 1.3). Then for a = −2 we obtain ρgD = 0 and the almost complex structure C on C(M ) is integrable, hence M is a η-Einstein Sasaki manifold with Einstein constants a = −2 and b = 2n + 2. c) For the proof of this assertion and for other related results see [BG01]. With the notions defined in Section 2.5 we have the following Theorem 3.8.22. [TKi88] If the total space M of the almost contact Riemannian fibered space with isometric fibers is η-Einstein and its fibers are invariant submanifolds then the base space M 0 is an Einstein manifold and the fibers are η-Einstein manifolds whose fundamental 2-forms are harmonic. It is easy to verify that for Lorentz Sasaki manifolds the following result similar to Theorem 3.8.21 is valid.

136 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Theorem 3.8.23. [BGM04] Let M be a compact quasi-regular η-Einstein Sasaki manifold of dimension 2n + 1 and suppose a < −2. Then on M there exists an Einstein Lorentz Sasaki structure so that g 0 = −αg − α(α − 1)η ⊗ η,

ξ 0 = α−1 ξ

where

α=

a+2 . 2(n + 1)

Let M be a manifold of dimension m ≥ 3. A Weyl structure on M is a pair W = ([g], ∇w ), where [g] is a class of conformal Riemannian metrics and ∇w is the unique symmetric connection which preserves [g], that is for any g 0 from the conformal class [g] there exists θ so that 0 0 (∇w X g ) (Y, Z) = −2θ(X)g (Y, Z)

for all X, Y, Z ∈ X (M ). ∇w is called the Weyl connection of the structure W. We see immediately that θ ∈ F 1 (M ) and the relation between this connection and the Levi-Civita connection ∇ of the metric g 0 is expressed under the form 0 ∗ ∇w X Y = ∇X Y + θ(X)Y + θ(Y )X − g (X, Y )θ

(3.79)

where θ∗ is the vector field dual to the form θ, that is g 0 (X, θ∗ ) = θ(X) for any X ∈ X (M ). The Weyl structure W = ([g], ∇w ), also denoted by W = (g, ∇w ), is called an Einstein-Weyl structure if there exists Λ ∈ F(M ) such that w

w

∇ Ric∇ g (X, Y ) + Ricg (Y, X) = Λg(X, Y )

(3.80)

for any X, Y ∈ X (M ). For a complete presentation of Einstein-Weyl structures see for example [CP88]. Theorem 3.8.24. Let M be a manifold of dimension 2n + 1 ≥ 3 with the Sasaki structure (F, ξ, η, g). If M admits an Einstein-Weyl structure of the form W = (g, f η) with f ∈ F(M ) a non constant function, then M is a η-Einstein manifold. Proof. From the equality (3.79) it results that if W is a Weyl structure on the almost contact Riemannian manifold M then the following relation is true w

Ric∇ g (X, Y ) = Ricg (X, Y ) − 2n (∇X θ) Y + (∇Y θ) X+ h i 2 + (2n − 1)θ(X)θ(Y ) + δθ − (2n − 1) kθk g(X, Y ) 2

(3.81)

where kθk = g ij θi θj . From (3.81) and (3.80) we deduce that if the Weyl structure W is Einstein-Weyl then Λ 2 − δθ + (2n − 1) kθk g(X, Y )+ Ricg (X, Y ) = 2 (3.82) 2n − 1 + [(∇X θ) Y + (∇Y θ) X] − (2n − 1)θ(X)θ(Y ) 2

3.9. LOCALLY F -SYMMETRIC SASAKI MANIFOLDS

137

for any X, Y ∈ X (M ). But θ = f η and from Proposition 1.5.5 we have δ(f η) = 0. Then (3.82) becomes Λ 2 Ricg (X, Y ) = + (2n − 1) kηk f 2 g(X, Y )+ 2 (3.83) 2n − 1 2 + [(Xf )η(Y ) + (Y f )η(X)] − (2n − 1)f η(X)η(Y ) 2 Taking Y = ξ and since in a Sasaki manifold we have Ricg (X, ξ) = 2nη(X) (see Proposition 3.2.1, e)), from (3.83) it follows Xf = 0 for any X ⊥ ξ. Now, if we consider the decompositions X = a ˜ξ + X ⊥ , Y = ˜bξ + Y ⊥ with X ⊥ , Y ⊥ ⊥ ξ, then a simple computation proves that (Xf )η(Y ) + (Y f )η(X) = 2(ξf )˜ a˜b = 2(ξf )η(X)η(Y ) Finally we obtain Λ 2 + (2n − 1) kηk f 2 g(X, Y ) + (2n − 1)(ξf − f 2 )η(X)η(Y ) Ricg (X, Y ) = 2 and thus M is η-Einstein. In [BGM04] it is proved that this result is also valid for K-contact manifolds. Example. [BGM04] The Sasaki structure (F, ξ, η, g) defined in Example 6, Section 1.5 on the Heisenberg group Hn (R) is η-Einstein. Moreover, each D˜ ˜ homothetic q deformation (F , ξ, η˜, g˜) with constant α > 0 is also η-Einstein and 2n−1 1 for α = 2(n+1) and f = α tan(t + c), c ∈ R, we obtain the Weyl-Einstein structure W = (˜ g , f η˜). In [Ma00], [Na98] other examples of Einstein-Weyl structures on almost contact manifolds are presented. By using arguments inspired from the proof of Theorem 3.8.24 we also obtain the following two results: Theorem 3.8.25. [BGM04] Let M be a K-contact manifold of dimension ≥ 5. M admits the Einstein-Weyl structure W = (g, kη) for some k ∈ R if and only if M is η-Einstein with Einstein constants a, b < 0. Theorem 3.8.26. [BGM04] Let M be a Sasaki manifold of dimension ≥ 5. M admits simultaneously the Einstein-Weyl structures W + = (g, θ) and W − = (g, −θ) if and only if M is η-Einstein with Einstein constants a, b and b < 0.

3.9

Locally F -symmetric Sasaki manifolds

A natural way to define the notion of locally F -symmetric Sasaki manifold M is to report to the orbit space Mξ . So, we understand by such a manifold a Sasaki manifold whose orbit space is a locally symmetric Hermitian space.

138 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Let M be a manifold with the Kähler structure (J, g). M is called locally (or globally) symmetric Hermitian space if it is a locally (or globally) symmetric manifold with respect to the Levi-Civita connection of the metric g. For the general theory of locally and globally Hermitian spaces see [KN69], vol. II, pg. 259–264. Let M be a Sasaki manifold. If U is an open neighborhood of the point x ∈ M then U is a Sasaki manifold with the induced structure. Moreover, we take U sufficiently small so that it is a regular manifold and denote by Uξ its orbit space and by (JU , gU ) the induced Kähler structure on Uξ (see Theorem 2.6.4, b)). The Sasaki manifold M is called locally F -symmetric if each point x ∈ M has a neighborhood U as above and such that Uξ is a locally symmetric Hermitian manifold with respect to the induced Kähler structure (JU , gU ). From Proposition 3.2.6 and from the formula (3.49) it follows Proposition 3.9.1. The Sasaki manifold M is locally F -symmetric if and only if F 2 [(∇W R) (X, Y )Z] = 0 for any X, Y, Z, W ∈ D. The notion of locally F -symmetric Sasaki manifold was introduced by T. Takahashi, [Ta77]. Examples. 1. From Theorem 3.4.4 and Proposition 3.5.3 it follows that any Sasaki space form is locally F -symmetric. 2. [KM69] Let G be a compact simply connected semi simple Lie group and G0 be a closed connected subgroup. Suppose that the Lie algebra g of G satisfies the conditions: 0 a) g = g0⊕ g1 ⊕ g2 , where g0 is the Lie algebra of G and dim g2 = 1; b) gi , gj ⊂ gi+j + g|i−j| and [g0 , g0 ] = g0 ; c) there exists u ∈ g2 such that [u, [u, X]] = −X for any X ∈ g1 ; d) ad(a)gi = gi , ad(a)u = u for any a ∈ G0 . Remark that the tangent space at 0 = [G0 ] ∈ G/G0 to the homogeneous space G/G0 can be identified with m = g1 ⊕ g2 such that we can define the tensor fields 1 F0 = ad u|m , ξ0 = u, g0 = − B|m 2n where dim G/G0 = 2n + 1 and B : g × g −→ R, defined by B(X, Y ) = trace (ad(X) ◦ ad(Y )), is the Killing form of g. g0 is obviously a Riemannian metric (see for instance [Jo97], pg. 233–235). If η0 is the dual form of the vector field u then (F0 , ξ0 , η0 , g0 ) is a normal almost contact Riemannian structure on G/G0 and it is G-invariant. Moreover, this structure satisfies the condition dη0 (X, Y ) = 21 g0 (X, F0 Y ) and then F = F0 , ξ = 2ξ0 , η = 12 η0 , g = 41 g0 is a G-invariant Sasaki structure on G/G0 . But the orbit space of G/G0 coincides with the symmetric Hermitian space G/H, where H is the subgroup of G


139

whose Lie algebra is g0 ⊕ g2 . It follows that the Sasaki manifold G/G0 is locally F -symmetric. Proposition 3.9.2. The Sasaki manifold M is locally F -symmetric if and only if (∇W R)(X, Y )Z = [g(X, W )g(Y, F Z)− −g(Y, W )g(X, F Z) + g (R(X, Y )W, F Z)] ξ+ + η(X) [g(Z, F W )Y − g(Y, Z)F W − R(Y, F W )Z] + + η(Y ) [−g(Z, F W )X + g(X, Z)F W + R(X, F W )Z] + + η(Z) [g(X, W )F Y − g(Y, W )F X + F R(X, Y )W ]

(3.84)

for any X, Y, Z, W ∈ X (M ). Proof. Assume that the condition (3.84) is fulfilled. Then we deduce F 2 [(∇W R) (X, Y )Z] = 0 for any X, Y, Z, W ∈ D and from Proposition 3.9.1 it follows that M is locally F -symmetric. For the converse, from Proposition 3.9.1 and from (1.1) we get (∇W R)(X, Y )Z = η ((∇W R) (X, Y )Z) ξ = [g (∇W (R(X, Y )Z), ξ) − −g (R(X, Y )∇W Z, ξ) − g (R (∇W X, Y ) Z, ξ) − g (R (X, ∇W Y ) Z, ξ)] ξ for any X, Y, Z, W ∈ D and by using (1.54) we have (∇W R)(X, Y )Z = [W (R(ξ, Z, X, Y )) + R(F W, Z, X, Y )− (3.85) −R (ξ, ∇W Z, X, Y ) − R (ξ, Z, ∇W X, ξ) − R (ξ, Z, X, ∇W Y )] ξ But by Proposition 3.2.1, a) we have R(X, Y )ξ = 0 for X, Y ∈ D. On the other hand ∇W X = g(F W, X)ξ + X D with X D ∈ D for any X, W ∈ D, hence taking into account Proposition 3.2.1, b) we obtain (∇W R) (X, Y )Z = [g(X, F W )g(Y, Z) − g(Y, F W )g(X, Z)− −g(W, F R(X, Y )Z)W + R(F W, Z, X, Y )] ξ From (3.86) and Proposition 3.2.1, d) it follows (∇F 2 W R) F 2 X, F 2 Y F 2 Z = g(X, F W ) [g(Y, Z)η(Y )η(Z)] − − g(Y, F W ) [g(X, Z)η(X)η(Z) − g(W, F R(X, Y )Z)] ξ

(3.86)

(3.87)

for any X, Y, Z, W ∈ X (M ). On the other hand, by using Proposition 3.2.5, a), the formulas (3.24) (the second equality) and (3.25) in the left hand-side of the equality (3.87) we obtain (3.84). The Okumura connection completely characterizes locally F -symmetric Sasaki manifolds, namely we have

140 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS Theorem 3.9.3. [Ok62] A Sasaki manifold is locally F -symmetric if and only o o

o

o

if ∇R= 0, where ∇ is the Okumura connection and R denotes its curvature tensor. Proof. From Proposition 3.2.14 we deduce o o ∇W R (X, Y )Z = (∇W R) (X, Y )Z + A(W )R(X, Y )Z− −R(A(W )X, Y )Z − R(X, A(W )Y )Z − R(X, Y )A(W )Z where A(X)Y = g(X, F Y )ξ − η(X)F Y + η(Y )F X. A straightforward compuo

o

tation shows that ∇W R= 0 if and only if ∇W R given by the above equality satisfies the condition (3.84) from Proposition 3.9.2. Theorem 3.9.4. [BeTV97] Any simply connected Sasaki space form is weakly symmetric. Proof. Let M (c) be a 2n + 1-dimensional Sasaki space form with the Sasaki structure (F, ξ, η, g) and denote by ∇ its Levi-Civita connection. Then the tensor field T of type (1,2) given by T (X, Y ) = g(F X, Y )ξ + η(X)F Y − η(Y )F X defines a homogeneous structure on M . On the other hand, ∇ = ∇−T is a connection on M and ∇g = 0, hence it is Riemannian. Since M has a homogeneous structure, ∇ is complete and simple computation shows that ∇ R = ∇ T . For x ∈ M let X ∈ Tx M be an arbitrary vector and choose a F -basis {Xi , Xi∗ , ξx } in Tx M such that X ∈ hXn∗ , ξx i and define an isometry IX : Tx M −→ Tx M by IX Xi = Xi , IX Xi∗ = −Xi∗ , IX ξx = −ξx . We obtain IX X = −X,

IX ◦ Fx = −Fx ◦ Rx ,

∗ IX ηx = −ηx

and then ∗ IX R x = I X ◦ Rx ,

∗ IX Tx = IX ◦ Tx

∗ ∗ Now, from the definition of the connection ∇ we deduce IX R x = R x , IX Tx = T x , hence we can apply Theorem 3.2.7.

Let M be a Sasaki manifold and denote by sx a local diffeomorphism around x ∈ M . sx is called a F -geodesic symmetry at x if for each F -geodesic γ such that γ(0) belong to the orbit of ξ passing through x and for any t we have sx ◦ γ(t) = γ(−t) By using this notion we obtain the following characterization of locally F symmetric Sasaki manifolds: Theorem 3.9.5. [Ta77] The regular Sasaki manifold M is locally F -symmetric if and only if at any point x of M there exists a F -geodesic symmetry and this is a local automorphism around x.


141

Proof. Let sx be a F -geodesic symmetry at x ∈ M and suppose that is a local automorphism. The canonical projection π : U −→ Uξ induces a geodesic symmetry s˜π(x) of Uξ . Moreover, s˜π(x) is an automorphism of Uξ with respect to the induced K¨ ahler structure (JUξ , gUξ ) on the orbit space Uξ and s˜π(x) ◦ π = x π ◦ s . Now, from (3.49) it follows that M is locally F -symmetric (see [KN69], vol. I, Note 7, pg. 300–304). The converse is proved in [Ta77] by using Theorem 3.9.3. The notion of ξ-preserving symmetry is studied in [IS93], [In92] and is used in order to obtain other characterizations of locally F -symmetric Sasaki manifolds. A ξ-preserving symmetry s˜x at the point x of the Sasaki (or K-contact) manifold M is a local diffeomorphism with the properties: a) s˜x ◦ γ = γ for any geodesic γ with initial velocity ξx ; b) for each x0 ∈ γ the map s˜x sends any F -geodesic through x0 to a F geodesic through x0 ; c) there exists a neighborhood U of x so that the points of γ in U are only fixed points of s˜x . If there exists k ≥ 2 such that (˜ sx )k = 1 then s˜x is a ξ-preserving symmetry of order k. Remark that a F -geodesic symmetry is a ξ-preserving symmetry of order 2. Theorem 3.9.6. [BeV99] A K-contact manifold M is locally F -symmetric if and only if the ξ-preserving symmetry of order 2 at any point are isometric, Ω-preserving or η-preserving. In this case M is a Sasaki manifold. Let M be a connected Sasaki manifold. If any F -geodesic symmetry can be extended to an automorphism of M and if the Reeb vector field of M generates a global Reeb group then M is called F -symmetric or globally F -symmetric, according to the initial definition given by T. Takahashi, [Ta77]. Of course, any F -symmetric Sasaki manifold is locally F -symmetric. Under some conditions the following converse of this result is also valid. Theorem 3.9.7. [Ta77] Any complete connected and simply connected locally F -symmetric Sasaki manifold is globally F -symmetric. Proof. Since the Levi-Civita connection is complete, from the formula (3.31) it results that the Okumura connection is also complete. Then, taking into account Theorem 3.9.3 and Proposition 3.2.14, any local affine transformation (relative o

to ∇) can be extended to an affine transformation of the manifold M (see also [KN69], vol. I, pg. 265, Corollary 7.9). Now we apply Theorem 3.9.5. J. Berndt and L. Vanhecke generalize Theorem 3.9.4 and obtain the following Theorem 3.9.8. [BeV99] Any simply connected F -symmetric Sasaki manifold is weakly symmetric. If the Sasaki manifold M is F -symmetric then the Reeb group is composed by automorphisms of M and it acts transitively on M . It follows that M is

142 CHAPTER 3. CURVATURE PROBLEMS IN CONTACT MANIFOLDS a regular manifold. On the other hand, because any F -geodesic symmetry of M can be extended to an automorphism of M it follows that any geodesic symmetry of the orbit space Mξ can be extended to an automorphism of Mξ and taking into account Theorem 2.6.3, b) we can state the following Proposition 3.9.9. Any F -symmetric Sasaki manifold is a principal bundle with structure group S1 over the symmetric Hermitian space Mξ .

