On $(H,\widetilde {H}) $-harmonic Maps between pseudo-Hermitian ...

e ON (H, H)-HARMONIC MAPS BETWEEN

arXiv:1610.01032v1 [math.DG] 4 Oct 2016

PSEUDO-HERMITIAN MANIFOLDS*

Yuxin Dong Abstract. In this paper, we investigate critical maps of the horizontal energy functional 2m+1 , H(M), J, θ) and EH,H e (f ) for maps between two pseudo-Hermitian manifolds (M e These critical maps are referred to as (H, H)-harmonic e e θ). e (N 2n+1 , H(N), J, maps. We 2 derive a CR Bochner formula for the horizontal energy density |dfH,H e | , and introduce a Paneitz type operator acting on maps to refine the Bochner formula. As a result, we e are able to establish some Bochner type theorems for (H, H)-harmonic maps. We also ine e troduce (H, H)-pluriharmonic, (H, H)-holomorphic maps between these manifolds, which e provide us examples of (H, H)-harmonic maps. Moreover, a Lichnerowicz type result is e established to show that foliated (H, H)-holomorphic maps are actually minimizers of EH,H (f ) in their foliated homotopy classes. We also prove some unique continuation ree e sults for characterizing either horizontally constant maps or foliated (H, H)-holomorphic e maps. Furthermore, Eells-Sampson type existence results for (H, H)-harmonic maps are

established if both manifolds are compact Sasakian and the target is regular with none positive horizontal sectional curvature. Finally, we give a foliated rigidity result for (H, H)harmonic maps and Siu type strong rigidity results for compact regular Sasakian manifolds with either strongly negative horizontal curvature or adequately negative horizontal curvature.

Introduction A smooth map f between two Riemannian manifolds M and N is called harmonic if it is a critical point of the energy functional (cf. [EL]) Z 1 |df |2 dvM . E(f ) = 2 M Harmonic maps became a useful tool for studying complex structures of Kähler manifolds through the fundamental work of Siu [Si1,2]. In his generalization of Mostow’s rigidity theorem for Hermitian symmetric spaces, Siu proved that a harmonic map of sufficiently high maximum rank of a compact Kähler manifold to a compact Kähler manifold with strongly negative curvature or a compact quotient of an irreducible bounded 1991 Mathematics Subject Classification. Primary: 32V20, 53C43, 53C55. e e Key words and phrases. pseudo-Hermitian manifold, (H, H)-harmonic map, (H, H)-pluriharmonic e map, (H, H)-holomorphic map. *Supported by NSFC grant No. 11271071, and LMNS, Fudan. Typeset by AMS-TEX

1

symmetric domain must be holomorphic or anti-holomorphic. It follows that if a compact Kähler manifold is homotopic to such a target Kähler manifold, then the homotopy equivalent map is homotopic to a biholomorphic or anti-biholomorphic map. The latter result is usually known as Siu’s strong rigidity theorem. Although harmonic maps are successful in the study of geometric and topological structures of Kähler manifolds, they are not always effective in other settings. For example, it is known that harmonic maps are no longer in force for investigating general Hermitian manifolds, since holomorphic maps between these manifolds are not necessarily harmonic. In recent years, some generalized harmonic maps were introduced and investigated in various geometric backgrounds (cf. [JY], [Kok], [BD], [BDU], [KW], [Pe], [CZ]). First, we recall the notion of transversally harmonic maps between Riemannian foliations. Let (M, g, F ) and (N, h, Fe) be two compact Riemann manifolds with Riemannian foliations F and Fe respectively, and f : M → N a smooth foliated map. Denote by υ(F ) and υ(Fe) the normal bundles of the foliations F and Fe respectively. The differential df gives rise naturally to a smooth section dT f of Hom(υ(F ), f −1 υ(Fe)). Then we may define the transverse energy 1 ET (f ) = 2

Z

M

|dT f |2 dvM

where dT f : υ(F ) → υ(Fe) is the induced map of the differential map df , called the transversally differential map. A smooth foliated map f : (M, g, F ) → (N, h, Fe) is called transversally harmonic if it is an extremal of ET (·) for any variation of f by foliated maps (cf. [BD], [KW]). Suppose now that the foliations F and Fe are two Kählerian foliations with complex structures J and Je on their normal spaces υ(F ) and υ(Fe) respectively. A foliated map f : M → N is said to be transversally holomorphic if dT f ◦ J = Je◦ dT f . It was proved in [BD] that any transversally holomorphic map f : (M, g, F ) → (N, h, Fe) of Kählerian foliations with F harmonic is transversally harmonic. Recall that a CR structure on an (2m + 1)-dimensional manifold M 2m+1 is an 2mdimensional distribution H(M ) endowed with a formally integrable complex structure J. The manifold M with the pair (H(M ), J) is called a CR-manifold. A pseudo-Hermitian manifold, which is an odd-dimensional analogue of Hermitian manifolds, is a CR manifold M endowed with a pseudo-Hermitian structure θ. The pseudo-Hermitian structure θ determines uniquely a global nowhere zero vector field ξ and a Riemannian metric gθ on M . The integral curves of ξ forms a foliation Fξ , called the Reeb foliation. From [Ta], [We], we know that each pseudo-Hermitian manifold admits a unique canonical connection ∇ (the Tanaka-Webster connection), which is compatible with both the metric gθ and the CR structure (see Proposition 1.1). However, this canonical connection always has nonvanishing torsion T∇ (·, ·), whose partial component T∇ (ξ, ·) is an important pseudo-Hermitian invariant, called the pseudo-Hermitian torsion. Pseudo-Hermitian manifolds with vanishing pseudo-Hermitian torsion are referred to as Sasakian manifolds which play an important role in AdS/CFT correspondence stemming from string theory (cf. [MSY]). An equivalent characterization for a pseudo-Hermitian manifold to be Sasakian is that Fξ is a Riemannian foliation. It is known that Sasakian geometry sits naturally in between two Kähler geometries. On the one hand, Sasakian manifolds 2

are the bases of metric cones which are Kähler. On the other hand, the Reeb foliation Fξ of a Sasakian manifold is a Kählerian foliation (cf. [BG], [BGS]). Due to these two aspects, a Sasakian manifold can be viewed as an odd-dimensional analogue of a Kähler manifold. Consequently, it is natural to expect similar Siu type rigidity theorems for Sasakian manifolds. In [CZ], an interesting Siu type holomorphicity result was asserted for transversally harmonic maps between Sasakian manifolds when the target has strongly negative transverse curvature. However, Siu type strong rigidity theorems have not been established for Sasakian manifolds yet. e between two pseudoe ), J, e θ) For a map f : (M 2m+1 , H(M ), J, θ) → (N 2n+1 , H(N Hermitian manifolds, Petit [Pe] defined a natural horizontal energy functional 1 EH,He (f ) = 2

Z

M

|dfH,He |2 dvθ

where |dfH,He |2 denotes the horizontal energy density (cf. §3), and he called a critical map of EH,He (f ) a pseudoharmonic map. The main purpose of [Pe] is to derive Mok-SiuYeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into pseudo-Hermitian manifolds and obtain some rigidity theorem for the horizontal pseudoharmonic maps. Note that the Euler-Lagrange equation for EH,He (f ) derived in [Pe] contains an extra condition on the pull-back torsion (see (3.10)). Besides, the authors in [BDU] introduced another kind of pseudoharmonic maps from a pseudo-Hermitian manifold to a Riemannian manifold. To avoid the extra torsion condition in the Euler-Lagrange equation of [Pe] and any possible confusion with the notion of pseudo-harmonic maps in [BDU], we modify Petit’s definition slightly by restricting the variational vector field to be horizontal and refer to the corresponding critical maps e as (H, H)-harmonic maps. Although the energy functional EH,He (f ) is defined in a way similar to that of the transversal energy functional, we would like to point out the differe ences between the notions of (H, H)-harmonic maps and transversally harmonic maps. First, the Reeb foliation of a pseudo-Hermitian manifold is not a Riemannian foliation e in general; secondly, a (H, H)-harmonic map is not a priori required to be a foliated map; thirdly, a horizontal variational vector field is not necessarily foliated too. Therefore the starting points for their definitions are different, though they may coincide for e foliated maps between Sasakian manifolds. Actually we will see that (H, H)-harmonic maps may display some geometric phenomenon which are invisible from transversally harmonic maps. e In this paper, we will study some basic geometric properties and problems for (H, H)harmonic maps such as Bochner-type, Lichnerowicz-type and Eells-Sampson-type, Siu e type holomorphicity results, etc. Our main aim is to utilize (H, H)-harmonic maps to establish Siu type strong rigidity theorems for Sasakian manifolds. The paper is organized as follows. Section 1 begins to recall some basic facts and notions of pseudoHermitian geometry, including some properties of the curvature tensor of a pseudoHermitian manifold. Next, we introduce the notions of strongly negative or strongly semi-negative horizontal curvature, including adequately negative horizontal curvature, for Sasakian manifolds. Some model Sasakian spaces with either strongly negative or adequately negative horizontal curvature are given. In Section 2, we introduce the 3

second fundamental form β(·, ·) with respect to the Tanaka-Webster connections for a map f between two pseudo-Hermitian manifolds, and derive the commutation relations e of its covariant derivatives. In Section 3, we recall the definition of a (H, H)-harmonic map and the corresponding Euler-Lagrange equation (see (3.4)) (0.1)

