Almost r - contact structures on the tangent bundle

Almost r - contact structures on the tangent bundle Lovejoy S. Das and Mohd. Nazul Islam Khan

Abstract The differential geometry of tangent bundles was studied by several authors, for example: D. E. Blair [1], E. T. Davies [3], P. Dombrowski [4], S. Ianus [5, 6], A. J. Ledger and K. Yano [7], V. Oproiu [8, 9], C. Udriste [10], Yano and Davies [11], Yano and Ishihara [13, 14] and among others. It is well known that different structures defined on a manifold M can be lifted to the same type of structures on its tangent bundle. Several authors cited herein obtained results in this direction. However, when we consider an almost contact structure not the same type of structure is obtained on the tangent bundle. In this case the base manifold must be of odd dimension while the tangent bundle is always of even dimension. Motivated by this fact, our goal is to see as to what kind of structure is defined on the tangent bundle T (M ) when we consider an almost r – contact structure on the base manifold.

M.S.C. 2000: 53D15, 53B35. Key words: Almost r-contact structure, vertical lifts, base space, horizontal and complete lifts, complex structure, Lie derivative.

1

Introduction

Let M be an n-dimensional differentiable manifold of class C ∞ and let Tp (M ) be the tangent space of M at a point p of M . Then the set [14] (1.1)

T (M ) = ∪ Tp (M ) pεM

is called the tangent bundle over the manifold M . For any point p˜ of T (M ), the correspondence p˜ → p determines the bundle projection π : T (M ) → M , Thus π(˜ p) = p, where π : T (M ) → M defines the bundle projection of T (M ) over M . The set π −1 (p) is called the fibre over p ε g and M the base space. Suppose that the base space M is covered by a system of coordinate neighbourhoods {U ; xh }, where (xh ) is a system of local coordinates defined in the neighbourhood U of M . The open set π −1 (U ) ⊂ T (M ) is naturally differentiably homeomorphic to the direct product U × Rn , Rn being the n-dimensional vector space over the real field R, in such a way that a point p˜εTp (M ) (p ε U )is represented by an ordered pair (P, X) of the point p ε U , and a vector X ε Rn whose components are given by the cartesian coordinates (y h ) of p˜ in the tangent space Tp (M ) with respect to the natural

Differential Geometry - Dynamical Systems, Vol.7, 2005, pp.

34-41. c Balkan Society of Geometers, Geometry Balkan Press 2005. °

Almost r - contact structures on the tangent bundle

35

base {∂h }, where ∂h = ∂x∂ h . Denoting by (xh ) the coordinates of p = π(˜ p) in U and establishing the correspondence (xh , y h ) → p˜ ε π −1 (U ), we can introduce a system of local coordinates (xh , y h ) in the open set π −1 (U ) ⊂ T (M ). Here we call (xh , y h ) the coordinates in π −1 (U ) induced from (xh ) or simply, the induced coordinates in π −1 (U ). of all tensor fields of class C ∞ and of type (r, s) in We denote by =rs (M ) the P set ∞ r M . We now put =(M ) = r,s=0 =s (M ), which is the set of all tensor fields in M . r Similarly, we denote by =s (T (M )) and =(T (M )) respectively the corresponding sets of tensor fields in the tangent bundle T (M ).

Vertical lifts If f is a function in M , we write f V for the function in T (M ) obtained by forming the composition of π : T (M ) → M and f : M → R,so that (1.2)

fV = f o π

Thus, if a point p˜ ε π −1 (U ) has induced coordinates (xh , y h ), then (1.3)

f V (˜ p) = f V (x, y) = f o π (˜ p) = f (p) = f (x)

Thus the value of f V (˜ p) is constant along each fibre Tp (M ) and equal to the value f (p). We call f V the vertical lift of the function f . ˜ ˜ V = 0 for all f ε =0 (M ). Then we say that X ˜ ε =1 (T (M ))be such that Xf Let X 0 0 h hi ˜ ˜ with respect to the induced is a vertical vector field. Let X be components of X ¯ ˜h X ˜ is vertical if and only if its components in π −1 (U ) satisfy coordinates. Then X h hi £ ¤ ˜ X (1.4) = 0X h¯ ¯ Xh Suppose that X ε =10 (M ), so that X is a vector field in M . We define a vector field X V in T (M ) by (1.5) X V (ι ω) = (ω(X))V ω being an arbitrary 1-form in M . We call X V the vertical lift of X. ˜ (X V ) = 0 for all X ε =10 (M ). Then we say that Let ω ˜ ε =01 (T (M )) be such that ω ω ˜ is a vertical 1-form in T (M ). We define the vertical lift ω V of the 1-form ω by (1.6)