Chapter 4

Differential forms and topology of contact manifolds 4.1

Harmonic forms

Laplace operator on Riemannian manifolds We recall some definitions and formulas concerning the Laplace operator. Let M be an oriented Riemannian manifold of dimension m and denote by g its Riemannian metric. Choose an orientation on all tangent spaces Tx M ∂ ∂ and assume that the natural bases ∂x are positively oriented. We 1 , . . . , ∂xm define the star (or Hodge) operator ∗ : F p (M ) −→ F m−p (M ) by ∗ (dxi1 ∧ . . . ∧ dxip ) = dxj1 ∧ . . . ∧ dxjm−p (4.1) i where j1 , . . . , jm−p are selected so that dx 1 , . . . , dxip , dxj1 , . . . , dxjm−p is a positive basis of the cotangent space Tx∗ M with the orientation naturally induced by the orientation of Tx M . The star operator has the following elementary properties √ Proposition 4.1.1. a) ∗1 = det g dxR1 ∧ . . . ∧ dxm , ∗(dx1 ∧ . . . ∧ dxm ) = 1; b) If M is compact then vol(M ) = M ∗1; 2 c) ∗∗ = (−1)p(m−p) and ω ∧ ∗θ = (−1)p ∗ ω ∧ θ for ω, θ ∈ F p (M ); d) If ω ∈ F p (M ) is locally given by X ω= ωi1 ...ip dxi1 ∧ . . . ∧ dxip i1 0 α and then from Theorem 3.8.12 it follows that M with the metric g ∗ is isometric to the sphere S 2n+1 . If ρ = 2n(2n + 1) then α = 1, hence g ∗ = g and the proof is complete. Concerning the Betti numbers of higher order we have the following result: Theorem 4.3.8. Let M be a compact connected Sasaki manifold of dimension 2n + 1. Then bp (M ) is even for odd p ∈ 1, n and for even p ∈ n + 1, 2n. Proof. For p = 1 the assertion holds by Theorem 4.3.1. Thus we assume that ˜ : Hp (M ) −→ Hp (M ) the map defined by Φ(ω) ˜ p ≥ 2 and denote by Φ = ω ˜, where ω ˜ is the harmonic form given by (4.12). Then we have X ˜ 2 ω (X1 , . . . , Xp ) = −pω + Φ ω(X1 , . . . , F Xi , . . . , F Xj , . . . , Xp ) i6=j

˜ 2 ω is harmonic and therefore P ω(X1 , . . . , F Xi , . . . , F Xj , . . . , Xp ) is But Φ i6=j ˜ 3 ω, . . . , Φ ˜ p ω we deduce that also harmonic. By the same argument applied to Φ the p-form θ defined by θ(X1 , . . . , Xp ) = ω(F X1 , . . . , F Xp ) is harmonic and then is well-defined the morphism Φ : Hp (M ) −→ Hp (M ) with Φω = θ. Moreover, we have Φ2 = (−1)p 1Hp (M ) so that if p is odd and p ∈ 1, n then bp (M ) is even. The second affirmation follows from Poincaré duality theorem. Given the Sasaki manifold M , we denote by C p (M ) the real vector space of C-harmonic forms and by cp (M ) its dimension. Theorem 4.3.9. Let M be a compact Sasaki manifold of dimension 2n + 1. Then: a) c0 (M ) = b0 (M ) = 1, c1 (M )= b1 (M ); b) c2k (M ) ≥ 1 for k = 1, . . . , n2 ; c) bp (M ) = cp (M)−c p−2 (M ); cp (M ) = bp (M )+bp−2 (M )+. . .+bp−2r (M ) for p ∈ 2, n and r = p2 .

4.3. BETTI NUMBERS OF A CONTACT MANIFOLD

159

Proof. a) follows from Remark 4.2.1. b) From Proposition 4.2.13 we deduce that the 2k-form Lk 1 is C-harmonic, hence dim C 2k (M ) ≥ 1 for k ≥ 1. c) From the decomposition Theorem 4.2.15 for C-harmonic forms we have C p (M ) = Hp (M ) ⊕ LHp−2 (M ) ⊕ . . . ⊕ Lr Hp−2 (M ), C p+2 (M ) = Hp+2 (M ) ⊕ LHp (M ) ⊕ . . . ⊕ Lr+1 Hp+2−2(r+1) (M ) We observe that the map L is injective and by comparing the above equalities we have C p+2 (M ) = Hp+2 (M ) ⊕ LC p (M ) and then the first equality c) follows. The second equality c) is a consequence of the first. From Theorem 4.2.14 we deduce cp (M ) = bp (Mξ ) and then by applying Theorem 4.3.9, c) we obtain Theorem 4.3.10. Let M be a compact regular Sasaki manifold of dimension 2n + 1 and let Mξ be its orbit space. Then bp (M ) = bp (Mξ ) − bp−2 (Mξ ) for any p ∈ 2, n. We already noticed that the orbit space of the product M1 ∗M2 is M1ξ ×M2ξ . Then, by using the classical relation between the cohomology groups H p (M1ξ × M2ξ ) and H i (M1ξ ), H j (M2ξ ) and b0 (M ) = 1, with the help of Theorem 4.3.10 we can compute the Betti numbers of the product M1 ∗ M2 . Proposition 4.3.11. [BG00] If M1 , M2 are compact regular Sasaki manifolds of dimensions 2n1 + 1 and 2n2 + 1 respectively, then: i) b2 (M1 ∗ M2 ) = b2 (M1 ) + b2 (M2 ) + 1 for n1 ≥ 1, n2 ≥ 1; ii) b3 (M1 ∗ M2 ) = b3 (M1 ) + b3 (M2 ) for n1 ≥ 3, n2 ≥ 3; iii) b4 (M1 ∗ M2 ) = b4 (M1 ) + b4 (M2 ) + b2 (M1 )b2 (M2 ) + b2 (M1 ) + b2 (M2 ) + 1 for n1 ≥ 4, n2 ≥ 4. By using these results, C. Boyer and K. Galicki, [BG00], construct simply connected compact Sasaki-Einstein manifolds of dimension 2n + 1 a) with b2 = 0 and b3 = 0 for any n ≥ 3; b) with arbitrary b2 and b3 6= 0 for any n ≥ 6. The constructions are given in [BG00], but here we only do the following remarks: Let S 3 be the sphere with the Sasaki space form structure from Example 1, Section 3.5. For α ∈ R∗+ , α 6= 1, it is a η-Einstein Sasaki manifold with Einstein constants a = α4 − 2, b = α4 − 4. But b0 (S 3 ) = b3 (S 3 ) = 1 and b1 (S 3 ) = b2 (S 3 ) = 0, so that from Proposition 4.3.11 we deduce b2 (S 3 ∗ S 3 ) = 1, b3 (S 3 ∗ S 3 ) = 2, or in the general case b 2 ∗k S 3 = b 3 ∗k S 3 = k − 1

160

CHAPTER 4. DIFFERENTIAL FORMS AND TOPOLOGY

where ∗k M denotes the ∗-product of k copies of M . By using the same argument for the product S 5 ∗ S 5 we have b2 S 5 ∗ S 5 = 0 and in the general case we obtain b2 ∗k S 5 = k − 1, b3 ∗k S 5 = 0

4.4

Basic forms on Sasaki manifolds

Let M be a 2n + 1-dimensional manifold with the almost contact structure (F, ξ, η). Recall that a horizontal and invariant form ω on M is called basic (see Section 1.3), i. e. it is characterized by the conditions ıξ ω = 0, Lξ ω = 0. Denote by FBp (M ) the subspace of basic p-forms on the manifold M . From Proposition 1.3.6, a) and from its proof it follows that if M is contact a Riemannian manifold then the forms η and dη are invariant forms. Besides this, the forms dη, (dη)2 , . . . , (dη)n are basic. A characterization of basic forms, depending on the forms of the orbit space Mξ is the following: Theorem 4.4.1. Let M be a regular Sasaki manifold and denote by π : M −→ Mξ the natural projection. Then π ∗ : F p (Mξ ) −→ FBp (M ) is an isomorphism for each p ∈ 1, 2n. Proof. Obviously, π ∗ is injective. We prove that for any ω ∈ FBp (M ) there exists ω ˜ ∈ F p (Mξ ) so that ω = π ∗ ω ˜. Since ω is horizontal, the values ω(X1 , . . . , Xp ) can be non zero only when the tangent vectors {X1 , . . . , Xp } ∈ Tx M are orthogonal on ξ. But the condition Lξ ω = 0 shows that ω is invariant by the Reeb group {Φt }t∈R , that is Φ∗t ω = ω. It follows that ω (Φt∗ X1 , . . . , Φt∗ Xp ) = ω (X1 , . . . , Xp ) and so at the point π(x) is well-defined a p-form ω ˜ π(x) with the property ωx = π∗ ω ˜ π(x) . Or x is arbitrary in M and then is well-defined the p-form ω ˜ ∈ F p (Mξ ) p ∗ ∗ with the property ω = π ω ˜ , which proves that Im π ⊇ FB (M ). ˜ ∈ It remains only to prove that Im π ∗ ⊆ FBp (M ). Remark that for any ω F p (Mξ ) we have Φ∗t π ∗ ω ˜ = π∗ ω ˜ , ıξ π ∗ ω ˜ = 0, hence Lξ π ∗ ω ˜ = 0 and then π ∗ ω ˜ ∈ FBp (M ). From the definition of horizontal forms it follows Proposition 4.4.2. On a 2n + 1-dimensional Sasaki manifold M there are no non zero horizontal 2n + 1-forms, i. e. FB2n+1 (M ) = 0. For invariant forms the following decomposition theorem holds: Theorem 4.4.3. [Re52] Let M be a regular contact Riemannian manifold. Any invariant form ω ∈ F p (M ) can be written under the form ω = ω1 ∧ η + ω2 for some forms ω1 ∈ ω1 is also closed.

FBp−1 (M )

and ω2 ∈ FBp (M ). Moreover, if ω is closed then

4.4. BASIC FORMS ON SASAKI MANIFOLDS

161

Proof. For p = 0 we can take ω1 = 0 and ω2 = ω. For p ≥ 1 we set ω1 = (−1)p−1 ıξ ω and then Lξ ω1 = (−1)p−1 Lξ ıξ ω = (−1)p−1 ıξ Lξ ω = 0 hence the form ω1 is invariant. Moreover, for p = 1 the form ω1 is basic. For p > 1 we have ıξ ω1 = 0, so that ω1 is horizontal and then it is also basic. By simple verification it follows that the form ω2 = ω − ω1 ∧ η is horizontal and from Proposition 1.3.6 we deduce that ω2 is also invariant. If ω is closed then we have dω1 = dıξ ω = Lξ ω − ıξ dω = 0, hence ω1 is closed. In the case of 1-forms the result from Theorem 4.4.3 can be improved, namely we have Theorem 4.4.4. Any closed invariant 1-form on a compact regular contact Riemannian manifold is basic. Proof. From Theorem 4.4.3 it results ω1 = c = constant and since ω is closed, we have cdη + dω2 = 0 and by its exterior multiplication with η ∧ (dη)n−1 we get cη ∧ (dη)n + dω2 ∧ η ∧ (dη)n−1 = 0, hence cη ∧ (dη)n = η ∧ d ω2 ∧ (dη)n−1 = = dη ∧ ω2 ∧ (dη)n−1 − d η ∧ ω2 ∧ (dη)n−1

(4.47)

But dη ∧ ω2 ∧ (dη)n−1 is a basic 2n + 1-form because dη, ω2 and (dη)n−1 are basic. According to Proposition 4.4.1 we have dη ∧ ω2 ∧ (dη)n−1 = 0 and then the equality (4.47) yields cη ∧ (dη)n = −d η ∧ ω2 ∧ (dη)n−1 Integrating this equality over M and taking into account the Green formula we obtain Z c η ∧ (dη)n = 0 M

Then from Theorem 1.3.9 it results c = 0, so that by applying Theorem 4.4.3 we deduce ω = ω2 ∈ FB1 (M ). Proposition 4.4.5. Let M be a regular contact Riemannian manifold. If ω is a basic (or invariant) p-form then its differential dω is basic (or invariant). Proof. We have ıξ dω = Lξ ω − dıξ ω = 0 and the analogous equalities.

162


From Proposition 4.4.5 it follows that we can consider the operator dB = d|FBp (M ) : FBp (M ) −→ FBp+1 (M ), called the basic differential. On the other hand, from the first formula (4.18) and using (4.16) and Proposition 1.5.5 we deduce that if ω ∈ FBp (M ) then Lξ ıξ ∗ ω = ıξ Lξ ∗ ω = −ıξ ∗ eη δω − ı ∗ δeη ω = = (−1)p ∗ eη δeη ω = (−1)p ∗ [η ∧ δ(η ∧ ω)] = 0 the (2n − p)-form ıξ ∗ ω is basic. Therefore we can define the basic star operator ∗B : FBp (M ) −→ FB2n−p (M ) by ∗B ω = (−1)p ıξ ∗ ω

(4.48)

Proposition 4.4.6. Let M be a Sasaki manifold. Then: ω ∧ ∗B ω = ν,

∗ω = ∗B ω ∧ η,

∗B ω = ∗eη ω

(4.49)

for any ω ∈ FBp (M ). Proof. As we already remarked the form (dη)n is basic, hence the transversal volume form (see the proof of Proposition 1.3.10) is basic and the first equality follows easily. The second results from the equality 0 = ıξ (?ω ∧ η) = ıη ∗ ω ∧ η + (−1)p+1 ∗ ω and the third is a consequence of (4.16). If we assume that the manifold M is compact then the scalar product h, i, given by the formula (4.2), restricts to the following scalar product of the basic forms ω, θ ∈ FBp (M ) Z hω, θiB =

ω ∧ ∗B θ ∧ η M

With respect to this product the operator δB : FBp (M ) −→ FBp−1 (M ), defined by δB = − ∗B dB ∗B , is the h, iB -adjoint of dB , i. e. hdB ω1 , ω2 iB = hω1 , δB ω2 iB

(4.50)

for any ω1 ∈ FBp−1 (M ) and ω2 ∈ FBp (M ). Proposition 4.4.7. Let M be a compact Sasaki manifold. Then δω = δB ω + eη Λω for any ω ∈ FBp (M ). Proof. It follows by direct verification from the formulas (4.14) and (4.16).

4.5. BASIC COHOMOLOGY OF SASAKI MANIFOLDS

163

Now, we define the basic Laplace operator ∆B : FBp (M ) −→ FBp (M ) by ∆B = δ B d B + d B δ B From Proposition 4.4.7 we get Proposition 4.4.8. Let M be a compact Sasaki manifold. Then ∆ω = ∆B ω + LΛω + eη [Λ, d]ω

(4.51)

for all ω ∈ FBp (M ). The basic form ω ∈ FBp (M ) is called transversally harmonic if ∆B ω = 0. p Denote by FBH (M ) the space of transversally harmonic basic forms. Then from Theorems 4.2.7 and 4.2.10 we easily obtain Proposition 4.4.9. Let M be a compact 2n + 1-dimensional Sasaki manifold and let ω ∈ F p (M ) be a C-harmonic form. a) If p ≥ n + 1 and ω is basic then it is transversally harmonic. b) If p ≤ n then it is a transversally harmonic basic form.