τH,He (f ) = 0,

where τH,He (f ) is called the horizontal tension field of f . Then some relationship among e (H, H)-harmonic maps, pseudo-harmonic maps and harmonic maps are discussed. It e turns out that although (H, H)-harmonic maps have many nice properties related to the pseudo-Hermitian structures, the PDE system (0.1) is too degenerate and thus its solutions may not be regular enough to detect global geometric properties, including the strong rigidity, of pseudo-Hermitian manifolds. Actually transversally harmonic maps between Riemannian foliations also have similar drawbacks. In order to repair these e drawbacks, we define a special kind of (H, H)-harmonic maps as follows. For a map f : M → N between two pseudo-Hermitian manifolds, we set τH (f ) = τH,He (f )+τH,Le (f ), where τH,Le (f ) is the vertical component of trgθ (β|H ). For our purpose, we introduce a nonlinear subelliptic system of equations (0.2)

τH (f ) = 0,

imposed on the map f . Since (0.2) implies (0.1), a solution of (0.2) is referred to as a e e special (H, H)-harmonic map (see Definition 3.2). Special (H, H)-harmonic maps will play an important role in our studying of the strong rigidity for Sasakian manifolds. In Section 4, we derive a CR Bochner formula for the horizontal energy density |dfH,He |2 , whose main difficulty in applications comes from a mixed term consisting of some contractions of dfH,He and β(·, ξ). In order to deal with this term, we introduce a Paneitz type operator acting on the map, which enables us to refine the Bochner formula. As e a result, we are able to establish some Bochner type theorems for (H, H)-harmonic e e maps. In Section 5, we first define the notions of (H, H)-pluriharmonic maps, (H, H)e e holomorphic maps and (H, H)-biholomorphisms. It turns out that foliated (H, H)e e holomorphic maps are (H, H)-pluriharmonic, and (H, H)-pluriharmonic maps are foe liated (H, H)-harmonic. Next, we give a unique continuation theorem which asserts e e that a foliated (H, H)-harmonic map between two Sasakian manifolds must be (H, H)e holomorphic on the whole manifold if it is (H, H)-holomorphic on an open subset. Some e examples of (H, H)-holomorphic maps are also given. From [BGS], we know that for a given Sasakian structure S = (ξ, θ, J, gθ) on M , the Reeb vector field ξ polarizes the Sasakian manifold (M, S), and the space S(ξ, Jυ ) of all Sasakian structures with the fixed Reeb vector field ξ and the fixed transverse holomorphic structure Jυ on υ(Fξ ) is an affine space. We show that idM : (M, S1 ) → (M, S2 ) for any S1 , S2 ∈ S(ξ, Jυ ) e is a foliated (H, H)-biholomorphism. In addition, we discuss the case when idM : e (M, S1 ) → (M, S2 ) is a special (H, H)-biholomorphism. In Section 6, we obtain a Lichnerowicz type result which asserts that the difference of horizontal partial energies for a foliated map is a smooth foliated homotopy invariant. As an application, we deduce 4

e that a foliated (H, H)-holomorphic map between two pseudo-Hermitian manifolds is an absolute minimum of the horizontal energy EH,He (f ). In Section 7, we study the existence problem for (0.2) by looking at the following subelliptic heat flow ∂ft = τH (ft ) ∂t (0.3) f |t=0 = h

where h : M → N is a smooth map. In order to show that a solution of this system exists for all t > 0 and converges to a solution of (0.2) as t → ∞, we impose a nonpositivity condition on the horizontal curvature of N . The main result of this section asserts that if h : M → N is a foliated map between two compact Sasakian manifolds and N is regular with non-positive horizontal sectional curvature, then there exists a e foliated special (H, H)-harmonic map in the same foliated homotopy class as h. In Section 8, we first give a foliated rigidity result which states that if f : M → N is a e (H, H)-harmonic map between two compact Sasakian manifolds and the target N has e non-positive horizontal curvature, then f must be foliated. Next, we obtain a (H, H)e holomorphicity result which asserts that a (H, H)-harmonic map of sufficiently high maximum rank of a compact Sasakian manifold to a compact Sasakian manifold with either strongly negative horizontal curvature or adequately negative horizontal curvature e e must be (H, H)-holomorphic or (H, H)-antiholomorphic. Besides, we establish some foliated strong rigidity theorems for Sasakian manifolds with either strongly negative horizontal curvature or adequately negative horizontal curvature (see Theorem 8.12 and Corollary 8.13). In Appendix A, we introduce another natural generalized harmonic maps between pseudo-Hermitian manifolds, called pseudo-Hermitian harmonic maps. First, we give a continuation theorem about the foliated property for pseudo-Hermitian harmonic maps. Next, we obtain a rigidity result which asserts that if f : M → N is a pseudo-Hermitian harmonic map between two compact Sasakian manifolds, and N e has non-positive horizontal curvature, then f is a foliated special (H, H)-harmonic map. e The latter result shows the rationality for using special (H, H)-harmonic maps as a tool in our study of global geometric and topological properties of Sasakian manifolds. In Appendix B, we give explicit formulations for both (0.2) and (0.3) , which are helpful for us to understand the existence theory in Section 7. The method for these formulations is possibly useful in studying the existence of other generalized harmonic maps. 1. Pseudo-Hermitian Geometry In this section, we collect some facts and notations concerning pseudohermitian structures on CR manifolds (cf. [DTo], [BG] for details). Definition 1.1. Let M 2m+1 be a real (2m + 1)-dimensional orientable C ∞ manifold. A CR structure on M is a complex rank m subbundle H 1,0 M of T M ⊗ C satisfying (i) H 1,0 M ∩ H 0,1 M = {0} (H 0,1 M = H 1,0 M ); (ii) [Γ(H 1,0 M ), Γ(H 1,0M )] ⊆ Γ(H 1,0 M ). The pair (M, H 1,0 M ) is called a CR manifold of CR dimension m. The complex subbundle H 1,0 M corresponds to a real subbundle of T M : (1.1)

H(M ) = Re{H 1,0 M ⊕ H 0,1 M } 5

which is called the Levi distribution. The Levi distribution H(M ) admits a natural complex structure defined by J(V + V ) = i(V − V ) for any V ∈ H 1,0 M . Equivalently, the CR structure may be described by the pair (H(M ), J). Let E be the conormal bundle of H(M ) in T ∗ M , whose fiber at each point x ∈ M is given by (1.2)

Ex = {ω ∈ Tx∗ M : ker ω ⊇ Hx (M )}.

Since M is assumed to be orientable, and the complex structure J induces an orientation on H(M ), it follows that the real line bundle E is orientable. Thus E admits globally defined nowhere vanishing sections. Definition 1.2. A globally defined nowhere vanishing section θ ∈ Γ(E) is called a pseudo-Hermitian structure on M . The Levi-form Lθ associated with a pseudoHermitian structure θ is defined by (1.3)

Lθ (X, Y ) = dθ(X, JY )

for any X, Y ∈ H(M ). If Lθ is positive definite for some θ, then (M, H(M ), J) is said to be strictly pseudoconvex. When (M, H(M ), J) is strictly pseudoconvex, it is natural to orient E by declaring a section θ to be positive if Lθ is positive. Henceforth we will assume that (M, H(M ), J) is a strictly pseudoconvex CR manifold and θ is a positive pseudo-Hermitian structure. The quadruple (M, H(M ), J, θ) is called a pseudo-Hermitian manifold. On a pseudo-Hermitian manifold (M, H(M ), J, θ), one may use basic linear algebra to derive that ker θx = Hx (M ) for each point x ∈ M , and there is a unique globally defined vector field ξ such that (1.4)

θ(ξ) = 1,

dθ(ξ, ·) = 0.