ω V = (ωi )V (dxi )V

in each open set π −1 (U ), where {U ; xh } is coordinate neighbourhood in M and ω is given by ω = ωi dxi in U . The vertical lift ω V of ω with local expression ω = ωi dxi has components of the form (1.7) ω V : (ω i , 0) with respect to the induced coordinates in T (M ). Vertical lifts to a unique algebraic isomorphism of the tensor algebra =(M ) into the tensor algebra = (T (M )) with respect to constant coefficients by the conditions (1.8)

(P ⊗ Q)V = P V ⊗ QV , (P + R)V = P V + RV

36

Lovejoy S. Das and Mohd. Nazul Islam Khan

P, Q and R being arbitrary elements of =(M ). The vertical lifts F V of an element F ε =11 (M ) with local components Fih has components of the form µ ¶ 0 0 V F : (1.9) Fih 0

Complete lifts If f is a function in M , we write f C for the function in T (M ) defined by (1.10)

f C = ι(df )

and call f C the complete lift of the function f . The complete lift f C of a function f has the local expression (1.11) f C = y i ∂i f = ∂f with respect to the induced coordinates in T (M ), where ∂f denotes y i ∂i f . Suppose that X ε =10 (M ). We define a vector field X C in T (M ) by (1.12)

X C f C = (Xf )C ,

f being an arbitrary function in M and call X C the complete lift of X in T (M ). The complete lift X C of X with components xh in M has components h h i (1.13) X C : x∂xh with respect to the induced coordinates inT (M ). Suppose that ω ε =01 (M ). Then a 1-form ω C in T (M ) defined by (1.14)

ω C (X C ) = (ω(X))C

X being an arbitrary vector field in M . We call ω C the complete lift of ω. The complete lift ω C of ω with components ωi in M has components of the form (1.15)

ω C : (∂ ωi , ωi )

with respect to the induced coordinates in T (M ). The complete lifts to a unique algebra isomorphism of the tensor algebra J(M ) into the tensor algebra =(T (M )) with respect to constant coefficients, is given by the conditions (1.16)

(P ⊗ Q)C = P C ⊗ QV + P V ⊗ QC , (P + R)C = P C + RC ,

P, Q and R being arbitrary elements of =(M ). The complete lifts F C of an element F of =11 (M ) with local components Fih has components of the form µ h ¶ Fi 0 C (1.17) F : ∂Fih Fih


37

Horizontal lifts The horizontal lift f H of f ε =00 (M ) to the tangent bundle T (M ) is given by f H = f C − ∇γ f

(1.18) where (1.19) Let X ε

∇γ f =γ (∇f ), =10 (M ).

Then the horizontal lift X H of X defined by X H = X C − ∇γ X,

(1.20) in T (M ), where (1.21)

∇γ X = γ(∇X) H

The horizontal lift X of X has the components ¶ µ h x (1.22) XH : −Γhi xi with respect to the induced coordinates in T (M ), where Γhi = y j Γhji

(1.23)

Let ω ε =01 (M ) with affine connection ∇. Then the horizontal lift ω H of ω is defined by (1.24) ω H = ω C − ∇γ ω in T (M ), where ∇γ ω =γ (∇ω). The horizontal lift ω H of ω has component of the form (1.25) ω H : (Γhi ωh , ωi ) with respect to the induced coordinates in T (M ). Suppose there is given a tensor field (1.26)

i...h S = Sk...j

∂ ∂ ⊗ ... ⊗ ⊗ ∂xk ⊗ ... ⊗ ∂xj ∂xi ∂xh

in M with affine connection ∇, and in T (M ) a tensor field ∇γ S defined by (1.27)

i...h ∇γ S = (y l ∇l Sk...j )