4.5 4.5.1

Basic cohomology of Sasaki manifolds Basic de Rham cohomology

The set FBp (M ) of all basic p-forms on the regular Sasaki manifold M is a module over the ring FB0 (M ) = FB (M ) of basic functions on M . Obviously we have f ∈ FB (M ) if and only if ξf = 0. Taking into account Proposition 4.4.5, we deduce that the basic forms constitute a subcomplex FB∗ (M ) = (⊕p≥0 FBp (M ), dB = (dpB )) of the de Rham complex F ∗ (M ) = (⊕p≥0 F p (M ), d). The cohomology of this subcomplex is called the basic cohomology of the Sasaki manifold M and is defined by p ∗ HB (M ) = ⊕p≥0 HB (M ),

p p−1 HB (M ) = ker dpB /Im dB

This cohomology play the role of de Rham cohomology of the orbit space of the Sasaki manifold M and we call it the basic de Rham cohomology or simply ∗ the basic cohomology of M . Moreover, the space of basic cohomology HB (M ) is an invariant of the characteristic foliation Fξ and therefore is an invariant of the Sasaki structure on the manifold M . The groups of basic cohomology of dimensions 0 and 1 have the following simple characterization: Proposition 4.5.1. If M is a regular and compact Sasaki manifold then 0 1 a) HB (M ) ∼ b) HB (M ) ∼ = R; = H 1 (M ).

164


Proof. a) is obvious. b) If ω ∈ FB1 (M ) is exact then there exists f ∈ F(M ) such that ω = df and we have ξf = ıξ df = ıξ ω = 0, that is f ∈ FB (M ) and then ω defines the basic 1 cohomology class [ω]B = 0 ∈ HB (M ), hence the inclusion of complexes ı = (ıp ) : (⊕p≥0 FBp , dB ) ,→ (⊕p≥0 F p (M ), d) 1 induces an injective map HB (M ) −→ H 1 (M ). But according to Theorem 4.4.1 any basic cohomology class contains at least one closed invariant 1-form. Now, let ω be a harmonic 1-form. ξ is a Killing vector field and then Lξ ω = 0. We deduce that ω is a closed invariant 1-form. Thus, we have proved that any basic cohomology class contains at least one harmonic 1-form and therefore the map defined above is also surjective. ∗ (M ) and de Rham cohomolThe relation between the basic cohomology HB ∗ ogy H (M ) of the Sasaki manifold M is the same as in the general case of a foliation generated by a nonsingular Killing vector field (see for instance [To88], Theorem 10.13, pg. 139). We have

Theorem 4.5.2. If M is a regular and compact Sasaki manifold then the following exact cohomology sequence holds ıp

∆∗

p p−1 p+1 ∗ · · · → HB (M ) → H p (M ) → HB (M ) → HB (M ) → · · ·

where ∆∗ [ω] = [dη ∧ ω] and ı∗ is the homomorphism induced by the inclusion ıp : FBp (M ) ,→ F p (M ). Proof. The manifold M is compact, hence the Lie group of isometries of the metric g is compact and then the closure of the subgroup {exp(tξ)}t∈R is a compact abelian Lie group, that is it is isomorphic to a torus T . Denoting by F ∗ (M )T the complex of T -invariant forms on M , an argument similar to the one used in the proof of Theorem 4.4.1 shows that FB∗ (M ) ⊂ F ∗ (M )T and it follows that the inner product ıξ : F p (M )T −→ FBp−1 (M ) induces a surjective map also denoted by ıξ : F ∗ (M )T −→ FB∗−1 (M ). So, we obtain the sequence of complexes ı

ıξ

0 → FB∗ (M ) → F ∗ (M ) → FB∗−1 (M ) → 0

(4.52)

FBp (M )

If ω ∈ then ıξ ıω = ıξ ω = 0, hence Im ı ⊂ ker ıξ . Conversely, let ω ∈ ker ıξ . Then ıξ ω = 0 and taking into account ω is T invariant, we have Lξ ω = 0, hence ω is basic , that is ker ıξ ⊂ Im ı. Thus we have proved that the sequence (4.52) is exact. Now, from the long exact cohomology sequence associated to the sequence (4.52) and taking into account the isomorphism of the cohomologies of complexes F ∗ (M )T and F ∗ (M ), obtained by an argument similar to the one used in the proof of Theorem 4.4.1, we get the announced exact sequence. From Theorem 4.5.2 it results that all basic cohomology groups of a Sasaki manifold are finitely dimensional. Moreover, we have


165

Proposition 4.5.3. Let M be a compact regular Sasaki manifold of dimension 2n + 1. Then: 2n+1 2n a) HB (M ) = 0; b) HB (M ) ∼ = R. Proof. a) is obvious taking into account Proposition 4.4.2. b) The cohomology sequence from Theorem 4.5.2 ends by the exact sequence 2n 0 −→ H 2n+1 (M ) −→ HB (M ) −→ 0, 2n hence HB (M ) ∼ = H 2n+1 (M ) and then b) follows from the compactness of the manifold M .

Example. Let S 2n+1 be the sphere with the usual Sasaki structure (see Example 2, Section 1.5). We know that H 0 (S 2n+1 ) = H 2n+1 (S 2n+1 ) ∼ =R and its other de Rham cohomology groups are equal to 0. Then from the p exact cohomology sequence given in Theorem 4.5.2 we obtain HB (S 2n+1 ) ∼ = p n H (P (C)) for p ∈ 0, 2n. Taking into account Proposition 1.3.10 and the basic Hodge-de Rham decomposition for general transversally oriented Riemannian foliations with minimal leaves on compact orientable Riemannian manifolds with bundle-like metric (see [KH86] or [Mol88], Appendix B), D. Chinea, M. de Leon and J. C. Marrero obtain the following decomposition for basic forms Theorem 4.5.4. [CLM97] Let M be a 2n+1-dimensional compact Sasaki manifold. For any p ∈ 0, 2n + 1 there exists the following orthogonal decomposition p FBp (M ) = Im dB ⊕ Im δB ⊕ FBH (M )

(4.53)

p Moreover, FBH (M ) has finite dimension.

Proof. The idea of the proof is a straightforward transcription of the general proof in our case. Namely one proves that there exist two linear operators p p HB : FBp (M ) −→ FBH (M ) and GB : FBp (M ) −→ FBH (M )⊥ , associated to the p p p ⊥ decomposition FB (M ) = FBH (M ) ⊕ FBH (M ) and having the property ω = HB ω + ∆ B G B ω

(4.54)

These operators commute with dB , δB and then (4.53) follows easily. The second affirmation of the Theorem also follows as a particular case from the same general results (see for instance [Mol88], Theorem 2.2, pg. 237). This result also holds for K-contact manifolds (see [CLM97]).

166


4.5.2

Basic Dolbeault cohomology

The study of complex valued forms on Sasaki manifolds was initiated by T. Fujitani [Fu66]. Now, we return to the decomposition p FBC (M ) = FBp (M ) ⊗ C = ⊕r+s=p FBr,s (M )

of the complexified of the space of basic p-forms (see Section 1.3). In particular, for the complexified of the space of basic 1-forms we have 1 FBC (M ) = FB1 (M ) ⊗ C = FB1,0 (M ) ⊕ FB0,1 (M )

and then, by applying the classical method used in the case of almost complex manifolds (see for instance [KN69], vol. II, pg. 125–126) and taking into account that on a Sasaki manifold we have NF (X, Y ) = 0 for all X, Y ∈ D, a simple computation proves that dB FBt,s (M ) ⊂ FBr+1,s (M ) + FBr,s+1 (M ) 10 For ω ∈ TBt,s (M ) we denote by DB ω and d01 B ω the (r + 1, s)-component and the (r, s + 1)-component of dB ω, respectively, and thus the following operators are well defined r,s r+1,s d10 (M ), B : FB (M ) −→ FB

r,s r,s+1 d01 (M ) B : FB (M ) −→ FB

From the formula d2B = 0 we deduce 10 d10 B dB = 0,

01 d01 B dB = 0,

01 01 10 d10 B dB + dB dB = 0

(4.55)

01 10 ω, induced ω + δB On the other hand, we have the decomposition δB ω = δB 10 01 by the decomposition dB ω = dB ω + dB ω of the basic differential and some formulas similar to (4.55), namely 10 10 δB δB = 0,

01 01 δB δB = 0,

10 01 01 10 δB δB + δB δB = 0

(4.56)

Moreover, from (4.55) and (4.56) we deduce 10 10 10 01 01 01 01 ∆B = 2 d10 B δB + δB dB = 2 dB δB + δB dB

(4.57)

The first equality (4.55) induces the complex d01

d01

r,s−1 B B 0 → FBr,0 (M ) → · · · → DB (M ) → FBr,s (M ) → FBr,s+1 (M ) → · · ·

and we can define its cohomology groups by h i r,s r,s+1 r,s−1 rs 01 HB (M ) = ker d01 : F (M ) → F (M ) /d F (M ) B B B B B These are the analogous of Dolbeault cohomology groups from Kähler geometry. C. P. Boyer, K. Galicki and M. Nakamaye, who introduced this cohomology called the basic Dolbeault cohomology of the Sasaki manifold M , proved the following characterization theorem:


167

Theorem 4.5.5. [BGN03] Let M be a compact regular Sasaki manifold of dimension 2n + 1. Then: p r,s a) the following decomposition holds HB (M ) = ⊕r+s=p HB (M ); r,s s,r ∼ b) HB (M ) = HB (M ); r,s n−r,n−s c) the following Serre type duality formula is valid HB (M ) ∼ (M ); = HB 1,1 d) the basic cohomology class [dη]B ∈ HB (M ) does not vanishes and if ω is a closed real form of type (1,1) such√that ω ∈ [dη]B then there exists a basic function f with the property ω = dη + −1∂∂f ; n,n e) HB (M ) ∼ = R; p,p f ) HB (M ) 6= 0 for each p ≤ n. Proof. a), b) are c) the adapted versions of the well-known properties of Dolbeault cohomology on K¨ ahler manifolds (see for instance [Do57]), while d) follows immediately from the proof of Theorem 4.4.4. Replacing dη by (dη)p in the formula cdη+dω2 = 0 (see the proof of Theorem 4.4.4), taking into account that (dη)p is a basic form of type (p, p) and following the same argument we prove that [(dη)p ]B 6= 0 and thus we obtain f). Besides this, by using Proposition 4.5.3 we obtain e). Now, we can complete the characterization of basic cohomology groups by the following Theorem 4.5.6. [BGN03] Let M be a compact regular Sasaki manifold of di2p+1 mension 2n + 1. The basic cohomology group HB (M ) has even dimension for any p ∈ 0, n − 1. Proof. From Theorem 4.5.5, a), b) we deduce 2p+1 dim HB (M ) =

p X

r,2p+1−r dim HB (M ) +

r=0

=

p X

r,2p+1−r dim HB (M ) +

r=0 p X

=2

2p+1 X

r,2p+1−r dim HB (M )

r=p+1 p X

0

0

2p+1−r ,r dim HB (M )

r 0 =0 r,2p+1−r dim HB (M )

r=0 2p+1 and thus it follows that dim HB (M ) is even.

4.5.3

Spectral sequences of Sasaki manifolds

First of all, we recall some notions concerning the spectral sequences. A bigraded module E is a family {E r,s }r,s∈N of modules over R and a differential d : E −→ E of bidegree (t, 1 − t) is a family of homomorphisms d = dr,s : E r,s −→ E r+t,s+1−t

168


such that d2 = 0, i. e. dr+t,s+1−t dr,s = 0. The cohomology module H(E) of the bigraded module E is the bigraded module defined by H r,s (E) = ker dr,s : E r,s → E r+t,s+1−t /dr−t,s+t−1 E r−t,s+t−1 01 Example. {FBr,s (M )} is a bigraded module and d10 are differentials B , dB of bidegree (1, 0) and (0, 1), respectively. Its cohomology with respect to the second differential is the basic Dolbeault cohomology. Now, by setting E p = ⊕r+s=p E r,s we see that the differential d defines a homomorphism ∂ = ⊕r+s=p dr,s : E p −→ E p+1 with the property ∂ 2 = 0. Moreover, the p-th cohomology module of the complex (⊕E p , ∂) is exactly ⊕r+s=p H r,s (E). A E k -spectral sequence E is a sequence {E p , dp }p≥k such that: a) E p is a bigraded module and dp is a differential of bidegree (t, 1 − t) on p E ; b) for p ≥ k there is an isomorphism H(E p ) ∼ = E p+1 . A filtration F on the graded module A = {Ap } is a sequence of graded submodules Ft A = {Ft Ap } such that Ft Ap ⊆ Ft+1 Ap . See for instance [Sp66] for a complete study of spectral sequences. Let M be a Sasaki manifold and denote by {Et (M )} the spectral sequence associated to the filtration 0

p Fr FBC (M ) = ⊕r0 +s=p,r0 ≥r FBr s (M )

We remark that it converges to the cohomology of the complex d

p p+1 B 0 0 → FBC (M ) → . . . → FBC (M ) → FBC (M ) → . . .

where we also denote by dB the extension of the basic differential to complex valued basic forms. We also observe that a form ω ∈ FBpq (M ) belongs to Etp,q if there exist the basic forms ω1 , . . . , ωt−1 such that 10 01 10 01 d01 B ω = 0, dB ω = dB ω1 , dB ω1 = dB ω2 , . . . 01 10 01 . . . , d10 B ωt−3 = dB ωt−2 , dB ωt−2 = dB ωt−1

(4.58)

If [ω]t is the cohomology class of ω in Etr,s (M ) then the operator dt : Etr,s (M ) −→ Etr+t,s−t+1 (M )

defined by dt [ω]t = d10 B ωt−1 t

(4.59)

r,s (M ) are isomorphic to the cohomolis a differential. Moreover, the groups Et+1 ogy groups of the sequence d

d

t t . . . −→ Etr−t,s+t−1 (M ) −→ Etr,s (M ) −→ Etr+t,s−t+1 (M ) −→ . . .

In particular we have


169

Proposition 4.5.7. [CLM97] Let M be a compact Sasaki manifold. Then r,s E1r,s (M ) ∼ = HB (M )

Theorem 4.5.8. [CLM97] The spectral sequence {Et (M )} of the compact Sasaki manifold M degenerates at the first level, i. e. E1 (M ) ∼ = ... ∼ = E∞ (M ) = E2 (M ) ∼ Proof. Let ω ∈ Etr,s (M ). Taking into account (4.58), (4.59) and the commuting relations between GB , HB and dB , δB , we have HB ω ∈ Etr,s (M ), dt [HB ω]t = 0 and 01 10 01 10 d10 B GB ω + d B GB ω = GB d B ω + GB d B ω = GB d B ω By comparing the degrees of the forms in the above equalities we get 10 d10 B GB ω = GB dB ω,

d01 B GB ω = 0

(4.60)

But from Theorem 4.5.4 and taking into account the definition of the basic Laplace operator ∆B and (4.54), (4.55) it follows that ω can be written as the sum 10 10 10 01 01 01 01 ω = HB ω + 2 d10 B δ B + δ B d B ω = HB ω + 2 d B δ B + δ B d B ω Now, by using the formulas (4.60) we obtain 01 ω = HB ω + 2d01 B δ B GB ω

and then [ω]1 = [HB ω]1 , hence [ω]t = [HB ω]t . Finally, we deduce that dt = 0 for each t ≥ 1. p (M ) does From Proposition 4.4.9, a) it follows that generally the space FBH p not coincide with the subspace KH (M ) of C-harmonic p-forms. However, this is true for complex valued forms in the following sense

Theorem 4.5.9. [CLM97] Let M be a 2n + 1-dimensional compact Sasaki manifold. Then: p a) the space FCBH (M ) of complex valued transversally harmonic basic forms on M is characterized by p rs FCBH (M ) = ⊕r+s=p KH (M ) rs rs b) KH (M ) ∼ (M ); = HB p (M ) the space of complex valued c) for p ∈ 1, 2n + 1 and denoting by FCH harmonic p-forms on M we have p rs rs ⊕r+s=p HB (M ) ∼ (M )) = FCH (M ) ⊕ (⊕r+s=p−2 LHB

170


p rs Proof. a) Obviously we have ⊕r+s=p KH (M ) ⊆ FCBH (M ). p rs Conversely, if ω ∈ FCBH (M ) then there is ωrs ∈ FBC (M ) such that

ω=

X

ωrs ,

0 = ∆B ω =

r+s=p

X

∆B ωrs

r+s=p

rs and from (4.57) it follows that ∆B ωrs ∈ FBC (M ) for each r and s. Then from rs the above relations we deduce ∆B ωrs = 0, that is ωrs ∈ KB (M ). b) is immediate since, by the proof of Theorem 4.5.8, each cohomology class rs from HB (M ) contains exactly one transversally harmonic basic form. c) follows from the equality p−2 p p (M ) (M ) ⊕ L FCBH (M ) = FCH FCBH

(see the proof of Proposition 4.2.13).