The vector field ξ is referred to as the Reeb vector field. The collection of all its integral curves forms an oriented one-dimensional foliation Fξ on M , which is called the Reeb foliation in this paper. Consequently there is a splitting of the tangent bundle T M (1.5)

T M = H(M ) ⊕ Lξ ,

where Lξ is the trivial line bundle generated by ξ. Let ν(Fξ ) be the vector bundle whose fiber at each point p ∈ M is the quotient space Tp M/Lξ , and let πν : T M → ν(Fξ ) be the natural projection. Clearly πν |H(M ) : H(M ) → ν(Fξ ) is a vector bundle isomorphism. ∗ Let πH : T M → H(M ) denote the natural projection morphism. Set Gθ = πH Lθ , that is, (1.6)

Gθ (X, Y ) = Lθ (πH X, πH Y )

for any X, Y ∈ T M . Let us extend J to a (1, 1)-tensor field on M by requiring that (1.7)

Jξ = 0. 6

Then the integrability condition (ii) in Definition 1.1 implies that Gθ is J-invariant. The Webster metric on (M, H(M ), J, θ) is a Riemannian metric defined by gθ = θ ⊗ θ + Gθ .

(1.8) It follows that (1.9)

θ(X) = gθ (ξ, X),

dθ(X, Y ) = gθ (JX, Y )

for any X, Y ∈ T M . We find that (1.5) is actually an orthogonal decomposition of T M with respect to gθ . In terms of terminology from foliation theory, H(M ) and Lξ are also called the horizontal and vertical distributions respectively. Clearly θ ∧ (dθ)m is, up to a constant, the volume form of (M, gθ ). On a pseudo-Hermitian manifold, we have the following canonical linear connection which preserves both the CR and the metric structures. Proposition 1.1 ([Ta], [We]). Let (M, H(M ), J, θ) be a pseudo-Hermitian manifold. Then there exists a unique linear connection ∇ such that (i) ∇X Γ(H(M )) ⊂ Γ(H(M )) for any X ∈ Γ(T M ); (ii) ∇gθ = 0, ∇J = 0 (hence ∇ξ = ∇θ = 0); (iii) The torsion T∇ of ∇ is pure, that is, for any X, Y ∈ H(M ), T∇ (X, Y ) = dθ(X, Y )ξ and T∇ (ξ, JX) + JT∇ (ξ, X) = 0. The connection ∇ in Proposition 1.1 is called the Tanaka-Webster connection. Note that the torsion of the Tanaka-Webster connection is always non-zero. The pseudoHermitian torsion, denoted by τ , is the T M -valued 1-form defined by τ (X) = T∇ (ξ, X). The anti-symmetry of T∇ implies that τ (ξ) = 0. Using (iii) of Proposition 1.1 and the definition of τ , the total torsion of the Tanaka-Webster connection may be expressed as (1.10)

T∇ (X, Y ) = (θ ∧ τ )(X, Y ) + dθ(X, Y )ξ

for any X, Y ∈ T M . Set (1.11)

A(X, Y ) = gθ (τ X, Y )

for any X, Y ∈ T M . Then the properties of ∇ in Proposition 1.1 also imply that τ (H 1,0 (M )) ⊂ H 0,1 (M ) and A is a trace-free symmetric tensor field. Lemma 1.2 (cf. [DTo]). The Levi-Civita connection ∇θ of (M, gθ ) is related to the Tanaka-Webster connection by (1.12)

1 ∇θ = ∇ − ( dθ + A)ξ + τ ⊗ θ + θ ⊙ J 2

where (θ ⊙ J)(X, Y ) = 12 (θ(X)JY + θ(Y )JX) for any X, Y ∈ T M . Remark 1.1. The Levi form in this paper is 2 times that one in [DTo]. Thus the coefficient of the term θ ⊙ J in (1.12) is different from that in Lemma 1.3 of [DTo]. 7

Lemma 1.3. Let (M 2m+1 , H(M ), J, θ) be a pseudo-Hermitian manifold with the associated Tanaka-Webster connection ∇. Let X and ρ be a vector field and a 1-form on M respectively. Then div(X) =

2m X

gθ (∇eA X, eA )

δ(ρ) = −

and

A=0

2m X

(∇eA ρ)(eA )

A=0

where {eA }A=0,1,...,2m = {ξ, e1 , ..., e2m} is any orthonormal frame field on M . Here div(·) and δ(·) denote the divergence and the codifferential respectively. Proof. According to (1.12) and using the property tr(A) = 0, it is easy to verify that ∇θξ ξ

(1.13)

= 0,

2m X

A=1

∇θeA eA

=

2m X

A=1

∇eA eA .

Since both ∇θ and ∇ are metric connections, we may employ (1.13) and the definition of divX to find divX =

2m X

gθ (∇θeA X, eA )

=

A=0

=

2m X

2m X

{eA gθ (X, eA ) − gθ (X, ∇θeA eA )}

A=0

gθ (∇eA X, eA ).

A=0

Similarly the codifferential of the 1-form ρ can be computed as follows δ(ρ) = − =−

2m X

(∇θeA ρ)(eA )

A=0 2m X

=−

2m X

{eA ρ(eA ) − ρ(∇θeA eA )}

A=0

(∇eA ρ)(eA )

A=0

in view of (1.13) again. For a pseudo-Hermitian manifold (M, H(M ), J, θ), the sub-Laplacian operator △H on a function u ∈ C 2 (M ) is defined by △H u = div(∇H u), where ∇H u = πH (∇u) is the horizontal gradient of u. In terms of a local orthonormal frame field {eA }2m A=1 of H(M ) on an open subset U ⊂ M , the sub-Laplacian can be expressed as (1.14)

△H u =

2m X

A=1

{eA (eA u) − (∇eA eA )u}. 8

The non-degeneracy of the Levi form Lθ on H(M ) implies that {eA , [eA , eB ]}1≤A,B≤2m spans the tangent space Tp M at each point p ∈ U . From [Hö], we know that △H is a hypoelliptic operator. For simplicity, we will denote by h·, ·i the real inner product induced by gθ on various tensor bundles of M . Recall that the curvature tensor R of the Tanaka-Webster connection ∇ is defined by (1.15)

R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z

for any X, Y, Z ∈ Γ(T M ). Set R(X, Y, Z, W ) = hR(Z, W )Y, Xi. Then R satisfies (1.16)

R(X, Y, Z, W ) = −R(X, Y, W, Z) = −R(Y, X, Z, W ),

where the second equality is because of ∇gθ = 0. However, the symmetric property R(X, Y, Z, W ) = R(Z, W, X, Y ) is no longer true for a general pseudo-Hermitian manifold due to the failure of the first Bianchi identity. The curvature tensor of (M, ∇) induces a morphism Q : ∧2 T M → ∧2 T M which is determined by (1.17)

hQ(X ∧ Y ), Z ∧ W i = R(X, Y, Z, W )

for any X, Y, Z, W ∈ T M . The complex extension of Q (resp. R) to a morphism from ∧2 T M C (resp. ⊗4 T M C ) is still denoted by the same notation. For a horizontal 2 plane σ = spanR {X, Y } ⊂ H(M ), we define the horizontal sectional curvature of σ by (1.18)

K H (σ) =

hQ(X ∧ Y ), X ∧ Y i . hX ∧ Y, X ∧ Y i

In particular, the horizontal holomorphic sectional curvature of a horizontal holomorphic 2-plane σ = span{X, JX} ⊂ H(M ) is given by (1.19)

H Khol (σ) =

hQ(X ∧ JX), X ∧ JXi . hX ∧ JX, X ∧ JXi

Definition 1.3. A pseudo-Hermitian manifold (M, H(M ), J, θ) is called a Sasakian manifold if its pseudo-Hermitian torsion τ is zero. If (M, H(M ), J, θ) is Sasakian, then the quadruple (ξ, θ, J, gθ ) is referred to as a Sasakian structure on M with underlying CR structure (H(M ), J|H(M )). It turns out that if (M, H(M ), J, θ) is a Sasakian manifold, then its curvature tensor satisfies the Bianchi identities. Consequently (1.20)