∂ ∂ ⊗ ... ⊗ ⊗ ∂xk ⊗ ... ⊗ ∂xj i ∂y ∂y h

with respect to the induced coordinates (xh , y h ) in π −1 (U ). The horizontal lift S H of a tensor field S of arbitrary type in M to T (M ) is defined by (1.28) S H = S C − ∇γ S For any P, Q ε =(M ). We have (1.29)

∇γ (P ⊗ Q) = (∇γ P ) ⊗ QV + P V ⊗ (∇γ Q)

or (P ⊗ Q)H = P H ⊗ QV + P V ⊗ QH

38


2

Complete lifts of almost complex structure and almost r-contact structure in the tangent bundle

¯ be an 2n+r dimensional differentiable manifold of Class C ∞ and T (M ¯ ) denotes Let M ¯ ¯ the tangent bundle of M . Suppose there is given in M , a tensor field F (1, 1), r vector fields Uα and r 1-forms ω α (r some finite integer and α = 1, 2, ...r) satisfying r X F 2 = −I + Uα ⊗ ω α (2.1) α=1

where (2.2)

F Ua = 0

(2.3)

ωα oF = 0

(2.4)

ω α (Uβ ) = δβα

where α, β = 1, 2, ...r and δβα denotes kronecker delta. ¯ satisfying conditions (2.1) and (2.2) will be said to possess Thus the manifold M an almost r-contact structure [2]. Now we will prove the following two theorems: ¯ be a differentiable Manifold endowed with almost r-contact Theorem 2.1. Let M structure (F, Uα , ω α ), then r ¡ ¢ P ¯ ). J˜ = F C + UαV ⊗ ω αV − UαC ⊗ ω αC is almost complex structure on T (M α=1

Proof : From (2.1) and (2.2), we have r X ¡ C ¢2 F = −I + UαV ⊗ wαC + UαC ⊗ ω αV

(2.5)

α=1

and (2.6)

F C UαV = 0,

(2.7) (2.8)

ω αV

F C UαC = 0

ω αV oF C = 0, ω αC oF V = 0 ω αC oF C = 0 ¡ V¢ ¡ ¢ ¡ ¢ Uβ = 0, ω αV UβC = δβα , ω αC UβV = δβα , ω αc (UβC ) = 0

¯ )) by Let us define an element J˜ of J11 (T (M (2.9)

J˜ = F C +

r X ¡

UαV ⊗ ω αV − UαC ⊗ ω αC

α=1

then we find by using (2.5), (2.6) and (2.9) that (2.10)

J˜2 = −I

¯ ). Thus J˜ is an almost complex structure in T (M

¢


39

In view of equation (2.9), we have ˜ V = (F X)V − (ω α (X))V UαC JX

(2.11)

(2.12)

˜ C = (F X)C + (ω α (X))V UαC − (ω α (X))C UαC JX

In particular, we have (2.13)

(2.14)

˜ V = (F X)V , JX

˜ C = (F X)C JX

˜ αC = δβα UαC = UβC ; αβ = 1, 2, ...r. JUαV = −δβα UαC = −UβC ; JU

X being an arbitrary vector field in M such that ω α (X) = 0 Theorem 2.2. Let the tangent bundle T (M ) of the manifold M admits J˜ defined in (2.5), then for vector fields X, Y such that ω α (Y ) = 0, we have ³ (2.15)

´ LX V J˜ Y V = 0 ,

(2.16)

˜ )Y C = ((L F )Y )V −((L ω α )Y )V UαC (LX V J X X

(2.17)

˜ )U V = −(L Uα )V (LX V J α X

(2.18)

˜ )U C = ((L F )U )V −((L ω α )(U ))V UαC (LX V J α X α X α

and (2.19)

˜ )Y V = ((L F )Y )V −((L ω α )(Y ))V UαC (LX C J X X

(2.20)

˜ )Y C = ((L F )Y )C +((L ω α )(Y ))V Uα −((L ω α )(Y ))C UαC (LX C J X X X

(2.21)

˜ )U V = ((L F )U C −[X , U ]C −((L ω α )(U )V UαC (LX C J α X α α X α

(2.22)

˜ )U C =((L F )U )C +[X , U ]V + ((LX ω α ) (Uα ))V Ua (LX C J α X α α

(2.23)

− ((LX ω α ) (Uα )) UαC

C

Proof : The proof follows in an obvious manner on using (2.6), (2.12) and (2.14).