4.6

Basic Chern classes

Chern classes of symplectic bundles We begin by a brief presentation of Chern classes of a symplectic bundle. Let (E, π, M ) be a symplectic vector bundle with the symplectic form ω, the fiber R2m and denote by J an almost complex structure compatible with the symplectic structure ω on E, i. e. ωx (Jx s1 , Jx s2 ) = ωx (s1 , s2 ) for any x ∈ M and s1 , s2 ∈ Ex . By putting gx (s1 , s2 ) = ω(s1 , Jx s2 ), we obtain a pseudoHermitian metric and it is well-known that there exist such metrics satisfying besides this, the condition gx (s, s) ≥ 0 for any s ∈ Ex , i. e. these are Hermitian. For x ∈ M we denote by U(Ex , Jx ) the set of unitary complex bases of the local fiber Ex equipped with the almost complex structure Jx and the √ Hermitian metric gx , that is the set of all bases of the form k = (ek − iJx ek )/ 2 k∈1,m , where {ek , Jx ek }k∈1,m is an orthonormal local basis adapted to the structure Jx on Ex . Then U(E, J) = ∪x∈M U(Ex , Jx ) has a natural structure of principal bundle over M , with the unitary group U (m) as structure group. Let ∇ be a U (m)-connection on U(E, J), that is a linear connection whose matrix θ = θkl of the connection 1-form is U (m)-valued. The local equations of this connection are ∇k = θkl l

(4.61)

and the corresponding curvature 2-forms Θlk are given by (see Section 2.4) Θlk = dθkl − θkh ∧ θhl

(4.62)

If ∇ is a O(m)-connection then the following conditions are fulfilled θkl + θlk = 0,

Θlk + Θkl = 0

(4.63)

4.6. BASIC CHERN CLASSES

171

The real valued 2k-forms √

∆(θ)ck =

k

−1 j1 ...jk i1 δ Θ ∧ . . . ∧ Θijkk (2π)k k! i1 ...ik j1

(4.64)

are the Chern forms of the symplectic bundle (E, π, M ). These are closed, hence [∆(θ)ck ] ∈ H 2k (M ) and do not depend on the choice of the almost complex structure J compatible with the symplectic structure ω and on the choice of the associated U (m)-connections. These cohomology classes are called Chern classes and we denote them by ck (E), i. e. ck (E) = [∆(θ)ck ]. In fact by their axiomatic definition, the Chern classes are elements of the cohomology groups with integer coefficients, that is ck (E) ∈ H 2k (M, Z) (see for instance [Hu66], Chapter 16), but taking into account the natural homomorphism H p (M, Z) −→ H p (M, R) induced (when M is paracompact) by the inclusion Z ,→ R , we usually write ck (E) ∈ H 2k (M ). For further results concerning the Chern classes of symplectic vector bundles see for instance [Va87], Chapter 4. Basic Chern classes of Sasaki manifolds Let M be a Sasaki manifold. Its contact distribution D has a natural structure of symplectic bundle (see Proposition 1.3.2) and then we have the following Theorem 4.6.1. All Chern classes of odd order of the contact distribution of a Sasaki manifold vanish. Proof. The Levi-Civita connection of the metric g|D×D is an O(n)-connection and then from the formula (4.63) it results that the determinant of the matrix Θlk vanishes when the matrix has odd order. It follows ∆(θ)c2k+1 = 0, hence c2k+1 (D) = 0. Now, by means of the Hermitian bundle D10 defined in Section 1.2 we associate to the vector bundle D some Chern type classes, following a construction given by C. P. Boyer, K. Galicki and M. Nakamaye, [BGN03]. For this purpose we state the following Proposition 4.6.2. Let M be a Sasaki manifold. a) The Hermitian connection ∇10 given by the formula (1.17) is of type (1, 0), that is its connection 1-forms θkl are of type (1, 0). Moreover, these forms are basic. b) The curvature 2-forms Θlk of the connection ∇10 are basic of type (1, 1). Proof. a) The connection forms θkl of the connection ∇10 are given by some formulas similar to (4.61), i. e. l ∇10 Z k = θk (Z)l

(4.65)

for any Z ∈ T c M . But {k } is a local basis in D01 and then we apply (4.65) for Z = s . By using Theorem 1.5.3 and the fact that on D we have N (1) = NF = 0,

172


a straightforward computation shows that θkl (s ) = 0. Hence θkl are forms of type (1, 0) and therefore ∇10 is a connection of type (1, 0). Then taking Z = ξ in (4.65), the same argument shows that θkl (ξ) = 0, hence the forms θkl are horizontal. b) The curvature tensor R10 of the connection ∇10 has the following expression with respect to the local basis {k } from D10 Θlk (X, Y ) = g R10 (X, Y )k , l (4.66) for X, Y ∈ T c M . Then, taking into account Proposition 3.2.1, b), the formulas (1.16), (1.17) and the symmetry properties of the Riemann-Christoffel curvature tensor (extended to vectors from Dc ), by a straightforward, but hard, computation we get ıξ Θlk (Y ) = Θlk (ξ, Y ) = g 10 R10 (ξ, Y )k , l = = g (R(ξ, Y )k , l ) = R(l , k , ξ, Y ) = = R(Y, ξ, k , l ) = g(R(k , l )ξ, Y ) = 0 and thus Θlk is a horizontal form. But Lξ Θlk = dıξ Θlk + ıξ dΘlk = 0

(4.67)

hence Θlk are basic forms. Now, by applying ıξ to the equality (4.61) and by using (4.67), we obtain Lξ θkl = 0, which means that the forms θkl are invariant and the proof of the affirmation a) is also complete. Finally, from (1.16) and (1.17) we also deduce that Θlk are complex forms of type (1, 1). From Proposition 4.6.2 and from the construction of Chern classes presented at the beginning of this section it follows that the first Chern class c1 (D10 ) of the Hermitian bundle D10 can be represented by a real closed basic form ωB 11 2 of type (1, 1), that is c1 (D10 ) = [ωB ]B ∈ HB (M ) ⊂ HB (M ) and we denote it B B by c1 (M ). Remark that c1 (M ) is independent on the metric g c and on the connection ∇c ; it is depending only on the characteristic foliation Fξ of M . cB 1 (M ) is called the first basic Chern class of the Sasaki manifold M . Denote by Ricg , RicgD the Ricci tensors of the metrics g, gD on the Sasaki manifold M and on its contact distribution D, respectively. We can associate the 2-forms ρg , ρgD ∈ F 2 (M ), defined by ρg (X, Y ) = Ricg (X, F Y ) for all X, Y ∈ X (M ), ρgD (X, Y ) = RicgD (X, F Y ) for all X, Y ∈ D Ricg , RicgD are called the Ricci form of the Sasaki manifold M and of its contact distribution D, respectively. Theorem 4.6.3. [BGN03] The first basic Chern class of a Sasaki manifold M is given by 1 cB [ρgD ]B 1 (M ) = 2π

4.6. BASIC CHERN CLASSES

173

˜ = Proof. Denote by θ˜ = (θ˜kl ) the matrix of the connection form and by Θ l D ˜ ) the matrix of the curvature form of the real connection ∇ obtained as (Θ k c the restriction to D of the connection ∇D to D, defined by a formula similar to (1.16). If we take Z ∈ X (M ) in (4.65) then by identification of the real √ l ˜l + , Θlk = Θ and imaginary parts, respectively, we obtain θkl = θ˜kl + −1θ˜k+n k √ c D ˜l −1Θ we have k+n for all k, l ∈ 1, n. As in the case of the connection ∇ ∇D F = 0 and then RD (X, Y )F Z = F RD (X, Y )Z On the other part we have the following formulas analogous to (4.66) ˜ βα (X, Y ) = gD RD (X, Y )eα , eβ Θ

(4.68)

(4.69)

where α, β ∈ 1, 2n and eα = F eα−n for α ≥ n + 1. From (4.68) and (4.69) we obtain ˜l = Θ ˜ n+l = −Θ ˜ k, Θ k l n+k

˜l ˜ n+l = Θ ˜k Θ n+k = −Θk n+l

(4.70)

˜ l = 0, so that for k = 1 the formula (4.64) yields for k, l ∈ 1, n and then Θ l n

∆(θ)c1 =

−1 X ˜ l Θl+n 2π l=1

Now, from Proposition 3.2.1, from the first Bianchi identity and taking into account (4.69) we obtain RD (X, el , F Y, el ) = −RD (F el , Y, el , X) = = RD (F el , el , X, Y ) + RD (F el , X, Y, el ) = = gD RD (X, Y )el , F el + RD (F el , X, F Y, F el ) ˜ l+n (X, Y ) − RD (X, F el , F Y, F el ) =Θ l

Summing up on l ∈ 1, n it results ρD = is a consequence of (4.70).

Pn

l=1

˜ l+n (X, Y ) and then our assertion Θ l

Two Sasaki structures (F, ξ, η, g), (F 0 , ξ 0 , η 0 , g 0 ) on the manifold M are homologous if ξ 0 = ξ and [dη 0 ]B = [dη]B . Proposition 4.6.4. [BGN03] Let M be a manifold with the Sasaki structure (F, ξ, η, g) and denote by S(ξ) the set of its ξ-deformations. Then: a) any two structures from S(ξ) are homologous; b) the first basic Chern class cB 1 (M ) is an invariant of the family S(ξ). Proof. a) follows easily from Theorem 4.5.5 and from the condition that dηt is a closed basic form of type (1, 1) (see Theorem 1.5.15). b) Let (Ft , ξ, ηt , gt ) be a ξ-deformation of the structure (F, ξ, η, g) and consider the isomorphism 1X (M ) − ξ ⊗ ηt : X (M ) −→ X (M ). For any X ∈ D we have gt (X − ηt (X)ξ, ξ) = 0, so that X −ηt (X)ξ ∈ Dt and the map 1X (M ) −ξ⊗ηt induces an isomorphism between D and Dt . From this and from the remark that the induced map between basic p-forms is the identity, we obtain b).

174


From the definition of the Ricci forms ρg and ρgD and from the formula (3.78) we obtain ρg = ρgD − 2dη

(4.71)

and then the following converse of Proposition 4.6.4 is also true Theorem 4.6.5. [BGN03] Let M be a manifold with the Sasaki structure (F, ξ, η, g). If the first basic Chern class of M is represented by a real basic form ρ of type (1, 1) then there exists an unique Sasaki structure (F 0 , ξ, η 0 , g 0 ) ∈ S(ξ) homologous to (F, ξ, η, g) and with the property that ρ − 2dη 0 is the Ricci form of the metric g 0 and η 0 = η + 21 dc f , where f is given in Theorem 4.5.5,d).

4.7

Positive Sasaki structures

Positive, negative and null Sasaki structures The Sasaki structure (F, ξ, η, g) on the manifold M is called positive (negative) if the first basic Chern class cB 1 (M ) can be represented by a real form of type (1, 1) positively (negatively) defined, that is by a 2-form ω ∈ F 2 (M ) whose matrix (ωij ) of components is positively (negatively) defined everywhere on M and (F, ξ, η, g) is null if cB 1 (M ) = 0. Sometimes we simply say positive B (negative) Sasaki manifold or structure and we write cB 1 (M ) > 0 (c1 (M ) < 0). From Theorem 4.6.5 and from Myers Theorem (see footnote ?, Section 3.2) and taking into account the formulas (4.71), (1.54) it results Theorem 4.7.1. [BGN03] a) The Sasaki structure (F, ξ, η, g) on the manifold M is positive if and only if there exists on M a Sasaki structure (F 0 , ξ, η 0 , g 0 ) homologous to (F, ξ, η, g) and such that Ricg0 > −2g 0 . b) Any positive Sasaki manifold is compact and has finite fundamental group. From Theorems 4.3.2 and 4.7.1 it follows Proposition 4.7.2. The first Betti number of a positive Sasaki manifold is zero. Theorem 4.7.1, a) has the following improvement Theorem 4.7.3. [BGN01] Let (F, ξ, η, g) be a positive Sasaki structure on the compact manifold M . There exists on M a Sasaki structure (F 0 , ξ, η 0 , g 0 ) homologous to (F, ξ, η, g) and such that Ricg0 > 0. Proof. From Theorem 4.6.5 it follows that for the positive structure (F 0 , ξ, η 0 , g 0 ), given in Theorem 4.7.1, we have (see the formula (3.78)) 0 0 RicgD = Ricg0 |D0 ×D0 + 2g|D 0 ×D 0 0

and we consider the following D-homothetic deformation F 00 = F 0 ,

ξ 00 = aξ

η 00 =

1 0 η a

g 00 =

1 0 g 0 + η 00 ⊗ η 00 , a D

a>0

4.7. POSITIVE SASAKI STRUCTURES

175

Then D00 = D0 , Fξ00 = Fξ0 and a simple computation shows that 2 00 00 Ricg|D = RicgD − gD00 00 ×D 00 00 a 00 But RicgD > 0 and since M is compact there exists a0 > 0 such that for 00 00 any a > a0 we have Ricg|D > 0. Moreover by Proposition 3.2.1 we have 00 ×D 00 00 00 Ricg00 (ξ , ξ ) = 2n and then Ricg00 > 0. On an Einstein Sasaki manifold we have Ric(X, X) = 2n for any unitary vector field X ∈ X (M ) and then by applying Theorem 4.7.1 we have Proposition 4.7.4. Any Einstein Sasaki manifold is positive. The negative or null Sasaki structures correspond to η-Einstein Sasaki manifolds, namely we have the following Theorem 4.7.5. [BGM04] If the Sasaki structure (F, ξ, η, g) on the manifold M is negative or null then there exists a ξ-deformation (F 0 , ξ, η 0 , g 0 ) which is ηB Einstein with Einstein constant a < −2 for cB 1 (M ) < 0 or a = −2 for c1 (M ) = 0. A simple consequence of Theorem 3.8.23 is the following Theorem 4.7.6. Any negative compact quasi-regular Sasaki manifold has an Einstein Lorentz Sasaki structure. We notice some recent results concerning Sasaki manifolds of low dimension. From Proposition 4.5.3, b) it follows that if M is 3-dimensional then the basic 2 (M ) has dimension 1, hence there are no indefinite forms cohomology group HB other than zero and then Theorem 4.7.7. [BGM04] Any compact regular Sasaki manifold of dimension 3 is positive, negative or null. By using this result the following classification of compact 3-dimensional Sasaki manifolds was obtained; see [G97a], [Bel01] and [BGM04]. Theorem 4.7.8. Let M be a compact connected manifold of dimension 3 with the Sasaki structure (F, ξ, η, g). a) If M is positive then it is diffeomorphic to S 3 /Γ and there exists a deformation of the given Sasaki structure, such that M becomes Sasaki space form with F -sectional curvature equal to 1. f b) If M is negative then it is diffeomorphic to SL(2, R)/Γ and there exists a deformation of the given Sasaki structure, such that M becomes Sasaki space form with F -sectional curvature equal to −4. c) If M is null then it is diffeomorphic to H 3 (R)/Γ and there exists a deformation of the structure (F, ξ, η, g), such that M becomes Sasaki space form with F -sectional curvature equal to −3. Γ is a discrete subgroup of the connected component I0 (P ) of the group of isometries of the manifold corresponding to each case, namely: a) P = S 3 ; b) f f P = SL(2, R), where SL(2, R) is the universal covering of SL(2, R); c) P is the 3 Heisenberg group H (R).

176


Spin Sasaki structures There are some interesting relations between positive Sasaki structures and spin structures. But firstly we shall present some basic notions concerning spin structures. Let V be a m-dimensional real vector space and assume that q is a non degenerate positive definite quadratic form on V . Then we can chooseP a basis for ∞ V such that q(x) = x21 + . . . + x2m for each x ∈ V . Denote by T (V ) = r=0 ⊗r V the tensor algebra of V and consider the ideal Iq (V ) = hv ⊗ v + q(v)1, v ∈ V i. The quotient space C`(V, q) = T (V )/Iq (V ) is an associative algebra with unity, called the Clifford algebra of V . In fact, C`(V, q) can be defined for any quadratic form on V . Denote by C`∗ (V, q) the group of all units in C`(V, q), that is the subset of invertible elements of Cl(V, q) and by P (V, q) the subgroup of C`∗ (V, q) spanned by the elements v ∈ V such that q(v) 6= 0. Then for m even the subgroup of P (V, q) defined by

Spinm = v1 , . . . , vm ∈ P (V, q) : q(vi ) = 1 for i ∈ 1, m is called the spin group associated to (V, q). There is an universal covering morphism Σ0 : Spinm −→ SO(V, m) = {α ∈ Gl(V ) : α∗ q = q, det(α) = 1} with the property ker Σ0 = {−1, +1} ∼ = Z2 . For m ≥ 3 let M be an oriented 2m-dimensional Riemannian manifold and denote by O(M ) its bundle of all oriented orthogonal frames. A spin structure on M is a principal bundle PSpin (M ) over M with the structure group Spinm and with a 2-sheeted1 covering map Σ : PSpin (M ) −→ O(M ) such that Σ(pg) = Σ(p)Σ0 (g) for all p ∈ PSpin (M ) and g ∈ Spinm . When m = 2 the definition of the spin structure is similar, but we replace Spinm by the group SO(2). We also recall the following classical characterization Theorem of spin structures Theorem 4.7.9. The oriented Riemannian manifold M admits a spin structure if and only if its second Stiefel-Whitney class w2 (M ) is zero. For the general study of spin structures and related subjects see for instance [LM89], pg. 78-93. Theorem 4.7.10. [BGN01] Any compact, simply connected positive Sasaki manifold carries a spin structure. ∗ Proof. Let ω ∗ ∈ cB ahler 1 (M ) be a positively defined form and denote by g a K¨ ∗ ∗ metric whose fundamental 2-form is ω and by J the associated almost complex structure. Like in Section 1.3, starting from J ∗ we can construct a positive Sasaki structure (F 0 , ξ, η 0 , g 0 ) for which g 0 = g ∗ + η 0 ⊗ η 0 and denote by D0 its contact distribution. 1 i. e. for each fr ∈ O(M ) there is an open neighborhood U with the property that Σ−1 (fr) has two mutually disjoint components.