R(X, Y, Z, W ) = R(Z, W, X, Y )

for any X, Y, Z, W ∈ T M . Since ∇ξ = 0, it follows that if one of the vectors X, Y, Z and W is vertical, then (1.21)

R(X, Y, Z, W ) = 0. 9

Furthermore, in terms of ∇J = 0 and the J-invariance of Gθ , we have (1.22)

R(JX, JY, Z, W ) = R(X, Y, JZ, JW ) = R(X, Y, Z, W )

for any X, Y, Z, W ∈ H(M ). When X, Y, Z and W vary in H(M ), R(X, Y, Z, W ) may be referred to as the horizontal curvature tensor, which will be denoted by RH . Hence we discover that all curvature information of the Sasakian manifold is contained in its horizontal curvature tensor RH which enjoys the same properties as the curvature tensor of a Kähler manifold. For a Sasakian manifold (M, H(M ), J, θ), it is known that its Reeb foliation Fξ defines a Riemannian foliation (cf. [BG], [DT]). In addition, Fξ is transversely holomorphic in the following sense. There is an open covering {Uα } of M together with a family of diffeomorphisms Φα : Uα → (−1, 1) × Wα ⊂ R × C m and submersions (1.23)

ϕα = π ◦ Φα : Uα → Wα ⊂ C m

where π : (−1, 1) × Wα → Wα is the natural projection, such that ϕ−1 α (ϕα (x)) is just the integral curve of ξ in Uα passing through the point x, and when Uα ∩ Uβ 6= ∅ the map ϕβ ◦ ϕ−1 α : ϕα (Uα ∩ Uβ ) → ϕβ (Uα ∩ Uβ ) is a biholomorphism. Such a triple (Uα , Φα ; ϕα ) is called a foliated coordinate chart. For every point x ∈ Uα the differential dϕα : Hx (M ) → Tϕα (x) Wα is an isomorphism taking the complex structure Jx on Hx (M ) to that on Tϕα (x) Wα . Definition 1.4. Let M be a compact Sasakian manifold and let Fξ be the Reeb foliation defined by ξ. Then the foliation Fξ is said to be quasi-regular if there is a positive integer k such that each point has a foliated coordinate chart (U, ϕ) such that each leaf of Fξ passes through U at most k times, otherwise it is called irregular. If Fξ is quasi-regular with the integer k = 1, then the foliation is called regular. It is known that the quasi-regular property is equivalent to the condition that all the leaves of the foliation are compact. In the quasi-regular case, the leaf space has the structure of a Kähler orbifold. In the regular case, the foliation is simple so that the Sasakian manifold can be realized as a S 1 -bundle over a Kähler manifold (the Boothby-Wang fibration [BW]), and the natural projection of this fibration is actually a Riemannian submersion. In general, in the irregular case, the leaf space is not even Hausdorff. Definition 1.5. A Sasakian manifold M 2m+1 with the Tanaka-Webster connection ∇ is said to have strongly negative horizontal curvature (resp. strongly seminegative horizontal curvature) if hQ(ζ), ζi < 0 (resp. ≤ 0)

for any ζ = (Z ∧ W )(1,1) 6= 0, Z, W ∈ HM C . Here ζ is the complex conjugate of ζ. In addition, we say that the horizontal curvature tensor RH of a Sasakian manifold M 2m+1 is negative of order k if it is strongly seminegative and it enjoys the following property. ), B = (Biα ) are any two m × k matrices (1 ≤ α ≤ m, 1 ≤ i ≤ k) with If A = (Aα i A B rank = 2k B A 10

and if

X

αβ δγ Rαβγδ ξij ξij = 0

α,β,γ,δ

for all 1 ≤ i, j ≤ k, where

αβ ξij = Aα Bjβ − Aα Bβ , i j i

then either A = 0 or B = 0. The horizontal curvature tensor RH is called adequately negative if it is negative of order m. By the J-invariant property (1.22), we find that the curvature operator Q annihilates any 2-vector of type (2,0) or (0,2). Therefore strongly negativity (resp. strongly seminegativity) of the horizontal curvature tensor implies negativity (resp. semi-negativity) of the horizontal sectional curvature. Example 1.1. H (i) A Sasakian manifolds (M 2m+1 , H(M ), J, θ) with Khol constant is called a Sasakian space form. For each real number λ, Webster [We] gave a model M (λ) for the Sasakian H space form with Khol = λ. The horizontal curvature tensor of M (λ) may be expressed as (1.24)

λ Rαβγδ = − (gαβ gγδ + gαδ gγβ ) 2

with respect to any frame {ηα } of H 1,0 M at every point. Following the method for complex ball in [Si1], one may verify that if λ < 0, then M (λ) has strongly negative horizontal curvature. (ii) By a theorem of Kobayashi ([Ko]), we know that if B is a compact Hodge manifold with integral Kähler form, there exists a Sasakian manifold M , which is the total space of a Riemannian submersion over B. Suppose M 2m+1 is a compact Sasakian manifold with a Riemannian submersion π : M → B over a compact Kähler manifold. Denote by ∇θ and ∇B the Levi-Civita connections of M and B. Recall that a basic vector field on M is one that is both horizontal and projectable. Suppose X, Y are basic e Ye the vector fields on B that are π-related to X vector fields on M . Denote by X, and Y . A basic property on the connections of a Riemannian submersion (cf.[O’N], [GW]) gives that dπ[πH (∇θX Y )] = (∇Xe Ye ) ◦ π, where πH : T M → H(M ) is the natural projection. Since M is Sasakian, we know from Lemma 1.2 that πH (∇θX Y ) = ∇X Y . Thus dπ(∇X Y ) = (∇Xe Ye ) ◦ π, which implies that (1.25)

R(X, Y, Z, W ) = RB (dπ(X), dπ(Y ), dπ(Z), dπ(W ))

for any X, Y, Z, W ∈ H(M ), where RB is the curvature tensor of ∇B . Hence M has strongly curvature horizontal curvature (resp. strongly semi-negative horizontal curvature with negative order k) if and only if B has strongly negative curvature (resp. strongly semi-negative curvature with negative order k) in the sense of [Si1]. In terms of [Si1], we find that if B is a compact quotient of an irreducible symmetric bounded domain, then M has adequately negative horizontal curvature. Since the foliation of a Sasakian manifold is locally a Riemannian foliation over a Kähler manifold, the above 11

discussion helps us to understand the general curvature properties of a Sasakian manifold from those of a Kähler manifold. For example, it is proved in [Si1] that if the curvature tensor of a Kähler manifold is strongly negative, then it is negative of order 2. Therefore we may conclude that if a Sasakian manifold has strongly negative horizontal curvature, then its horizontal curvature tensor is negative of order 2. 2. Second fundamental forms and their covariant derivatives e be two pseudo-Hermitian manie ), J, e θ) Let (M 2m+1 , H(M ), J, θ) and (N 2n+1 , H(N e the Tanaka-Webster connections of M and N respectively. folds. Denote by ∇ and ∇ Let f : M → N be a smooth map. Then the bundle T ∗ M ⊗ f −1 T N has the induced e where f −1 ∇ e is the pull-back connection in f −1 T N . For simplicconnection ∇ ⊗ f −1 ∇, e as ∇ e when the meaning is clear. The second fundamental form ity, we will write f −1 ∇ e is defined by: of f with respect to (∇, ∇) e Y df ](X) β(X, Y ) = [(∇ ⊗ f −1 ∇)

(2.1)

e Y df (X) − df (∇Y X) =∇

for X, Y ∈ Γ(T M ). In what follows, we shall use the summation convention for repeated indices. e be a map. Then e ), J, e θ) Lemma 2.1. Let f : (M, H(M ), J, θ) → (N, H(N e X df (Y ) − ∇ e Y df (X) − df ([X, Y ]) = Tee (df (X), df (Y )) ∇ ∇

for any X, Y ∈ Γ(T M ), where Te∇ e denotes the torsion of the Tanaka-Webster connection e ∇ on N . e X df (Y ) − ∇ e Y df (X) − df ([X, Y ]). It is easy to show that S is Proof. Set S(X, Y ) = ∇ C ∞ (M )-bilinear. Choose a local coordinate chart (xA ) around p and a local coordinate e chart (uA ) around f (p). Then e

(2.2)

df ( e

∂ ∂f C ∂ ) = ∂xA ∂xA ∂uCe

e

e and using (2.2) , we deduce that where f C = uC ◦ f . By the definition of f −1 ∇ S(

∂ ∂ e ∂ df ( ∂ ) − ∇ e ∂ df ( ∂ ) , ) = ∇ ∂xA ∂xB ∂xA ∂xB ∂xB ∂xA e e ∂ ∂f C ∂f C ∂ e e = ∇ ∂A ( B ) − ∇ ∂B ( A ) ∂x ∂x ∂x ∂uCe ∂x ∂uCe e e ∂ ∂ ∂f C e ∂f C e ∇ ∂A ∇ ∂B − = B A e ∂x ∂x ∂x ∂x ∂uC ∂uCe e e ∂f D ∂f C e ∂ ∂ −∇ e ∂ ∂ ] [∇ = B A e e e ∂uC e ∂uD ∂x ∂x ∂uD ∂uC ∂ ∂ = Te∇ ), df ( B )). e (df ( A ∂x ∂x 12

Hence this lemma is proved.