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3

Horizontal lifts of an almost r – contact structure

¯ with an affine connection then Let (F, Uα , ω α ) be an almost r-contact structure in M in view of (2.18) and (2.22) we have ¡

(3.1)

F

¢ H 2

Ã =

−I +

r X

!H Uα ⊗ ω α

r=1 r X ¡ H ¢2 H F = −I + (Uα ⊗ ω α )

(3.2)

α=1

¡

(3.3)

FH

¢2

= −I +

r X

¡

UαH ⊗ ω αV + UαV ⊗ ω αH

¢

α=1

Also, (3.4) (3.5)

ω

¡ αH

¢ H

Uβ

F H UαH = 0, = 0,

¡ ¢ ω αV UβH = δβα

ω αH (UαV ) = δβα ,

ω αH of

(3.6)

F H UαV = 0

H

= 0, ω αV of

H

= 0.

˜∗

¯ ) by Let us define a tenser field J of type (1,1) in T (M (3.7)

J˜∗ = F H +

r X ¡

UαV ⊗ ω αV − UαH ⊗ ω αH

¢

r=1

then it is easy to show that (3.8)

2 J˜∗ = −I

¯ ) and we have Thus J ∗ is an almost complex structure inT (M ¯ with an Theorem 3.1. Let (F, Uα , ω α ) be an almost r-contact structure in M ¯ ). affine connection ∇. Then J˜∗ is an almost complex in T (M Acknowledgement. The authors are grateful to the referee for their comments and suggestions toward the improvement of this work.

References [1] Blair, D.E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Math, 509, Springer Verlag, New York, (1976). [2] Das, Lovejoy S., Fiberings on almost r-contact manifolds, Publicationes Mathematicae, Debrecen, Hungary 43 (1993), 161-167. [3] Davies, E. T., On the curvature of the tangent bundle, Annali di Mat. (IV) 81 (1969), 193-204.


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[4] Dombrowski, P., On the geometry of the tangent bundles, Jour. reine und angew. Math. 210 (1962), 73-88. [5] Ianus, S., On certain distributions on the tangent bundle, Stud. Cerc. Mat. 23 (1971), 873-880. [6] Ianus, S. and Udriste, C., On the tangent bundle of a differentiable manifold, Stud. Cerc. Mat. 22 (1970), 599-611. [7] Ledger, A. J., and Yano, K., Almost complex structures on complex bundles, Jour. Diff. Geometry, 1 (1967), 355-368. [8] Oproiu, V., Some remarkable structures and connexions defined on the tangent bundle, Rendiconti di Matematica (3), 6 VI, (1973). [9] Oproiu, V., Some remarks on the almost cocomplex connexions, Rev. Roum. Math. Pures et. Appe. Tome XVI no. 3, (1971), 383-393. [10] Udriste, C., Congruences on the tangent bundle of a differentiable manifold, Rev. Roumaine Math. Pures Appe. 15 (1970), 1079-1096. [11] Yano, K., and Davies, E. T., Metrics and connections in the tangent bundle, Kodai. Math. Sem. Rep., 23 (1971), 493-504. [12] Yano, K. and Ishihara, S., Almost complex structures induced in tangent bundles. Kˆodai Math. Sem. Rep., Vol. 19, (1967), 1-27. [13] Yano, K. and Ishihara, S., Differential geometry in tangent bundles. Kˆodai Math. Sem. Rep., Vol. 18 (1996), 271-292. [14] Yano, K. and Ishihara, S., Tangent and Contangent Bundles. Marcel Dekker, Inc. New York.

Authors’ addresses: Lovejoy S. Das Kent State University, Tuscarwas, Department of Mathematics, New Philadelphia, OH 44663, U.S.A. e-mail : [email protected] Mohd. Nazul Islam Khan Azad Institute of Engineering & Technology, Department of Applied Sciences Natkur, Post-Chandrawal, Bangla Bazar Road, Lucknow-226002, (U.P.), INDIA e-mail : [email protected]