4.8. MORSE THEORY ON SASAKI MANIFOLDS

177

Now we write the following part of the exact sequence from Theorem 4.5.2: ∆∗

ı

∗ 0 2 H 2 (M ) → → H 1 (M ) → HB (M ) → HB (M ) →

0 But by Proposition 4.5.1 we have HB (M ) ∼ = R and from Proposition 4.3.2 we 1 ∗ obtain H (M ) = 0. Hence ∆ (c) = c[ω ∗ ]B and then c1 (D0 ) = ı∗ ∆∗ (1) = ı∗ cB 1 (M ) = 0. Now, from the sequence of equalities which gives the second Stiefel-Whitney class we deduce (see for instance [Hu66], Chapter 16)

w2 (M ) = w2 (T M ) = w2 (D) = cB 1 (M )(mod 2) our assertion results easily.

4.8

Morse theory on Sasaki manifolds

This section is based on the results of P. Rukimbira, who initiated this study. Let M be a 2n + 1-dimensional compact regular Sasaki manifold and denote by S1 its circle group (see Theorem 2.6.3). A critical submanifold of the function f ∈ F(M ) is a connected submanifold of M with the property that its points are critical for f . Let T ∈ R. If the Reeb vector field ξ of M has a closed orbit γ of period T then γ is a closed characteristic of period T of ξ (or of the contact form η). This means that the reduced γ : R/T Z −→ M satisfies the condition γ˙ = ξγ. Proposition 4.8.1. Let M be a compact regular Sasaki manifold and Z be the infinitesimal generator of the circle action. Then the function S = η(Z) is invariant by this action and its critical circles are closed characteristics of η. Proof. Since Z leaves η invariant we have dS = dıZ η = LZ η − ıZ dη = −ıZ dη hence x ∈ M is a critical point for S if and only if Zx and ξx are collinear. But Z and ξ commute and then theirs orbits through x are the same. The Lyusternik-Schnirelman category of the compact space M is the minimum number of contractible open sets which can cover M and is denoted by cat(M ). The cup-length of M , denoted cuplength(M ), is the maximum number of cohomology classes [ω1 ], [ω2 ], . . ., [ωm ] of degrees ≥ 1 with the property that ω1 ∧ ω2 ∧ . . . ∧ ωm nowhere vanishes on M . The following inequality holds cat(M ) ≥ cuplength(M ) + 1

(4.72)

(See the Lyusternik-Schnirelman theory in J. T. Schwartz, Nonlinear functional analysis. Gordon and Breach Publ., 1969.) Proposition 4.8.2. [Ru99a] Let M be compact regular Sasaki manifold. Then any circle invariant function on M has at least cat(Mξ ) critical circles.

178


Proof. Let γ1 , . . . , γp be the critical circles of the circle invariant function f ∈ F(M ) and choose circle invariant tubular neighborhoods2 γ1∗ , . . . , γp∗ of them, each γi∗ being contractible to γi . Then γi∗ projects to a contractible open neighborhood Ui of the projection γ˜i of γi in the orbit space Mξ . ^f on Mξ . If {Ψt } grad f is circle invariant and induces a vector field grad ^f then the sets Vi = ∪t≥0 Ψt (Ui ) is the 1-parameter group generated by grad are open, contractible to Ui and hence to γ˜i in Mξ . But for each x ˜ ∈ Mξ there x) = γ˜i (˜ x) and then Mξ ⊂ ∪pi=1 Vi , hence is i ∈ 1, p such that limt→−∞ Ψt (˜ p ≥ cat(Mξ ). Proposition 4.8.3. [Ru99a] Let M be a compact regular 2n + 1-dimensional Sasaki manifold and denote by Z the infinitesimal generator of the circle action. If η(Z) nowhere vanishes on M then η has at least n + 1 closed characteristics. 1 η is a contact form and ıZ dη 0 = 0, hence Proof. First, we remark that η 0 = η(Z) 0 Z is the Reeb vector field of η . Then dη 0 defines a symplectic form on MZ , hence cuplength(MZ ) ≥ n and by Proposition 4.8.2 and using the inequality (4.72) it follows that the function S = η(Z) has at least n + 1 critical circles and these are closed characteristics of η (see the end of the proof of Proposition 4.8.1).

The critical submanifold N ⊂ M is called a nondegenerate critical submanifold of the function f ∈ F(M ) if the Hessian of f is nondegenerate in all directions normal to N . The function f ∈ F(M ) is clean if any critical submanifold is nondegenerate. Let f be a clean function on the Sasaki manifold M , N a nondegenerate − critical submanifold and for x ∈ N denote by ⊥+ x N , ⊥x N the positive and the negative eigenspaces of Hessx f , respectively (i. e. the direct sum of eigenspaces corresponding to positive and negative eigenvalues of Hessx f , respectively). Then the normal space Tx⊥ N of N has the following decomposition − Tx⊥ N =⊥+ x N ⊕ ⊥x N

We obtain two vector bundles ⊥+ N , ⊥− N on N . The fiber dimension λN of ⊥− N is called the index of N . ˇ k (M, R) be the k-th Cech ˇ Let H cohomology group of the manifold M . We call X ˇ k (M, R) Pt (M, R) = tk dim H k

the Poincaré series of M . We can associate to M another series with respect to a clean function f ∈ F(M ), namely X Mt (f ) = tλN Pt (N, θ− ⊗ R) N 2A

tubular neighborhood N of the manifold M is a pair (f, ξ0 ), where ξ0 = (E, π, N ) is a vector bundle over N and f : E −→ M is an embedding such that: a) f|N = 1N , where N is identified with the zero section of E, i. e. 0x ≡ x; 2. f (E) is an open neighborhood of N in M .


179

where N are the critical submanifolds of f and θ− is the orientation bundle of ⊥− N . Mt (f ) is the Morse series of f and the following Morse type inequalities hold for all t. Pt (M, R) ≤ Mt (f )

(4.73)

Theorem 4.8.4. [Ru99a] On any compact regular Sasaki manifold there is a clean function whose critical submanifolds have even index. Proof. We prove that the function S from Proposition 4.8.1 satisfies these conditions. Assertion 1. All critical submanifolds of S are non degenerate. First we compute the Hessian of S in directions orthogonal to a critical manifold N of S. For this purpose let x be a point of N and u, v ∈ Tx⊥ N . Denote by U, V two local vector fields in M , obtained by parallel translation along the geodesics of M passing through x. Hence we have ∇U = ∇V = 0 and using (1.54) and the property of Z to be a Killing vector field commuting with ξ (see the proof of Proposition 4.8.1) at x we obtain Hessx S(u, v) = (U (V g(ξ, Z))) = U (−g(F V, Z) + g(ξ, ∇V Z)) from Theorem 1.5.3 and Proposition 3.2.1, a) and since ξ is tangent to N , we have U (g(F V, Z)) = g (∇U (F V ), Z) + g (F V, ∇U Z) = = g ((∇U F ) V, Z) − g (V, F (∇U Z)) = = g(R(ξ, U )V, Z) − g (V, F (∇U Z)) = = g(U, V )η(Z) − g (V, F (∇U Z))

(4.74)

U (g (ξ, ∇V Z)) = g (∇U ξ, ∇V Z) + g (ξ, ∇U ∇V Z) = = −g (F U, ∇V Z) + g (ξ, ∇U ∇V Z)

(4.75)

But ∇U W = [U, W ] for any W ∈ X (M ) and from the Jacobi identity for the vector fields U, V, Z we obtain R(U, V )Z = 2∇U ∇V Z and (4.75) yields 1 U (g (ξ, ∇V Z)) = −g (F U, ∇V Z) + g(ξ, R(U, V )Z) = 2 (4.76) 1 = g (U, F (∇V Z)) − g(R(U, V )ξ, Z) = g (U, F (∇V Z)) 2 Now, since dS(U ) = 0 at x, from (4.74) and (4.76) we deduce Hessx S(u, v) = 2g (v, F ∇u Z) − 2g(u, v)S(x)

(4.77)

Setting Z = η(Z)ξ + δ, where δ is a Kiling vector field orthogonal to ξ and vanishing along N , we have ∇v Z = −η(Z)F v + ∇v δ and since ∇v δ is non zero and orthogonal to N we get Hessx S (v, F ∇v δ) = 2g (∇v δ, ∇v δ)

180


Therefore Hessx S is nondegenerate in the directions normal to N . Assertion 2. Any critical submanifold of f has even index. From (4.77) we deduce Hessx S(F u, F u) = 2g (F u, F ∇F u Z) − 2g(F u, F u)S(x) = = 2g (u, ∇F u Z) − 2g(u, u)S(x)

(4.78)

Now, using Theorem 1.5.3 and δ|N = 0 in T (F U, Z) = 0, we obtain g (∇F u Z, u) = g (F ∇u Z, u) + g([F U, Z], u) and taking into account (1.19) we get g(∇F u Z, u) = g(F ∇u Z, u). Hence from (4.78) we deduce Hessx S(F u, F u) = Hessx S(u, u) and this shows that if Hessx S is negative definite in the direction of u then it is also negative definite in the direction of F u and therefore the index of N is even. Theorem 4.8.5. [Ru99a] If M is a 2n + 1-dimensional compact regular Sasaki manifold with n + 1 closed characteristics then bk (M ) ≤ 1 for any k ∈ 0, 2n + 1. Proof. The manifold M hasPn + 1 closed characteristics, one for each even index n 0, 2, . . . , 2n and Mt (S) = i=0 t2i Pt (Ni , θi− ⊗ R). But ⊥− Ni are orientable 1 vector bundles over S , hence Pt (Ni , θi− ⊗ R) = 1 + t and (4.73) yields 2n+1 X i=0

bi (M )ti ≤

n X

t2i (1 + t)

i=0

The proof is complete by comparing the terms of these polynoms. Notice that from this result we also obtain another proof for Theorem 4.3.8. From Proposition 4.8.1 and Theorem 4.8.4 we deduce that any 2n + 1dimensional compact Sasaki manifold M has at least n + 1 closed characteristics and equivalently its characteristic foliation has at least n+1 closed leaves. Moreover, there are Sasaki manifolds with minimum number of closed characteristics as the following example shows. Example. [Ru00] Consider the sphere S 2n+1 with the deformed Sasaki structure defined in Section 1.5 (Linear deformations of Sasaki structures). If λ1 , λ2 , . . . , λn are rationally independent then from the general theory of differential equations it follows that the characteristic foliation has exactly n + 1 compact leaves and these are circles. Proposition 4.8.6. [Ru00] Let M be a compact Sasaki manifold of dimension 2n + 1. If M has n + 1 closed characteristics then it is simply connected.


181

For the proof of this result see [Ru00]. Let M be a 2n + 1-dimensional compact Sasaki manifold with n + 1 closed characteristics N0 , N1 , . . ., Nn . Taking into account Theorem 4.8.4 there are critical circles of the function S, each of index 0, 2, . . . , 2n, respectively. For k ∈ 0, n denote by M 2k+1 the closure of the set {x ∈ M : limt→∞ Ψt (x) ∈ Nk }, where {Ψt } is the gradient flow of the function S. For any X ∈ X (M ) we have dS(X) = dıZ η(X) = g(F Z, X), hence {Ψt } is the 1-parameter transformations group generated by F Z. Moreover, the sets M 2k+1 constitute a stratification of M , that is we have M 1 ⊂ M 3 ⊂ M 5 ⊂ . . . ⊂ M 2n−1 ⊂ M

(4.79)

and each M 2k+1 is a compact 2k+1-dimensional submanifold of M and has k+1 nondegenerate critical circles. On the other hand, M 2k−1 and Nk are disjoint compact submanifolds of M 2k+1 , hence each has a tubular neighborhood in M 2k+1 , that is we can find two open neighborhoods U and V for M 2k−1 and Nk , respectively and such that M 2k−1 , Nk ≡ S 1 and S 2k−1 ×S 1 are deformation retracts for U , V and U ∩ V , respectively. From the Mayer-Vietoris sequence associated to (U, V ) we obtain the following integer coefficients homology exact sequence . . . −→ H∗ S 2k−1 × S 1 −→ H∗ (M 2k−1 ) ⊕ H∗ (S 1 ) −→ (4.80) −→ H∗ (M 2k+1 ) −→ H∗−1 S 2k−1 × S 1 −→ . . . Proposition 4.8.7. [Ru00] Let M be a 2n + 1-dimensional compact Sasaki manifold with n + 1 closed characteristics and the stratification (4.79). Then: a) H1 (M 2k+1 ) = 0 for k ∈ 1, n and H2 (M 2k−1 ) ≡ H2 (M 2k+1 ) for k ∈ 2, n; b) if M 2k−1 is a homology sphere then M 2k+1 is also a homology sphere. Proof. a) For k ∈ 2, n the sequence corresponding to ∗ = 2 in (4.80) yields 0 −→ H2 (M 2k−1 ) −→ H2 (M 2k+1 ) −→ Z −→ H1 (M 2k−1 ) ⊕ Z −→ 0 and then H1 (M 2k−1 ) = 0. Now, the second part of the affirmation a) follows from the first part of the above exact sequence. b) We have H1 (M 2k+1 ) = 0, hence it remains to show that Hi (M 2k+1 ) = 0 for i ∈ 2, 2k. For ∗ ∈ 2, l and l ∈ 2, k − 1 the exact sequence (4.80) yields 0 −→ H2l (M 2k+1 ) −→ 0 −→ 0 −→ H2l−1 (M 2k+1 ) −→ 0 and then Hi (M 2k+1 ) = 0 for i ∈ 3, 2k − 2. Similarly, for ∗ = 2 and using a) we get 0 −→ H2 (M 2k+1 ) −→ Z −→ Z −→ 0 and then H2 (M 2k+1 ).

182


We have H0 (M 2k+1 ) ∼ = Z, H1 (M 2k+1 ) = 0 and by duality it follows H2k (M 2k+1 ) ∼ = H 1 (M 2k+1 ) ∼ = Hom H1 (M 2k+1 ), Z = 0 By the same argument, but taking into account H1 (M 2k+1 ) = 0, H2 (M 2k+1 ) = 0 we deduce H2k−1 (M 2k+1 ) ∼ = H 2 (M 2k+1 ) ∼ = Hom H2 (M 2k+1 ), Z = 0 and the proof is complete. Finally, using Proposition 4.8.7 one can prove that, up to homeomorphisms, the spheres are the unique compact manifolds with minimal number of closed characteristics (see the above Example). Theorem 4.8.8. [Ru00] A compact Sasaki manifold of dimension 2n + 1 with n + 1 closed characteristics is homeomorphic to the sphere S 2n+1 . Remark that Theorem 4.8.5 is a simple corollary of Theorem 4.8.8. We also remark that P. Rukimbira proves all results presented in this Section for K-manifolds.