We will use the moving frame method to perform local computations on maps between pseudo-Hermitian manifolds. Let us now recall the structure equations of the Tanaka-Webster connection on a pseudo-Hermitian manifold. For the pseudo-Hermitian manifold (M 2m+1 , H(M ), J, θ), we choose a local orthonormal frame field {eA }2m A=0 = {ξ, e1 , ..., em , em+1 , . .., e2m } with respect to gθ such that {em+1 , ..., e2m} = {Je1 , ..., Jem}. Set (2.3)

1 ηj = √ (ej − iJej ), 2

1 ηj = √ (ej + iJej ) 2

(j = 1, ..., m).

Then {ξ, ηj , ηj } forms a frame field of T M ⊗ C. Let {θ, θ j , θ j } be the dual frame field of {ξ, ηj , ηj }. From Proposition 1.1, we have (2.4)

∇X ξ = 0,

∇X ηj = θji (X)ηi ,

∇X ηj = θji (X)ηi

for any X ∈ T M , where {θ00 = θ0i = θ0i = θi0 = θi0 = 0, θji , θji } are the connection 1-forms of ∇ with respect to the frame field {ξ, ηj , ηj }. According to (iii) of Proposition 1.1, the pseudo-Hermitian torsion may be expressed as (2.5)

τ = Ajk θ k ⊗ ηj + Akj θ k ⊗ ηj .

The symmetry of A implies that Ajk = Akj = Ajk . Since ∇ preserves H 1,0 M , we may write (2.6)

i ηi . R(ηk , ηl )ηj = Rjkl

From [We], we know that {θ, θ i, θ i , θji , θji } satisfies the following structure equations (cf. also §1.4 of [DTo]): √  dθ = −1θ j ∧ θ j   dθ i = −θji ∧ θ j + Aji θ ∧ θ j (2.7)   i dθj = −θki ∧ θjk + Ψij

where (2.8)

√ i k i k θ ∧ θ + −1θ i ∧ Ajk θ k Ψij =Wjk θ ∧ θ − Wjk √ i θk ∧ θl − −1Aik θ k ∧ θ j + Rjkl

and (2.9)

j Wkl = Akl,j ,

13

j Wkl = Alj,k .

e ηeα , ηeα }α=1,...,n be a local frame field on the pseudo-Hermitian manifold N 2n+1 , Let {ξ, e θeα , θeα }α=1,...,n be its dual frame field. We will denote the connection 1-forms, and let {θ, e on N by the same torsion and curvature, etc., of the Tanaka-Webster connection ∇ e are valid notations as in M , but witheon them. Then similar structure equations for ∇ in N too. Henceforth we shall make use of the following convention on the ranges of indices: A, B, C, ... = 0, 1, ..., m, 1, ..., m; i, j, k, ... = 1, ..., m, i, j, k, ... = 1, ..., m; e B, e C, e ... = 0, 1, ..., n, 1, ..., n; A,

α, β, γ, ... = 1, ..., n, α, β, γ, ... = 1, ..., n. As usual repeated indices are summed over the respective ranges. For a map f : M → N , we express its differential as e

(2.10) Therefore

(2.11)

 ∗e f θ    f ∗ θeα    ∗ eα f θ

df = fBA θ B ⊗ ηeAe. = f00 θ + fj0 θ j + fj0 θ j = f0α θ + fjα θ j + fjα θ j = f0α θ + fjα θ j + fjα θ j .

By taking the exterior derivative of the first equation in (2.11) and making use of the structure equations in M and N , we get (2.12) where

Df00 ∧ θ + Dfj0 ∧ θ j + Dfj0 ∧ θ j + if00 θ j ∧ θ j + fj0 θ ∧ Akj θ k + fj0 θ ∧ Ajk θ k − if ∗ θeα ∧ f ∗ θeα = 0 0 0 j 0 j θ Df00 = df00 = f00 θ + f0j θ + f0j

(2.13)

0 0 l 0 l θ Dfj0 = dfj0 − fk0 θjk = fj0 θ + fjl θ + fjl 0 0 l 0 l θ + fjl θ + fjl Dfj0 = dfj0 − fk0 θjk = fj0 θ.

Then (2.12) gives 0 0 fj0 − f0j + fk0 Akj = i(f0α fjα − f0α fjα )

0 0 fj0 − f0j + fk0 Ajk = i(f0α fjα − f0α fjα )

(2.14)

0 fjl − flj0 = i(fjα flα − fjα flα ) 0 fjl − flj0 = i(fjα flα − fjα flα )

0 − flj0 − if00 δjl = i(fjα flα − fjα flα ). fjl 14

bα = f ∗ A eα , Ψ b α = f ∗Ψ e α , etc. To simplify the notations, we will set θbβα = f ∗ θeβα , A β β β β Similar computations for the second equation in (2.11) yield (2.15) where

(2.16)

Df0α ∧ θ + Dfjα ∧ θ j + Dfjα ∧ θ j + if0α θ j ∧ θ j

bα f ∗ θe ∧ f ∗ θeβ + fjα Ajk θ ∧ θ k + fjα Ajk θ ∧ θ k = A β α α j α j θ + f0j θ + f0j Df0α = df0α + f0β θbβα = f00 θ ,

α α l α l θ + fjl θ + fjl Dfjα = dfjα − fkα θjk + fjβ θbβα = fj0 θ, α l α l α θ + fjl θ + fjl Dfjα = dfjα − fkα θjk + fjβ θbβα = fj0 θ.

From (2.15), it follows that

α α bα (f 0 f β − f 0 f β ) f0j − fj0 − fkα Akj = A 0 j β j 0

(2.17)

α α bα (f 0 f β − f00 f β ) f0j − fj0 − fkα Akj = A β j 0 j α bα (f 0 f β − f 0 f β ) fjl − fljα = A j l β l j

α bα (f 0 f β − f β f 0 ) − fljα + if0α δlj = A fjl l j β l j

α bα (f 0 f β − f 0 f β ). − fljα = A fjl j l β l j

α Likewise we may deduce those commutative relations of fAB from the third equation of (2.11) by taking its exterior derivative, or directly from (2.17) by taking bar of each index. Clearly the second fundament form β can be expressed as

(2.18)

e

A θ B ⊗ θ C ⊗ ηeAe. β = fBC

Due to the torsions of the Tanaka-Webster connections, β is not symmetric. Although the non-symmetry of β causes a little trouble, we will see that it may also lead to some unexpected geometric consequences. By taking the exterior derivative of the first equation of (2.13) and making use of the structure equations, we get (2.19)

0 j 0 0 0 ∧ θ j + if00 θ ∧ θj Df00 ∧ θ + Df0j ∧ θ j + Df0j 0 0 + f0j Akj θ ∧ θ k + f0j Ajk θ ∧ θ k = 0

where 0 0 0 0 0 θj Df00 = df00 = f000 θ + f00j θ j + f00j

(2.20)

0 0 0 k 0 0 0 θl Df0j = df0j − f0k θj = f0j0 θ + f0jl θ l + f0jl 0 0 0 k 0 0 0 Df0j θj = f0j0 θl . = df0j − f0k θ + f0jl θ l + f0jl 15

It follows from (2.19) that 0 0 0 Akj f00j − f0j0 = f0k 0 0 0 f00j − f0j0 = f0k Akj 0 0 f0jl = f0lj