4.9

Construction of contact manifolds

First we recall that all manifolds are assumed to be without boundary and the contrary case is especially emphasized. In Chapter 1 we presented some examples of contact manifolds, but their construction used some particularities of the involved manifolds. However, in Chapter 3 we studied the ∗-product of Sasaki manifolds and proved that, under so few restrictive conditions, the resulting manifold is also Sasakian. The contact topology furnishes many topological operations which can be performed on contact manifolds, having as result a contact structure. In this Section we present the construction of some contact structures on 3dimensional manifolds, on the connected sum of two manifolds. We also study two other constructions of contact manifolds by branched covers and by contact reduction. But we have to notice that the most powerful method to construct contact manifolds in higher dimensions is by contact surgery. Since it uses many advanced knowledges of differential topology we do not present here (see for instance [El90] and [We91]). Even for the next constructions some basic background in differential topology is necessary. Contact structure on 3-dimensional manifolds The existence of contact structures on 3-dimensional manifolds has a long history which begin by S. S. Chern’s question [Che53]: Does every compact orientable 3-dimensional manifold admits a contact structure? The affirmative answer was given by R. Lutz [Lu71] and J. Martinet [M71] and it is contained in the following

4.9. CONSTRUCTION OF CONTACT MANIFOLDS

183

Theorem 4.9.1. On every compact orientable 3-dimensional manifold there exists a contact structure. Proof. We present W. P. Thurston and H. E. Winkelnkemper’s proof, [TW75]. Denote by M the given manifold, by P a orientable 2-dimensional compact manifold with boundary and let µ : P −→ P be a diffeomorphism which is the identity map on some collar neighborhood V of ∂P . This means that V is a neighborhood of ∂P with the property that there exists a diffeomorphism f : ∂P × [0, 1) −→ V so that f (x, 0) = x for any x ∈ ∂P . The existence of collar neighborhoods for the boundary of any manifold is guaranteed by a classical result from the differential topology (see for instance [Hi76], pg. 113). Let t be the collar parameter (i. e. the image by f of the natural parameter of the interval [0, 1)) and denote by dθ the volume form of ∂P . If ω ∈ F 1 (P ) is a 1-form such that ω = (1 + t)dθ near ∂P then we can consider a volume form Ξ on P , equal to dt ∧ dθ near ∂P . By Poincaré Lemma we have Ξ − dω = dλ, where λ is a 1-form vanishing near ∂P . It follows that the 1-form ω 0 = ω + λ has the properties: a) dω 0 is a volume form for P ; b) ω 0 = (1 + t)θ near ∂P . 0 The set of 1-forms ω satisfying a) and b) is convex and then, denoting by Pµ the manifold obtained from P ×[0, 1] by identifying the point (x, 0) and (µ(x), 1) for each x ∈ P , there exists a 1-form α ∈ F 1 (Pµ ) such that its restriction to any fiber satisfies a) and b). Moreover, we have α = (1 + t)dθ on a neighborhood of ∂Pµ . Pµ has a bundle structure over S 1 and denote by dΦ the pullback of the volume form of its base space. Then for k ∈ R the 1-form η = α+kdΦ ∈ F 1 (Pµ ) satisfies the condition η ∧ dη = kdΦ ∧ dα + α ∧ dα and since dΦ ∧ dα is a nondegenerate 3-form on Pµ it follows that for k sufficiently large η is a contact form on Pµ . Now we extend η to ∂Pµ × D2 , where D2 is the unit disk in R2 . For this purpose remark that we can take Φ as the polar angle on S 1 and if r is the polar radius in D2 then θ, r, Φ are local coordinates in ∂Pµ × D2 . We identify V ×S 1 with ∂Pµ ×(D2 (2) − D2 ), where D2 (2) is the disk of radius equal to 2 and remark that on ∂Pµ × (D2 (2) − D2 ) we can consider the same coordinates θ, Ψ and r = 1 + t, hence with respect to these coordinates we can set η = rdθ + kdΦ. On the other hand on ∂Pµ × D2 but near r = 0 we put η = −dθ + r2 dΦ and remark that for > 0 and r ∈ (0, ), η is a contact form. Hence on ∂Pµ × D2 (2) we search the 1-form η with the expression (−1, r2 ) for r ∈ (0, ) η = f1 (r)dθ + f2 (r)dΦ and (f1 (r), f2 (r)) = (r, k) for r ∈ [1, 2] Moreover, η is a contact form if and only if f (r) f2 (r) η ∧ dη = det 10 dθ ∧ dr ∧ dΦ 6= 0 f1 (r) f20 (r) This last condition is fulfilled for r ∈ (0, ) ∪ [1, 2] and obviously we can extend f1 , f2 to [0, 2] in such a manner that f1 (r)f20 (r) − f10 (r)f2 (r) 6= 0, hence η is a contact form on ∂Pµ × D2 .

184


Now, under the proposed hypothesis on M , by Alexander adjunction Theorem3 the manifold M is diffeomorphic to the manifold Pµ ∪id ∂P × D2 , obtained by gluing Pµ with ∂P × D2 along the identity map of ∂P × S 1 . The next question is about the existence of nonequivalent contact forms in the sense of the definition given in Section 1.3. First we prove the following Proposition 4.9.2. [Go87] Let M be a compact 3-dimensional contact manifold with the contact form η1 . The following assertions are equivalent: a) there exist the contact forms η2 , η3 such that η1 ∧η2 ∧η3 nowhere vanishes; b) there exist two 1-forms α, β such that η1 ∧ α ∧ β nowhere vanishes; c) the vector bundle D = ker η1 is trivial; d) M has a parallelization by the vector fields X, Y, Z such that [X, Y ] = Z. Proof. a) ⇒ b) is obvious. b) ⇒ a) For each > 0 the forms η1 , η1 + α, η1 + β are independent on M and for small these are also contact forms. b) ⇔ c) is obvious because a k-dimensional vector bundle is trivial if and only if there exist k sections linearly independent at any point of its base space. c)⇒ d) If X, Y ∈ D are independent on M then from the condition η∧dη 6= 0 it follows that η[X, Y ] 6= 0 at any point of M and then Z = [X, Y ] is independent of X and Y . d) ⇒ c) By setting ω(X) = ω(Y ) = 0, ω(Z) = 1 it follows that ω is a contact form and ker ω is a trivial bundle. Now, the answer to our second question is given by J. Gonzalo, who proves the following Theorem 4.9.3. [Go87] Each compact orientable 3-dimensional manifold has a parallelization by three contact forms. The proof uses techniques of contact topology and it can be found in [Go87]. The existence of contact structures on higher dimensional manifolds was studied by several authors by using refined techniques. In particular, for 5dimensional manifolds H. Geiges proves the following Theorem 4.9.4. [G91] Any compact orientable simply connected 5-dimensional manifold admits a contact structure in every homotopy class of almost contact structures. 3 Alexander Theorem. Every compact orientable manifold M of dimension 3 and without boundary is diffeomorphic to Wh ∪id ∂W × D2 , where W is a 2-dimensional orientable manifold with boundary and h : W −→ W is a diffeomorphism whose restriction to ∂W is the identity. For the proof see [Ale23].


185

Connected sum of two contact manifolds Let S 2n (a) be the sphere of radius a centered at the origin of R2n+1 and consider two tubular neighborhoods Σ1 and Σ2 of S 2n (a) such that neither of them contains the other. Assume that there are the contact forms η1 , η2 on Σ1 and Σ2 , respectively, and denote by dV a volume form on R2n+1 . Our purpose is to glue these forms in order to obtain a contact form on Σ1 ∪ Σ2 . Proposition 4.9.5. [Me82] If the following conditions are fulfilled: a) there exist the real valued functions f1 > 0, f2 > 0, g1 , g2 such that the following equalities hold on Σ1 ∩ Σ2 ηi ∧ dni = fi dV,

η1 ∧ η2 ∧ dηin−1 ∧ dr = (r − a)gi dV

√ where i ∈ {1, 2} and r = < x, x > is the radial coordinate; b) there exists a 1-form ω ∈ F 1 (Σ1 ∩ Σ2 ) such that η2 − η1 = (r − a)ω; c) fi − n3 |hi | > 0 for i = 1, 2. Then there exists a contact form η on Σ1 ∪ Σ2 so that for i = 1, 2 we have η|Σi −Σ1 ∩Σ2 = ηi|Σi −Σ1 ∩Σ2 Proof. By the definition of the tubular neighborhood, we can take Σ1 and Σ2 under the form Σi = (a − i , a + 0i ) × S 2n (a) with i > 0, 0i > 0. If ≤ min(1 , 2 , 01 , 02 ) then Σ0 = (a − , a + ) × S 2n (a) ⊂ Σ1 ∩ Σ2 and consider a smooth real valued function f defined on Σ1 ∩ Σ2 and satisfying the conditions 1 1 for r ≤ a − f (r) = , |(r − a)f 0 (r)| ≤ 0 for r ≥ a + 3 The 1-form η = f η1 + (1 − f )η2 satisfies the conditions η|Σi −Σ0 = ηi for i = 1, 2, hence it is a contact form on Σ1 ∪ Σ2 if it is a contact form on Σ0 . We have n

η ∧ (dη)n = [f η1 + (1 − f )η2 ] ∧ [f dη1 + (1 − f )dη2 ] + n−1

+ nf 0 [f dη1 + (1 − f )dη2 ]

∧ η1 ∧ η2 ∧ dr

(4.81)

But by recurrence on k it follows that for each k ∈ N there exists a 1-form αk such that k

[f dη1 + (1 − f )dη2 ] = f dη k + (1 − f )dη2k − f (1 − f )d(η2 − η1 )2 ∧ αk and then from (4.81) we deduce η ∧ (dη)n = −f (1 − f )θ+ + {f [f1 + nf 0 (r − a)h1 ] + (1 − f ) [f2 + nf 0 (r − a)h2 ]} dV where θ = (η2 − η1 ) ∧ d (η2 − η1 ) ∧

n−1 X

(dα2 )i ∧ (dα1 )n−i−1 +

i=0 2

+ [f η1 + (1 − f )η2 ] ∧ d (η2 − η1 ) ∧ αn + 2

+ nf 0 η1 ∧ η2 ∧ d (η2 − η1 ) ∧ ωn−1 ∧ dr

(4.82)

186

CHAPTER 4. DIFFERENTIAL FORMS AND TOPOLOGY 2

The forms (η2 − η1 ) ∧ d (η2 − η1 ), d (η2 − η1 ) and θ vanish on Σ0 and then taking into account the hypothesis c) in (4.82) we deduce that for small the form η ∧ (dη)n nowhere vanishes on Σ0 . Now, take two disjoint connected contact manifolds M1 , M2 of dimension 2n+1 and let µ : S 1 ×D2n+1 −→ M1 ∪M2 be an embedding with µ(1×D2n+1 ) ⊂ M1 , µ(−1× D2n+1 ) ⊂ M2 , where D2n+1 is the 2n + 1-dimensional disk. The manifold M ∪ N − Int µ S 0 × D2n+1 ∪µ D2n × S 1 is called the connected sum of M1 and M2 and denote it by M1 ]M2 . We can view M1 ]M2 as being obtained by gluing together M1 − Int D1 and M2 − Int D2 by a diffeomorphism between the boundaries of the disks D1 ∈ M1 and D2 ∈ M2 . Of course M1 ]M2 is orientable if and only if M1 and M2 are orientable. Moreover, we can state the following Theorem 4.9.6. [Me82] Let M1 and M2 be compact connected contact manifolds of dimension 2n + 1 with the contact forms η1 and η2 , respectively. On the connected sum M1 ]M2 there exists a contact form η and two open neighborhoods U1 , U2 of the boundaries of the gluing disks D1 and D2 , respectively, all these satisfying the conditions η|M1 −U1 = η1|M1 −U1 ,

η|M2 −U2 = η2|M2 −U2

Branched covers Let us consider the compact oriented 2n+1-dimensional manifolds M and M0 . A branched covering along N0 is a map µ : M −→ M0 such that there exists a 2n − 1-dimensional submanifold N0 of M0 with the property that N = µ−1 (N0 ) is a 2n − 1 dimensional submanifold of M and µ|M −N : M − N −→ M0 − N0 is a covering map. With these notations we can state the following Theorem 4.9.7. [G97b] If M0 admits a contact form η0 such that η0|X (N0 ) is a contact form on N0 then M admits a contact form η with the property that η = µ∗ η0 outside a neighborhood of N . Proof. We remark that η ∗ = µ∗ η0 is a contact form on M − N and moreover η ∗ defines a contact form on N . Let ω be a connection form on the normal sphere bundle of N and N (δ) a tubular neighborhood of radius δ of N with respect to some Riemannian metric on M . If we denote by r the radial coordinate in N (δ) then, as in the proof of Proposition 4.9.5, we can define a smooth function f (r) equal to 1 if r is near 0 and equal to 0 if r ≥ δ. Then for a large constant k > 0 the 1-form η = kη ∗ + f (r)r2 ω is globally defined on M and we have η ∧ (dη)n = k n+1 η ∗ ∧ (dη ∗ )n + + nk n (rf 0 + 2f )η ∗ ∧ (dη ∗ )n−1 ∧ rdr ∧ ω + 0,

k n+1 η ∗ ∧ (dη ∗ )n|N ≥ 0,

nk n (rf 0 + 2f )η ∗ ∧ (dη ∗ )n−1 ∧ rdr ∧ ω|N > 0,

0. On the other hand on the compact set K0 = {x ∈ M : δ1 ≤ r(x) ≤ δ} all terms in the right hand-side of (4.83) are bounded from below and k n+1 η ∗ ∧ (dη ∗ )n > 0, hence for k sufficiently large we have η ∧ (dη)n > 0 on K0 . Since this inequality holds for r > δ, it follows that η is a contact form on M and satisfies the imposed condition. By using Proposition 4.9.2, H. Geiges obtains the following application of this Theorem (see also Theorem 4.9.4): Theorem 4.9.8. [G97b] The product of any compact orientable manifold of dimension 3 with a surface of genus 3 admits a contact form. Reduction of contact manifolds Let M be a contact manifold with the contact form η and consider the compact Lie group G whose action R : G × M −→ M preserves the form η, i. e. (Rg )∗ η = η for any g ∈ G. We will prove that there exists a moment map µ : M −→ g∗ so that the reduced manifold µ−1 (0)/G can be equipped with a contact structure. In the case when G = S1 this result is the following Theorem 4.9.9. [G97b] Let M be a manifold with the contact form η and assume that the action of the group S1 on M is generated by a vector field X ∈ X (M ) and preserves the contact form η. 0 is a regular value for µ = η(X) : M −→ R if and only if X nowhere vanishes on M . Moreover, if 0 is regular value for µ then X generates a free action on µ−1 (0) and the quotient manifold µ−1 (0)/S1 admits a contact structure. Proof. We have dµ = dıX η = LX η − ıX dη = −ıX dη It follows that dx µ = 0 at the point x ∈ µ−1 (0) if and only if Xx = 0, hence the first part of the theorem is proved. From the above equalities we also deduce dµ(X) = 0 and therefore X is tangent to level surfaces of µ. On the other hand we have LX η = 0 and η(X) = 0 along µ−1 (0), so that η projects into a 1-form η0 on the quotient manifold µ−1 (0)/ hXi. But ker dη|X (µ−1 (0))∩ker η = hXi, hence η0 is contact form on µ−1 (0)/S1 .

188


Let G be a Lie group acting on the contact manifold M and denote by g its Lie algebra. Also we denote by X ∈ X (M ) the vector field generated by X ∈ g. Then the above result admits the following generalization Theorem 4.9.10. Let M be a contact manifold and let G be a compact Lie group whose action over M preserves the contact form η. Then: a) The moment map µ : M −→ g∗ defined for any x ∈ M by µ(x)(X) = ηx (Xx ) is G-equivariant; b) G acts over µ−1 (0); c) if 0 is a regular value of µ then the action of G over µ−1 (0) is free and the reduced manifold µ−1 (0)/G has a contact structure. We notice a remarkable result obtained by M. de Leon, J. C. Marrero, G. H. Tynman [LMT97]. This asserts that any contact manifold can be obtained by reduction from the standard contact manifold R2n+1 . G. Grantcharov and L. Ornea proved that if M is a Sasaki manifold then the reduced manifold µ−1 (0)/G is also Sasakian. More precisely we have Theorem 4.9.11. [GOr99] Let M be a compact Sasaki manifold of dimension 2n + 1 and G a compact k-dimensional Lie group acting on M by contact isometries. If 0 ∈ g∗ is a regular value of the moment map µ : M −→ g∗ , µ(x)(X) = ηx (Xx ) then the reduced manifold P = µ−1 (0)/G is Sasakian of dimension 2(n − k) + 1. Proof. Let {X 1 , . . . , X k } be a base of the algebra g and denote by X1 , . . . , Xk the corresponding vector fields on M . Since 0 is a regular value for µ it follows that {X1x , . . . , Xkx } is an independent system in Tx µ−1 (0). But ηx (Xi ) = 0, hence Xi ⊥ ξ and since G acts by contact isometries, we have LXi g = 0, LXi η = 0 and then [Xi , ξ] = 0. On the other hand, Y ∈ Tx µ−1 (0) if and only if µ∗,x Y = 0. It follows that ξ ∈ Tx µ−1 (0) and g(F Xi , Y ) = 0, hence {F X1 , . . . , F Xk } is a local basis in the normal bundle of µ−1 (0). Let ∇0 be the Levi-Civita connection induced on µ−1 (0) by the Levi-Civita connection on M . Taking into account Theorem 1.5.3 we have −1 −1 g (AF Xi Y, Z) = −g ∇Y kXi k F Xi , Z = − kXi k , (4.84) −1 g (∇Y (F Xi ), Z) = kXi k [g(Xi , Y )η(Z) − g (F ∇Y Xi , Z)] If we set Y = Z = ξ then from (4.84) we deduce −1

g(h(Y, ξ), F Xi ) = kXi k

g(Xi , Y ),

g(h(ξ, ξ), F Xi ) = 0

(4.85)

It follows that ξ|µ−1 (0) is a Killing vector field on µ−1 (0). By using the Gauss equation and taking into account (4.85), for X, Y, Z ∈ X (µ−1 (0)) orthogonal to ξ we have g(R0 (X, ξ)Y, Z) − g(R(X, ξ)Y, Z) = =−

k X i=1

−2

kXi k

[g(Xi , Z)g (∇X Xi , F Y ) − g(Xi , Y )g (∇X Xi , F Z)]

(4.86)


189

Let π : µ−1 (0) −→ P be the projection map and denote by g P the projection of the metric g on P . Since G acts by isometries, it follows that π becomes a Riemannian submersion. The vector fields Xi span the vertical distribution of the submersion π and since LXi ξ = 0 it follows that ξ is horizontal and projectable (see Section 2.5). Its projection ξ P on P is unitary and LξP g P (Y 0 , Z 0 ) = Lξ g(Y, Z), where X 0 , Y 0 are the vector fields on P for which X and Y , respectively, are π-related (see Section 2.3). Hence ξ P is a Killing vector field on P . On the other hand, from (4.85) and taking into account the Gauss formula we obtain g (∇Z ξ, Xi ) = g(F Z, Xi ) = −g(Z, F Xi ) = 0 for Z horizontal and orthogonal to ξ. Taking into account X 0 , Y 0 are orthogonal to all Xi , from the Gauss equation of the submersion π (see the formula (2.23)) and (4.86) we get −1

RP (X 0 , ξ P )Y 0 = Rµ

(0)

(X, ξ)Y = RM (X, ξ)Y

for all X, Y ∈ X (µ−1 (0)) and from Theorem 3.2.9 applied to the manifold M we deduce RP (X, Y )ξ P = g(ξ, Y )X − g(ξ, X)Y = g P (ξ P , Y P )X P − g P (ξ P , X P )Y P and thus P is Sasakian.

190


Chapter 5

Integral submanifolds 5.1

General properties

Let M be a contact manifold of dimension 2n + 1 and denote by η its contact form. The contact distribution D is never integrable since dη 6= 0. Therefore it is natural to find the maximum dimension of its integral submanifolds and to study theirs characteristic properties. The integral submanifolds of the distribution D, called integral submanifolds of the manifold M or C-totally real submanifolds, have the following elementary properties: Theorem 5.1.1. Let N be an integral submanifold of the 2n + 1-dimensional contact Riemannian manifold M . Then: a) F Tx N ⊂ Tx⊥ N for any x ∈ N ; qquad b) dim N ≤ n. Proof. a) For any X, Y ∈ Tx N we have dη(X, Y ) = 0 and from dη = Ω we obtain a). b) F|D is injective and ξ ∈ Tx⊥ M so that from a) we deduce 2dim Tx M + 1 ≤ 2n + 1. From the local expression of the contact form (see Theorem 1.3.11) it follows that the equations xi = constant and z = constant locally define an integral submanifold of dimension n of the manifold M . More generally, for any r ≤ n and for all real constants cr+1 , . . . , cn , c, the equations xr+1 = cr+1 , . . . , xn = cn , y 1 = . . . = y n = 0, z = c

(5.1)

locally represent an integral submanifold of dimension r. Hence the contact manifold M admits integral submanifolds in any dimension r ≤ n. Taking into account D = ker η and Proposition 5.1.1, a), on any integral submanifold the following equalities hold η = 0,

dη = 0

(5.2)

But it is easy to show that these formulas completely characterize the integral submanifolds, namely we have 191

192

CHAPTER 5. INTEGRAL SUBMANIFOLDS

Proposition 5.1.2. Let N be a submanifold of the contact Riemannian manifold M . The following assertions are equivalent: a) N is an integral submanifold of M ; b) the equalities (5.2) hold on N ; c) ξ ∈ X ⊥ (N ) and F X ∈ X ⊥ (N ) for any X ∈ X (N ). Proof. a) ⇔ b) is an immediate consequence of the formulas (5.2) and the definition of integral submanifolds. a) ⇒ c) follows from Proposition 5.1.1 and c) ⇒ b) is obvious. The existence of integral submanifolds in any dimension ≤ n and the following theorem shows how rich is the family of integral submanifolds of a contact manifold. This assertion is also proved by the following three results: Theorem 5.1.3. Let M be a contact manifold of dimension 2n+1 and r ∈ 1, n. For each X ∈ D and x ∈ M there exists an integral submanifold N of dimension r passing through x and such that X|N ∈ X (N ). Proof. It is enough to choose a local chart at the point x with the coordinates xi , y i , z and such that the contact form η has the canonical expression from ∂ Theorem 1.3.11 and X = ∂x 1 at x. Then the searched submanifold is locally given by equations of the type (5.1). In the case of the standard contact manifold R2n+1 this result can be improved as follows: Proposition 5.1.4. Let u = (xi0 , y0i , z0 ) be a point of the contact manifold R2n+1 . For r ≤ n the linearly independent vectors Xα aiα , biα , cα , α ∈ 1, r are tangent at x to a r-dimensional integral submanifold of R2n+1 if and only if cα =

n X i=1

y0i aiα ,

n X i=1

aiα biβ =

n X

aiβ biα

i=1

Proof. If we consider the canonical expression of the contact form (see the proof of Proposition 5.1.3), then the first set of equalities follows from the condition η(Xα ) = 0. The others equalities are obtained by simple computation from the condition dη(Xα , Xβ ) = 0. We can state the following characterization of the contact transformations by their behavior with respect to integral submanifolds: Theorem 5.1.5. The diffeomorphism µ of the contact manifold M is a contact transformation if and only if for each r-dimensional integral submanifold N , µ(N ) is an integral submanifold of dimension r. Proof. If µ is a contact transformation then from Proposition 1.3.13 and from the equality µ∗ [X, Y ] = [µ∗ X, µ∗ Y ] it follows that µ(N ) is also an integral submanifold.

5.1. GENERAL PROPERTIES

193

Conversely, let us consider x ∈ M and X ∈ Dx . By applying Theorem 5.1.3 there exists an integral submanifold N passing through x and having Xx as tangent vector at x. But µ(N ) is an integral submanifold, hence µ∗,x Xx ∈ Dµ(x) , that is µ∗ D ⊂ D and then the assertion follows from the same Proposition 1.3.13. Lemma 5.1.6. Let N be an integral submanifold of a Sasaki manifold M . Then for all X, Y ∈ X (N ) we have Aξ = 0,

AF X Y = AF Y X,

T

AF Y X = (F h(X, Y )) , ⊥

N ∇⊥ X (F Y ) = g(X, Y )ξ + F ∇X Y + (F h(X, Y )) ,

∇⊥ X ξ = −F X

where ∇N is the Levi-Civita connection induced on N by the Riemannian structure of M and ∇⊥ is the normal connection. Proof. These equalities follow by using the Gauss-Weingarten formulas in Theorem 1.5.3 and in the formula (1.54) from its proof and by equalizing the components of the same type. Denote by R⊥ the curvature tensor of the connection ∇⊥ in the normal bundle T ⊥ M of the submanifold N . Proposition 5.1.7. Let N be an integral submanifold of the Sasaki manifold M . Then: a) R⊥ (X, Y )ξ = 0 for any X, Y ∈ X (N ); b) If N is totally geodesic and if M is a Sasaki space form with F -sectional curvature c then N has constant sectional curvature c+3 4 . Proof. a) follows from Theorem 1.5.3 and from the Weingarten formula. b) is a immediate consequence of the Gauss equation written for the vectors of an orthonormal local basis in N . Theorem 5.1.8. Let N be a p-dimensional minimal integral submanifold of the Sasaki space form M (c) of dimension 2n + 1. c+3 a) RicN − c+3 4 (p − 1)g is negatively semi-defined and ρN ≤ 4 (p − 1)p. The equality holds if and only if N is totally geodesic. b) If N is compact, c ≥ 1 and if the second fundamental form h of N satisfies 2 the condition khk < p(2n+1−p)(c+3) then N is totally geodesic. 4(4n−2p+1) Proof. Taking an orthonormal local basis {X1 , . . . , Xp } in X (N ), from the Gauss equation it results RicN (X, Y ) = +

p X

c+3 (p − 1)g(X, Y )+ 4

[g (h(Xa , Xa ), h(X, Y )) − g (h(Xa , X), h(Xa , Y ))] =

a=1 p X c+3 = (p − 1)g(X, Y ) + pg(H, h(X, Y )) − g(h(Xa , X), h(Xa , Y ) 4 a=1

194


where H is the mean curvature vector of N and then the first affirmation a) follows easily. On the other hand, from this equality we obtain p h i X c+3 2 g (h(Xa , Xa ), h(Xb , Xb )) − kh(Xa , Xb )k = ρN = (p − 1)p + 4 a,b=1

=

p X c+3 2 (p − 1)p + p2 g(H, H) − kh(Xa , Xb )k 4 a,b=1

and then the second affirmation a) is obvious. b) A laborious computation (see [LOK77], Lemmas 6, 7 and Theorem 15) 2 2 shows that ∆ khk ≥ 0 and since N is compact it follows ∆ khk = 0, hence 2 khk = 0. A result similar to Theorem 5.1.8, a) is obtained by K. Matsumoto and I. Mihai, in absence of the minimality condition of the submanifold. Theorem 5.1.9. [MM02] Let N be a p-dimensional integral submanifold of the Sasaki space form M (c). Then c+3 p2 2 (p − 1)g − kHk g 4 4 is negatively semi-defined. This form vanishes identically if and only if N is totally geodesic or if p = 2 and if N is totally umbilical. RicN −

The mean curvature vector H of the submanifold N is said parallel if ∇⊥ H = 0. In this case we say that N has parallel mean curvature. Theorem 5.1.10. Let N be a compact integral submanifold of a Sasaki manifold. If N has nontrivial parallel mean curvature then b1 (N ) ≥ 1. Proof. Denote by α~n the dual form of the vector field F ~n. It is defined by α~n (X) = g(F ~n, X) and α~n is closed if and only if g ∇⊥ n, F Y = g ∇⊥ n, F X (5.3) X~ Y~ for any X, Y ∈ X (N ). Therefore we have dαH = 0. On the other hand, from the Gauss formula we deduce (∇Xa F ) H = 0,

g(h(X, Y ), ξ) = 0

for each vector Xa of the local basis {X1 , . . . , Xp } considered in the proof of Theorem 5.1.8 and then by applying the Weingarten formula we obtain p X

p X g ∇N g (∇Xa (F H), Xa ) = Xa (F H), Xa =

a=1

a=1 p X

=

g ((∇Xa F ) H + F ∇Xa H, Xa ) =

a=1

=−

p X a=1

g (∇Xa H, F Xa ) = −

p X a=1

g ∇⊥ Xa H, F Xa = 0

5.1. GENERAL PROPERTIES

195

Hence δαH = 0, that is αH is harmonic and since H is nontrivial we obtain the result. At the end of this section we write the structure equations of a p-dimensional integral submanifold Sasaki manifold M of dimension 2n + 1. For this purpose we consider a F -basis B = {Xi , Xi∗ = F Xi , X2n+1 = ξ}i∈1,n such that the restrictions to N of the vector fields X1 , . . . , Xp are tangent to the submanifold ∗ N . Denoting by B ∗ = ω i , ω i , ω 2n+1 = η the dual basis, at the points of N we have ωi = 0

∗

ωj = 0

ω 2n+1 = 0

(5.4)

for i ∈ p + 1, n and j ∈ 1, n. Let ω = (ωba ) be the connection form of the Levi-Civita connection ∇ on M . With respect to the basis B the connection ∇ is given by the formulas ∇X X a =

n h X

i ∗ ωaj (X)Xj + ωaj (X)Xj ∗ + ωa2n+1 (X)ξ,

(5.5)

j=1

for a ∈ i, i∗ , 2n + 1; i ∈ 1, n and X ∈ X (M ). But B is an orthonormal basis hence g (∇X Xa , Xb ) = −g (Xa , ∇X Xb ) and by applying (5.5) it results that at the points of the submanifold N the components of the connection form with respect to B satisfy the conditions ∗

β β α ω2n+1 = 0, ω2n+1 = ω2n+1 (X) = g(X, Xα ), = 0, ωα2n+1 ∗ ∗

∗

∗

ωαi = ω α ,

∗

∗

∗

ωβi = ωiβ ,

ωαi ∗ = ωαi ,

ωβi ∗ = ωβi

for all α, β ∈ 1, p and i ∈ p + 1, n. ¯ = (Ω ¯ b ) of the manifolds Now, we consider the curvature forms Ω = (Ωβα ), Ω a M and N , respectively. These forms are given by Ωγδ =

2n+1 1 X α Rβγδ ω γ ∧ ω δ , 2 γ,δ=1

p X ¯a = 1 ¯ a ωc ∧ ωd Ω R b bcd 2

(5.6)

c,d=1

α ¯ a are the components (with where α, β ∈ 1, 2n + 1, a, b ∈ 1, p and Rβγδ , R bcd respect to B) of the curvature tensors of the manifolds M and N , respectively. Taking into account (5.1), at the points of the integral submanifold N we have

¯ a = Ωa − Ω b b ¯λ Ω µ

=

n X

ωλa ∧ ωbλ −

λ=p+1 p X Ωλµ − ωaλ a=1

n X

∗

ωja∗ ∧ ωbj ,

j=1

∧

ωµa

p 1 X λ a = Rµab ω ∧ ω b 2 a,b=1

so that by using them in the equations (2.28) we obtain

(5.7)

196


Proposition 5.1.11. The structure equations of a p-dimensional integral submanifold in a Sasaki manifold of dimension 2n + 1 are the following a

dω = −

p X

ωba ∧ ω b ,

b=1

dωba = − dωµλ = −

p X

¯ a, ωca ∧ ωbc + Ω b

c=1 n X

ωνλ ∧ ωµν −

ν=p+1

n X

∗

¯λ ωjλ∗ ∧ ωµj + Ω µ

j=1

where a, b ∈ 1, p, λ, µ, ν ∈ p + 1, n.

5.2

Chern classes of integral submanifolds

Let M be a Sasaki manifold M of dimension 2n + 1. Taking into account Proposition 5.1.1, a), the normal space at the point x to the p-dimensional integral submanifold N has the following orthogonal decomposition Tx⊥ N = F (Tx N ) ⊕ τx (N ) ⊕ hξx i

(5.8)

where τx (N ) is the orthogonal complement of the subspace F (Tx N ) ⊕ hξx i in Tx⊥ N . Then dim τx (N ) = 2(n − p) and τ (N ) = ∪x∈N τx (N ),

π : τ (N ) −→ N, π(~nx ) = x

define a subbundle (τ (N ), π, N ) of the normal bundle T ⊥ N , also denoted by τ (N ). Moreover, let {Xi , Xi∗ , ξ}i∈1,n be a F -basis in M with the property that X1 , . . . , Xp are tangent to N and Bτ = Xp+1 , . . . , Xn , X(p+1)∗ , . . . , Xn∗ is a local basis of the module Γ(τ ) of all sections in the subbundle τ (N ). Of course, this construction has a sense only if p < n and we suppose fulfilled this condition. The subbundle τ (N ) is called the maximal invariant normal subbundle of the integral submanifold N . It has the following elementary properties: Proposition 5.2.1. Let N be an integral submanifold of the Sasaki manifold M and p < n. The maximal invariant normal subbundle of N has a natural structure of complex vector bundle and it is invariant with respect to F , i. e. F (τx (N )) = τx (N ) for any x ∈ N . ∗ Proof. Denote by nλ , nλ the components of the vector ~nx ∈ τx (N ) with respect to the basis Bτ . Then we can construct vector charts of the form (U, Φ, Cn ), where (U, φ) is a local in N and Φ : π −1 (U ) −→ U × Cn−p √ chart∗ is given by Φ(~nx ) = x, nλ + −1nλ for x ∈ U . These charts define a structure of complex vector bundle. The second part of the Proposition follows from the decomposition (5.8).