(2.21)

0 0 0 j f0jl − f0lj + if00 δl = 0

0 0 = f0lj . f0jl

Similar computations for the second equation of (2.13) give 0 k 0 0 0 θ ∧ θk Dfj0 ∧ θ + Dfjl ∧ θ l + Dfjl ∧ θ l + ifj0

(2.22)

0 l 0 l + fjl Ak θ ∧ θ k + fjl Ak θ ∧ θ k = −fk0 Ψkj

where 0 0 0 k 0 0 0 Dfj0 = dfj0 − fk0 θj = fj00 θ + fj0l θ l + fj0l θl

(2.23)

0 0 0 k 0 k 0 0 0 Dfjl = dfjl − fkl θj − fjk θl = fjl0 θ + fjlk θ k + fjlk θk 0 0 k 0 k 0 0 0 0 = dfjl − fkl θj − fjk θl = fjl0 θ + fjlk θ k + fjlk θk . Dfjl

Then (2.22) implies 0 0 0 fj0l − fjl0 = fjk Akl − fk0 Wjlk

(2.24)

0 0 0 − fjl0 = fjk Alk + fk0 Wjlk fj0l √ √ 0 0 fjkl − fjlk = −1fk0 Ajl − −1fl0 Ajk

0 0 l t 0 − fjlk − fj0 δk = ft0 Rjkl fjkl √ √ 0 0 fjkl − fjlk = −1δkj ft0 Alt − −1δlj ft0 Atk .

0 The commutative relations for fjAB may be derived similarly from the third equation in (2.13) or directly from (2.24) by taking bar of each index. By taking the exterior derivative of the first equation of (2.16), we deduce that

( 2.25) where

(2.26)

α j α α α ∧ θ j + if00 θ ∧ θj Df00 ∧ θ + Df0j ∧ θ j + Df0j α j α j bα + f0j Ak θ ∧ θ k + f0j Ak θ ∧ θ k = f0β Ψ β β bα α α α α α θβ = f000 θ + f00j θ j + f00j Df00 = df00 + f00 θj

β bα α α l α l α α α k θ θβ = f0j0 θ + f0jl θ + f0jl Df0j = df0j − f0k θj + f0j

α α α k α bα α α l α l Df0j θj + f0j θ. = df0j − f0k θβ = f0j0 θ + f0jl θ + f0jl 16

Consequently (2.27)

α α α k bα (f γ f δ − f δ f γ ) + f β W c α (f γ f 0 − f 0 f γ ) f00j − f0j0 − f0k Aj = f0β R j 0 βγ j 0 j 0 0 βγδ j 0

(2.28)

α α k α bα (f γ f δ − f δ f γ ) + f β W c α (f γ f 0 − f 0 f γ ) − f0j0 − f0k Aj = f0β R f00j βγ j 0 0 j 0 j 0 βγδ j 0

(2.29)

α α α cβγ bα (f γ fjδ − f γ flδ ) + f β W (flγ fj0 − fjγ fl0 ) f0jl − f0lj = f0β R 0 j βγδ l

(2.30)

α α bα (f γ f δ − f γ f δ ) + f β W c α (f γ f 0 − f γ f 0 ) − f0lj = f0β R f0jl βγ l j 0 βγδ l j j l j l

(2.31)

α α α j α cβγ bα (f γ f δ − flδ f γ ) + f β W (flγ fj0 − fjγ fl0 ) − f0lj + if00 δl = f0β R f0jl 0 βγδ l j j

β bα γ β β γ δ α c α (f γ f 0 − f 0 f γ ) + if β A bβ α δ − f0β W βγ j 0 j 0 0 δ (fj f0 − fj f0 ) − if0 Aγ (fj f0 − fj f0 ),

β γ α cβγ bβ (f α f0δ − f δ f0α ) − if β A bα γ β (fjγ f00 − fj0 f0γ ) + if0β A − f0β W 0 γ (fj f0 − fj f0 ), δ j j

β bα γ β γ β α δ c α (f γ f 0 − f γ f 0 ) + if β A bβ α δ − f0β W βγ l j 0 δ (fl fj − fj fl ) − if0 Aγ (fl fj − fj fl ), j l

β bα γ β γ β α δ c α (f γ f 0 − f γ f 0 ) + if β A bβ α δ − f0β W βγ l j 0 δ (fl fj − fj fl ) − if0 Aγ (fl fj − fj fl ), j l

β bα γ β γ β α δ c α (f γ f 0 − f γ f 0 ) + if β A bβ α δ − f0β W βγ l j 0 δ (fl fj − fj fl ) − if0 Aγ (fl fj − fj fl ). j l

Applying the exterior derivative to the second equation of (2.16), we obtain

(2.32)

where

(2.33)

α α α α k Dfj0 ∧ θ + Dfjl ∧ θ l + Dfjl θ ∧ θk ∧ θ l + ifj0 α l α l bα + fjl Ak θ ∧ θ k + fjl Ak θ ∧ θ k = −fkα Ψkj + fjβ Ψ β β bα α α α k α α l α l θ Dfj0 = dfj0 − fk0 θj + fj0 θβ = fj00 θ + fj0l θ + fj0l

β bα α α α k α k α α k α k Dfjl = dfjl − fkl θj − fjk θl + fjl θβ = fjl0 θ + fjlk θ + fjlk θ

β bα α α α k α k α k α k α Dfjl = dfjl − fkl θj − fjk θl + fjl θ + fjlk θ + fjlk θ . θβ = fjl0

Let us substitute (2.33) into (2.32) to get

(2.34)

α α α k bα (f γ f δ − f γ f δ ) fj0l − fjl0 − fjk Al = −ftα Wjlt + fjβ R 0 l βγδ l 0

c α (f γ f 0 − f γ f 0 ) c α (f γ f 0 − f γ f 0 ) − f β W + fjβ W βγ l 0 βγ l 0 0 l 0 l j bα (f γ f β − f γ f β ), bβ (f α f δ − f α f δ ) − if β A + ifjβ A l 0 0 l γ l 0 0 l j δ 17

(2.35)

α α α k bα (f γ f δ − f γ f δ ) fj0l − fjl0 − fjk Al = ftα Wjlt + fjβ R 0 l βγδ l 0

c α (f γ f 0 − f γ f 0 ) c α (f γ f 0 − f γ f 0 ) − f β W + fjβ W βγ l 0 βγ l 0 0 l 0 l j

bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ), + ifjβ A 0 l γ l 0 0 l j δ l 0

(2.36)

α α bα (f γ f δ − f γ f δ ) fjlk − fjkl = i(Ajk flα − fkα Ajl ) + fjβ R l k βγδ k l

c α (f γ f 0 − f γ f 0 ) c α (f γ f 0 − f γ f 0 ) − f β W + fjβ W βγ k l βγ k l j l k l k bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ), + ifjβ A γ k l k l l k j l k δ

(2.37)

α α α k t bα (f γ f δ − f γ f δ ) fjlk − fjkl + ifj0 δl = −ftα Rjkl + fjβ R βγδ k l l k

c α (f γ f 0 − f γ f 0 ) − f β W c α (f γ f 0 − f γ f 0 ) + fjβ W βγ k l βγ k l j l k l k bα (f γ f β − f γ f β ), bβ (f α f δ − f α f δ ) − if β A + ifjβ A γ k l k l j δ l k l k

(2.38)

α α bα (f γ f δ − f γ f δ ) − fjkl = iftα (Akt δlj − Alt δkj ) + fjβ R fjlk βγδ k l l k

c α (f γ f 0 − f γ f 0 ) c α (f γ f 0 − f γ f 0 ) − f β W + fjβ W βγ k l βγ k l j l k l k bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ). + ifjβ A γ k l j δ k l l k l k

Next, computing the exterior derivative of the third equation of (2.16), we derive that α k α α α ∧ θ l + ifj0 Dfj0 ∧ θ + Dfjl ∧ θ l + Dfjl θ ∧ θk

(2.39)

α l α l bα + fjl Ak θ ∧ θ k = −fkα Ψkj + fjβ Ψ Ak θ ∧ θ k + fjl β

where

(2.40)

β bα α α α α k α l α l Dfj0 θj + fj0 θβ = fj00 θ = dfj0 − fk0 θ + fj0l θ + fj0l

β bα α α α α k α k α k α k Dfjl θβ = fjl0 θj − fjk θ = dfjl − fkl θl + fjl θ + fjlk θ + fjlk