5.2. CHERN CLASSES OF INTEGRAL SUBMANIFOLDS

197

On τ (N ) we can define a connection as follows: If ∇⊥ is the induced connection by the Levi-Civita connection ∇ of the manifold M in the normal bundle T ⊥ N of the integral submanifold N then we have g ∇⊥ n, ξ = 0 for all X ∈ X (N ) and ~n ∈ Γ(τ ). Hence the normal vector X~ field ∇⊥ ~ n admits the decomposition X ∇⊥ n = B~n X + ∇τX ~n X~

(5.9)

where B~n X ∈ Γ(F T N ) and ∇τX ~n ∈ Γ(τ ). Moreover, the maps B : Γ(τ ) × X (N ) −→ Γ(F T N ),

∇τ : X (N ) × Γ(τ ) −→ Γ(τ )

have the following properties: Lemma 5.2.2. a) ∇τ is an almost complex connection in the maximal invariant normal subbundle of the integral submanifold N , that is (∇τX F ) ~n = 0 for all X ∈ X (N ) and ~n ∈ Γ(τ ). b) B~n X = F AF ~n X Proof. It follows from (5.9) by using the Weingarten formula. Taking into account Proposition 5.2.1 and Lemma 5.2.2, it follows that τ (N ) is a complex vector bundle equipped with the connection ∇τ , so that we can study its Chern classes. These are represented by the Chern forms √ k −1 µ1 ...µk λ1 γk (τ ) = δ K ∧ . . . ∧ Kµλkk (5.10) (2π)k k! λ1 ...λk µ1 where Kµλ are the curvature forms of the connection ∇τ (see the formula (4.64)). γk (τ ) is called the normal Chern form of order k and the cohomology class [γk (τ )] is the normal Chern class of order k of the integral submanifold N . Theorem 5.2.3. Let N be an integral submanifold of dimension p of the 2n+1dimensional Sasaki manifold and that p < n. If n − p is even then assume [γ2k+1 (τ )] = 0 for k = 0, 1, . . . , n−p−1 . 2 Proof. From Proposition 5.2.1 it results that Γ(τ ) can be considered as a complex vector space of dimension n − p with the scalar amplification defined by √ (α + −1β)~n = α~n + βF ~n for any α, β ∈ R and ~n ∈ Γ(τ√ ). Now, we define the map √ F τ : Γ(τ ) −→ Γ(τ ) by putting F τ ~n = F ~n∗ − −1F ~n∗∗ for each ~n = ~n∗ + −1~n∗∗ with ~n∗ , ~n∗∗ ∈ Γ(τ ). Moreover, F τ has the following obvious properties F τ (~n1 + ~n2 ) = F τ ~n1 + F τ ~n2 ,

F τ (λ~n) = λF τ ~n,

2

(F τ ) ~n = −~n

(5.11)

for any ~n, ~n1 , ~n2 ∈ Γ(τ ) and λ ∈ C, that is τ (N ) is a complex vector bundle with quaternionic structure. For the description of these bundles see [Va90a]. Now, we naturally extend the metric g to Γ(τ ) with the complex structure defined above, by putting ¯ τ (~n1 , ~n2 ), g τ (~n1 , λ~n2 ) = λg

g τ (~n2 , ~n1 ) = g τ (~n1 , ~n2 )

198


where λ is the conjugate of λ ∈ C and ~n1 , ~n2 ∈ Γ(τ ). Simple verification shows that g τ (F τ ~n1 , F τ ~n2 ) = g τ (~n1 , ~n2 ), hence g τ is a Hermitian scalar product over the complex vector bundle τ (N ) and the vector fields √ 1 Yλ = √ (Xλ + −1Xλ∗ ), 2

√ 1 F τ Yλ = √ (Xλ∗ + −1Xλ ), 2

λ ∈ p + 1, n

constitute a local basis Γ(τ ), orthonormal with respect to the metric g τ and with respect to the basis {Zλ = −F τ Yλ , Zλ∗ = Yλ }, the complex extension of the normal connection ∇τ has the following local expression ∇τ Zλ = αλµ Zµ + βλµ Zµ∗ ,

∇τ Zλ∗ = −βλµ Zµ + αλµ Zµ∗

(5.12)

Then the curvature matrix of ∇ is Λ, where Aµλ Bλµ Λ= −Bλµ Aµλ and the functions Aµλ , Bλµ are determined by the formulas (5.12), computing the curvature tensor of the connection ∇τ . Moreover, from the definition of the metric g τ it results ∇τ g τ = 0 and then from the formulas (5.12) we deduce1 αµλ = −αλµ ,

βλµ = βµλ

and analogous equalities for Aµλ , Bλµ , therefore the matrix Λ has the property Λ = −Λtr . But the forms γk (τ ) do not depend on the choice of the basis in Γ(τ ) and then, replacing the forms Kλµ with the corresponding Aµλ and Bλµ , we obtain γk (τ ) = (−1)k γk (τ ). Hence γk (τ ) vanishes for odd k. Generally, the calculation of Chern classes is difficult, so that we only deal with the study of the first normal Chern class of an integral submanifold in the case when n − p is odd. Proposition 5.2.4. If n − p = odd then the first normal Chern form of the integral submanifold N of dimension p in the 2n+1-dimensional Sasaki manifold of M is given by n ∗ 1 X γ1 (τ ) = Ωλλ 2π λ=p+1

1 If M is a manifold with the almost Hermitian structure (J, G) then we call the Hermitian connection the unique linear connection ∇ with the properties

∇J = 0,

∇G = 0,

T (JX, Y ) = T (X, JY )

where T is the torsion tensor of ∇. If dim M = 2m and {X1 , . . . , Xm , F X1 , . . . , F Xm } is an orthonormal basis of vector fields in the domain U of a local chart then ∇ has the following equations are valid on U ∇Xi = aji Xj + bji F Xj , aji

−aij , bji

where = [Va87], pg. 185).

=

bij

∇(F Xi ) = −bji Xj + aji F Xj

(5.13)

are the components of the connection form of ∇ (see for instance

5.2. CHERN CLASSES OF INTEGRAL SUBMANIFOLDS

199

Proof. From (5.9) and Theorem 1.5.3 and by applying the Weingarten formula we deduce the following expression of the components of the curvature tensor τ R of the normal connection ∇τ τ

∗

∗

λ λ Rλab = Rλab + g (BXλ Xb , BXλ∗ Xa ) − g (BXλ Xa , BXλ∗ Xb ) , ∗

for all a, b ∈ 1, p and λ ∈ p + 1, n. Then the curvature forms Kµλ of ∇τ are ∗ given by Kλ∗ µ = ωµλ . On the other hand, from Proposition 5.1.11 we deduce the complex form of the second structure equation of the connection ∇τ dΦλµ = −

n X

Φλν ∧ Φνµ + Ψλµ ,

Φλµ = ωµλ +

√

−1ωµλ∗ ,

Ψλµ = Kµλ∗

ν=p+1

But Ψλλ =

√

−1 Kλλ∗ and then from (5.10) the announced formula follows easily.

Theorem 5.2.5. Let N be an integral submanifold of the Sasaki form M (c). If the mean curvature vector of N is parallel then γ1 (τ ) = 0. Proof. From the expressions of the curvature and Ricci tensors for a Sasaki space form (see Theorem 3.4.4 and Proposition 3.5.2) it follows ∗

a Rabc = 0,

Ricbc∗ = 0

(5.14) 1

for a, b, c ∈ 1, p. Now, we consider the 1-form θ ∈ F (N ) defined by θ = P p b a∗ a=1 ωa , where ωa are the components of the connection form defined by the formulas (5.5). From (5.1) and from the second structure equation (2.28) we obtain dθ = 2

p n X X

ωaλ

∧

∗ ωaλ

+

a=1 λ=p+1

p X

∗

Ωaa

(5.15)

a=1

By applying the Gauss formula in (5.5) and by equalizing the terms of the same type we obtain ωaµ (X) = g (h(X, Xa ), Xµ ) =

p X

hµba ω b (X)

b=1

hα ac

where X ∈ X (N ) and are the componentsPof h(Xa , Xc ) with respect to the p c ∗ local basis Bτ in Γ(τ ). Then we have ωaα = a=1 hα ac ω for α = λ or α = λ and λ ∈ p + 1, n. From this equality and from (5.15) it follows dθ =

p X

p X ∗ ∗ ∗ hλab hλac − hλac hλab ω b ∧ ω c + Ωaa

(5.16)

a=1

a,b,c=1

¯ λ in the formula (5.7) we deduce From (5.4) and from the first expression of Ω µ p X a=1

∗

Ωaa =

p 1 X a∗ b Rabc ω ∧ ωc 2 a,b,c=1

(5.17)

200


at any point of N . From the first Bianchi identity and from Proposition 3.2.1, c), d) we obtain p X

∗

a Rabc = Ric(Xb , Xc∗ ) −

a,b,c=1

n X

R (Xλ∗ , Xλ , Xb , Xc )

λ=1

and taking into account the Ricci equation we have p X

a∗ Rabc

= Ricbc∗ −

a,b,c=1

n X

λ∗ Rλbc

p n X X

+

λ=p+1

Adλc Abλ∗ d − Adλ∗ c Abλd

λ=p+1 d=1

where Abλa are the components of the Weingarten operator with respect to the F -basis {Xi , Xi∗ , ξ}i∈1,n . From this equality and from (5.15), (5.6), (5.14) we obtain dθ = −2

n X

∗

Ωλλ

(5.18)

λ=p+1

But taking into account the definition of the 1-form α~n (see the proof of Theorem 5.1.10) we have θ = −pαH and then our assertion follows from (5.18) and from Proposition 5.2.4. Theorem 5.2.6. The first normal Chern class of an integral submanifold in a Sasaki space form always vanishes. Proof. From Proposition 5.2.4 and from the equality (5.18) it results 4π 1 γ1 (τ ) dαH = − dθ = p p and then [γ1 (τ )] = 0.

5.3

Chen invariants of an integral submanifold

B. Y. Chen [Ch93] showed that if N is a minimal submanifold of an Euclidean space then K(π) ≥ ρN for any point x ∈ N and for any plane π ⊂ Tx N . Now, if N is an arbitrary p-dimensional submanifold of the Riemannian manifold M then, inspired by the above inequality, it is natural to consider the following Riemannian invariant δN (x) = ρN (x) − (inf K)x where (inf K)x = inf {K(π); π ⊂ Tx N, dim π = 2}. δN is called the Chen invariant (or the first Chen invariant or the δ-invariant) of the submanifold N . The first Chen invariant can be generalized as follows:

5.3. CHEN INVARIANTS OF AN INTEGRAL SUBMANIFOLD

201

Let k ∈ N and consider the numbers p1 , . . . , pk ∈ N, p1 ≥ 2, . . ., pk ≥ 2, p1 < p, p1 + . . . + pk ≤ p. On N we can define the following real valued function S (p1 , . . . , pk ) (x) =

inf

π1 ,...,πk

{K(π1 ) + . . . + K(πk )} ,

where π1 , . . . , πk ⊂ Tx N are reciprocally orthogonal subspaces of dimensions dim πi = pi for i ∈ 1, k. Then the Riemannian invariant δN (p1 , . . . , pk )(x) = ρN (x) − S (p1 , . . . , pk ) (x) is called the k-th Chen invariant of the submanifold N . F. Defever, I. Mihai and L. Verstraelen obtained the following sharp inequality of the Chen invariant in the case of Sasaki space forms: Theorem 5.3.1. [DMV01] Let N be a p-dimensional integral submanifold of the Sasaki form M (c). If p > 2 then 2 (p + 1)(c + 3) p p−2 2 kHk + δN ≤ 2 p−1 4 Proof. By using Theorem 3.4.4, from the Gauss equation we deduce 2

2

2ρN = p2 kHk − khk + and we set e = 2ρN −

p(p − 1)(c + 3) 4

(5.19)

p(p − 1)(c + 3) p2 (p − 2) 2 − kHk 4 p−1

then (5.19) yields 2 2 (p − 1) e + khk = p2 kHk

(5.20)

Let π be a plane of the tangent space Tx N at the point x ∈ N . Assume that H 6= 0 at x and consider an orthonormal basis {e1 , . . . , ep , ep+1 , . . . , e2n , e2n+1 = ξ} so that π = he1 , e2 i and ep+1 is collinear to H. Taking into account 2

khk =

p X

g (h(ea , eb ), h(ea , eb )) ,

a,b=1

from the equality (5.20) we deduce   !2 p p p 2n X X X X 2 2 hp+1 = (p − 1) e + hp+1 + hiab  aa aa a=1

a=1

i=p+2 a,b=1

where hiab = g (h(ea , eb ), ei ). Now, we use the following algebraic result [Ch93]:

(5.21)

202


If the real numbers a1 , . . . , ap and c are so that !2 ! p p X X 2 ai = (p − 1) c + ai i=1

i=1

then 2a1 a2 ≥ c. The equality holds if and only if a1 + a2 = a3 = . . . = ap . Then from (5.21) we deduce 2n 2n X X

p+1 2hp+1 11 h22 ≥ e +

hiab

2

(5.22)

i=p+2 a,b=1

On the other hand, by putting X = Z = e1 , Y = U = e2 in the Gauss equation and taking into account Lemma 5.1.6 and the inequality (5.22) we obtain KN (π) =

2n h X 2 i c+3 + hi11 hi22 − hi12 ≥ 4 i=p+1

p 2n 2n 2n X X 2 c + 3 e 1 X X i 2 hab + hi11 hi22 − hi12 = ≥ + + 4 2 2 i=p+2 i=p+2 i=p+1 a,b=1

p 2n 2n 2 c+3 e 1 X 1 X X i i 2 = hiab + + + h + h22 + 4 2 2 i=p+2 11 2 i=p+2 a,b≥3 X p+1 2 p+1 2 c + 3 e h1a + + h2a + ≥ 4 2 a≥3

But taking into account the expression of e we get inf KN (π) ≥ π

p(p + 1)(c + 3) p2 (p − 2) c+3 2 + ρN − − kHk 4 8 2(p − 1)

and then our inequality follows from the definition of the invariant δN . The case when H = 0 at x can be treated similarly, with the minor changes imposed by the condition H = 0. The integral submanifold N with the property that in Theorem 5.3.1 the equality holds for any x ∈ N is called an ideal submanifold. From the proof of Theorem 5.3.1 we deduce the following technical characterization of these submanifolds: Proposition 5.3.2. Let N be a p-dimensional integral submanifold of the Sasaki space form M (c)of dimension 2n + 1. N is ideal if and only if at any point x ∈ N there exists a basis {e1 , . . . , e2n , ξ} of Tx M so that {e1 , . . . , ep } ⊂ Tx N , {ep+1 , . . . , e2n , ξ} ⊂ Tx⊥ N and the Weingarten operators have the following forms   a 0 h11 ha12 . . . 0  Aea :  ha12 −ha11 . . . 0 0 . . . 0p−2

5.3. CHEN INVARIANTS OF AN INTEGRAL SUBMANIFOLD

203

for any a ∈ p + 2, 2n and 

Aep+1

α :  0 0

0 β 0

 ... 0  ... 0 . . . (α + β)Ip−2

Proof. From the proof of Theorem 5.3.1 we deduce that in (5.22) the equality holds if and only if p+1 p+1 p+1 hp+1 11 + h22 = h33 = . . . = hpp

(5.23)

Moreover, in the expression of KN (π) we have equality if and only if hi11 + hi22 = 0,

p+1 hp+1 1a = h2a = 0,

hiab = 0

(5.24)

for i ∈ p + 2, 2n and a ≥ 3, b ≥ 3. Now, our Proposition is a simple corollary of (5.23) and (5.24). The converse follows by a straightforward computation. Proposition 5.3.3. Let M (c) be a 2n + 1-dimensional Sasaki space form and n ≥ 3. Any n-dimensional ideal submanifold of M (c) is minimal. T

Proof. Since N has dimension equal to n, we have [F h(X, Y )] = F h(X, Y ) for any X, Y ∈ X (N ), hence by Lemma 5.1.6 and Proposition 5.3.2 and taking into account (5.23), (5.24), we obtain nH = −(α + β)F e1 . On the other hand we have g(nH, nH) =

n X

g (AF e1 ei , ei ) = (n − 1)(α + β)

i=1

and then α + β = 0. By an argument similar to the one used in order prove Proposition 5.3.1 we obtain the following Proposition 5.3.4. [DMV01] If N is a p-dimensional integral submanifold of the Sasaki space form M (c) then " # k X p2 (p∗ − 1) 1 2 kHk + p(p − 1) − pi (pi − 1) (c + 3) δN (p1 , . . . , pk ) ≤ 2p∗ 8 i=1 where p∗ = p + k −

Pk

i=1

pi .

Now we define an analogous invariant in the case when N is a 2n+1-dimensional submanifold with a contact Riemannian structure in the m-dimensional real space form M (c) of constant sectional curvature c. For this purpose let k ∈ 2, 2n + 1 and denote by (F, ξ, η, g) the associated almost contact Riemannian

204


structure on N . For each k-dimensional subspace L ⊂ Tx N we consider the scalar X τN (L) = K(eα , eβ ) 1≤α