β bα α α α k α k α k α α k Dfjl = dfjl − fkl θj − fjk θl + fjl θ + fjlk θ + fjlk θ . θβ = fjl0

Therefore we have

(2.41)

α α k α bα (f γ f δ − f γ f δ ) − fjl0 − fjk Al = ftα Wjlt + fjβ R fj0l 0 l βγδ l 0

c α (f γ f 0 − f γ f 0 ) − f β W c α (f γ f 0 − f γ f 0 ) + fjβ W βγ l 0 βγ l 0 0 l 0 l j

bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ), + ifjβ A γ l 0 l 0 0 l 0 l δ j 18

(2.42)

α α k α bα (f γ f δ − f γ f δ ) Al = −ftα Wjlt + fjβ R − fjl0 − fjk fj0l 0 l βγδ l 0

c α (f γ f 0 − f γ f 0 ) − f β W c α (f γ f 0 − f γ f 0 ) + fjβ W βγ l 0 βγ l 0 0 l 0 l j bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ), + ifjβ A 0 l γ l 0 0 l δ l 0 j

(2.43)

α α bα (f γ f δ − f γ f δ ) − fjkl = −iftα Atk δlj + iftα Atl δkj + fjβ R fjlk l k βγδ k l

c α (f γ f 0 − f γ f 0 ) − f β W c α (f γ f 0 − f γ f 0 ) + fjβ W βγ k l βγ k l l k l k j bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ), + ifjβ A k l l k γ k l δ l k j

(2.44)

α α k t α bα (f γ f δ − f γ f δ ) − fjkl + ifj0 + fjβ R δl = −ftα Rjkl fjlk βγδ k l l k

c α (f γ f 0 − f γ f 0 ) − f β W c α (f γ f 0 − f γ f 0 ) + fjβ W βγ k l βγ k l j l k l k bβ (fkα f δ − f α fkδ ) − if β A bγα (f γ f β − f γ f β ), + ifjβ A k l δ l l j l k

(2.45)

α α bα (f γ f δ − f γ f δ ) fjlk − fjkl = ifkα Ajl − iflα Akj + fjβ R βγδ k l l k

c α (f γ f 0 − f γ f 0 ) − f β W c α (f γ f 0 − f γ f 0 ) + fjβ W βγ k l βγ k l j l k l k

bβ (f α f δ − f α f δ ) − if β A bα (f γ f β − f γ f β ). + ifjβ A γ k l δ k l l k j l k

α Similarly the commutative relations of fABC can be deduced from (2.27)-(2.31), (2.34)(2.38) and (2.41)-(2.45) by taking bar of each index.

e 3. (H, H)-harmonic maps

e be two pseudo-Hermitian manie ), J, e θ) Let (M 2m+1 , H(M ), J, θ) and (N 2n+1 , H(N e respectively. Suppose Ψ is folds endowed with the Tanaka-Webster connections ∇ and ∇ k −1 e )) a section of Hom(⊗ T M, f T N ). Let ΨH,He be a section of Hom(⊗k H(M ), f −1 H(N defined by (3.1)

ΨH,He (X1 , .., Xk ) = πHe ◦ Ψ(X1 , ..., Xk )

e ) is the natural projection morfor any X1 , ..., Xk ∈ HM , where πHe : T N → H(N phism. For convenience, one may extend ΨH,He to a section of Hom(⊗k T M, f −1 T N ) by requiring that ΨH,He (Z1 , .., Zk ) = 0 if one of Z1 , ..., Zk ∈ T M is vertical. For any smooth map f : M → N between the pseudo-Hermitian manifolds, Petit ([Pe]) introduced the following horizontal energy functional Z 1 (3.2) EH,He (f ) = |df e |2 dvθ 2 M H,H

where dvθ = θ ∧ (dθ)m and then he derived its first variational formula. Since our notations are slightly different from those in [Pe], we will derive the first variational formula of EH,He again for the convenience of the readers. 19

Proposition 3.1 ([Pe]). Let {ft }|t| −2mi(Ajk fjα fkα − Ajk fjα fkα ) dvθ . = m M 32

i

We multiply (4.33) by (m − 1) and minus (4.24) to get Z α α α α − fjα fj0 (m − 2)i (fjα fj0 + fjα fj0 − fjα fj0 )dvθ M Z Z Z 2(m − 1) m−1 2 α α |τH,He | dvθ − 2m f0 f0 dvθ + hP (f ) + P (f ), dfH,He idvθ = m m M M M Z − 2mi (Ajk fjα fkα − Ajk fjα fkα )dvθ M

that is, (4.34) Z

α α α α + fjα fj0 − fjα fj0 ) + 2mi(Ajk fjα fkα − Ajk fjα fkα ) dvθ − fjα fj0 (m − 2)i(fjα fj0 M Z Z Z 2(m − 1) m−1 2 α α |τH,He | dvθ − 2m f0 f0 dvθ + hP (f ) + P (f ), dfH,He idvθ . = m m M M M

Substituting (4.34) into (4.32) gives (4.35) Z α α 2(m − 1) 1 α α fjk ) − |τH,He |2 + hP (f ) + P (f ), dfH,He i dvθ 4(fjk fjk + fjk m m M Z Z e − 2m f0α f0α dvθ − 4 R(df e (ηk ))dvθ = 0. e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H H,H M

M

Using the Cauchy-Schwarz inequality, the parallelogram law and the fourth equation of (2.17), we discover

(4.36)

α α α α α α α α fjk + fjk fkk + fkk fkk fjk fjk ≥ fkk X 1 X X α 2 α 2 ≥ fkk | | fkk | + | m α k k X 1 X X α α 2 α α 2 = )| + | (fkk − fkk )| | (fkk + fkk 2m α k

k

m 1 |τH,He |2 + f0α f0α = 4m 2

P P α α 2 + fkk )| . It follows from (4.35), (4.36) and the curvature where |τH,He |2 = 2 α | k (fkk assumption on N that Z − hP (f ) + P (f ), dfH,He idvθ M Z 2m e ≥− R(df e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H e (ηk ))dvθ H,H m−1 M ≥ 0. The first claim is proved. 33

Assume now that M is a Sasakian manifold with m ≥ 1 and N is an arbitrary Sasakian manifold . Using (4.10) and the integration by parts, we get from (4.22) that −

Z

M

Z

α fjα fkkj

α fjα fkkj

α fjα fkkj

α fjα fkkj

hP (f ) + P (f ), dfH,He idvθ = − + + + dvθ M Z α α α α α α α α = fjj fkk + fjj fkk + fjj fkk + fjj fkk dvθ M

≥ 0.

This gives the second claim. e is called horizontally e ), J, e θ) Definition 4.2. A map f : (M, H(M ), J, θ) → (N, H(N constant if it maps the domain manifold into a single leaf of the pseudo-Hermitian foliation on N . e be a map between two pseudoe ), J, e θ) Lemma 4.5. Let f : (M, H(M ), J, θ) → (N, H(N Hermitian manifolds. Then f is horizontally constant if and only if dfH,He = 0.

Proof. If f is horizontally constant, then df (X) is tangent to the fiber of N for any X ∈ T M . Clearly we have dfH,He = 0. Conversely, we assume that dfH,He = 0, which is equivalent to fjα = fjα = fjα =

fjα = 0. Then the fourth equation in (2.17) yields f0α = f0α = 0. Hence πHe df (X) = 0 eq is the integral curve of ξe passing for any X ∈ T M . Suppose that f (p) = q and C through the point q. For any point p′ ∈ M , let c(t) be a smooth curve joining p and p′ . Obviously f (c(t)) is a smooth curve passing through p and df (c′ (t)) = λ(t)ξe for some function λ(t), which means that f (c(t)) is the reparametrization of the integral curve of e Therefore f (c(t)) ⊂ C eq . In particular, f (p′ ) ∈ C eq . Since p′ is arbitrary, we conclude ξ. eq . that f (M ) ⊂ C e It is easy to see that a horizontally constant map is foliated and (H, H)-harmonic. The following result is also obvious by intuition.

e is a horizontal map. If f is e θ) Lemma 4.6. Suppose f : (M 2m+1 , J, θ) → (N 2n+1 , J, horizontally constant, then f is constant. Proof. Since f is horizontal and horizontally constant, we have fj0 = fj0 = 0 and fjα = fjα = fjα = fjα = 0. Hence df ◦ iH = 0. Then the fifth equation of (2.14) implies f00 = 0. This shows that df (ξ) = 0, since f is foliated. Therefore we conclude that f is constant. Lemma 4.7. If N has non-positive horizontal sectional curvature, then

(4.37)

e R(df e (ηk )) e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H H,H

e + R(df e (ηk )) ≤ 0. e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H H,H 34

Proof. Write dfH,He (ηj ) = Xj + iYj (j = 1, ..., m). We find that (4.38) the l.h.s. of (4.37) e j , Xk , Xj , Xk ) − R(X e j , Xk , Yj , Yk ) + R(X e j , Yk , Xj , Yk ) + R(X e j , Yk , Yj , X k ) = R(X

e j , Xk , Xj , Yk ) + R(Y e j , Xk , Yj , Xk ) − R(Y e j , Yk , Xj , Xk ) + R(Y e j , Yk , Yj , Yk ) + R(Y

e j , Xk , Xj , Xk ) + R(X e j , Xk , Yj , Yk ) + R(X e j , Yk , Xj , Yk ) − R(X e j , Yk , Yj , X k ) + R(X e j , Xk , Xj , Yk ) + R(Y e j , Xk , Yj , Xk ) + R(Y e j , Yk , Xj , Xk ) + R(Y e j , Yk , Yj , Yk ) − R(Y

e j , Xk , Xj , Xk ) + R(X e j , Yk , Xj , Yk ) + R(Y e j , Xk , Yj , Xk ) + R(Y e j , Yk , Yj , Yk )}, = 2{R(X

e H ≤ 0. which is nonpositive by the assumption that K

Now we want to give some consequences of the Bochner formula in Theorem 4.1.

e be a (H, H)-harmonic e θ) e Theorem 4.8. Let f : (M 2m+1 , J, θ) → (N 2n+1 , J, map from a compact pseudo-Hermitian manifold with CR dimension m ≥ 2 to a Sasakian manifold with strongly semi-negative horizontal curvature. Let σ0 (x) be the maximal eigenvalue of the symmetric matrix (|Ajk |x )m×m at x ∈ M . Suppose that (4.39)

RicH − (m + 2)σ0 Lθ ≥ 0,

where Lθ is the Levi-form (see Definition 1.2). Then (i) βH,He = 0; (ii) If RicH − (m + 2)σ0 Lθ > 0 at a point in M , then f is horizontally constant; (iii) If N has negative horizontal sectional curvature, then f is either horizontally constant or of horizontal rank one. Proof. At each point, let λ be the minimal eigenvalue of the Hermitian matrix (Rjk ). Therefore hdfH,He (RicH (ηj )), dfH,He (ηj )i = Rjk fkα fjα + fkα fjα Rjk

≥ λ(fkα fkα + fkα fkα ) X =λ |fkα |2 + |fkα |2 .

(4.40)

α,k

By the definition of σ0 , one has |i(fjα fkα Ajk − fjα fkα Ajk )| ≤ 2τ0 (4.41)

≤ τ0

X α,k

X α,k

|fkα fkα | |fkα |2 + |fkα | .

From Theorems 4.1, 4.4, (4.39), (4.40) and (4.41), we immediately get (i). Clearly, βH,He = 0 implies that eH,He =const. Besides, we have (4.42)

e R(df e (ηk )) e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H H,H

e + R(df e (ηk )) = 0. e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H H,H 35

Next assume that RicH −(m+2)τ0 Lθ > 0 at a point p in M . This additional condition clearly implies that fkα (p) = fkα (p) = 0. Therefore eH,He = 0, that is, dfH,He = 0. It follows from Lemma 4.5 that f is horizontally constant. This proves (ii). Now we consider the claim (iii). If K H < 0, then (4.42) implies that rank(dfH,He ) is zero or one in view of (4.38). Since eH,He is constant, the rank is constant. In the first case, f is horizontally constant; and in the second case, we say that f is of horizontal rank one. Remark 4.1. (a) On a pseudo-Hermitian manifold, the interchange of two covariant derivatives with respect to the Tanaka-Webster connection yields not only the curvature terms, but also the pseudo-Hermitian torsion term. Hence it seems natural that the conditions for Bochner-type results include both ingredients. (b) We have already known that a horizontally constant map is foliated. So the maps in cases (ii) and (iii) are foliated. Using the Sasakian assumption on the target manifold, the fourth equation of (2.17) implies that f is also foliated for the case (i) of Theorem 4.8. The following two results show that if the domain manifold in Theorem 4.8 is also Sasakian, then the condition m ≥ 2 is not necessary, and the curvature condition on the target manifold may be slightly weakened. e be a (H, H)-harmonic e θ) e Corollary 4.9. Let f : (M 2m+1 , J, θ) → (N 2n+1 , J, map from a compact Sasakian manifold to a Sasakian manifold with non-positive horizontal sectional curvature. Suppose RicH ≥ 0. Then (i) βH,H = 0; (ii) If RicH > 0 at a point p in M , then f is horizontally constant; (iii) If N has negative horizontal sectional curvature, then f is either horizontally constant or of horizontal rank one. Proof. Using Lemma 4.7 and the second claim in Theorem 4.4, the remaining arguments are similar to that for Theorem 4.8. Corollary 4.10. Let M , N and f be as in Corollary 4.9. If f is horizontal, then (i) β = 0 (This property is called totally geodesic); (ii) If RicH > 0 at a point, then f is constant; (iii) If N has negative horizontal sectional curvature, then f is either constant or of horizontal rank one. Proof. From Corollary 4.9 and Remark 4.1 (b), we know that f is a foliated map with βH,He = 0, and thus Lemma 3.6 implies that (4.43)

e df (ξ) = λξ,

f ∗ θe = λθ

for some constant λ. Clearly the first equation of (4.43) yields that (4.44)

β(ξ, X) = 0

0 0 for any X ∈ T M . Since f is horizontal, we have fjl = fjl = 0. Therefore we conclude that β = 0. 36

The results for cases (ii) and (iii) follow immediately from Lemma 4.6 and Corollary 4.9. Remark 4.2. Corollary 4.10 improves a similar theorem of [Pe2] in two aspects. It not only slightly strengthens the corresponding results, but also weakens the curvature condition for the target manifold. e e 5. (H, H)-pluriharmonic and (H, H)-holomorphic maps

e e In this section, we first introduce two special kinds of (H, H)-harmonic maps: (H, H)e pluriharmonic maps and foliated (H, H)-holomorphic maps. Secondly, we give a unique e e continuation theorem which ensures that a (H, H)-harmonic map must be (H, H)e holomorphic on the whole manifold if it is (H, H)-holomorphic on an open subset. e Clearly a similar unique continuation result holds true for (H, H)-antiholomorphicity. As a result, we easily deduce a unique continuation theorem for horizontally constant maps. e The (H, H)-harmonicity equation (3.11) suggests us to introduce the following e between two pseudoe ), J, e θ) Definition 5.1. A map f : (M, H(M ), J, θ) → (N, H(N e Hermitian manifolds is called a (H, H)-pluriharmonic map if it satisfies (5.1)

(βH,He + f ∗ θ ⊗ f ∗ τe)(1,1) = 0,

where the left hand side term of (5.1) denotes the restriction of βH,He + f ∗ θ ⊗ f ∗ τe to H 1,1 (M ). Here H 1,1 (M ) denotes the (1, 1)-part of H(M )C ⊗ H(M )C with respect to the complex structure on H(M ). Clearly (5.1) implies that trGθ (βH,He + f ∗ θ ⊗ f ∗ τe) = 0,

e e that is, a (H, H)-pluriharmonic map is automatically (H, H)-harmonic. It follows from e (3.12), (3.13) and (5.1) that a map f is (H, H)-pluriharmonic if and only if (5.2)

and (5.3)

α bα f 0 f β = 0 fjk +A β j k

α bα f 0 f β = 0 +A fkj β k j

α bα f 0 f β = 0 and f α + A bα f 0 f β = 0 +A for 1 ≤ α ≤ n, 1 ≤ j, k ≤ m or equivalently, fjk β j k β k j kj for 1 ≤ α ≤ n, 1 ≤ j, k ≤ m. e J) e ), θ, e is a (H, H)e Proposition 5.1. Suppose that f : (M, H(M ), θ, J) → (N, H(N e pluriharmonic map. Then f is a foliated (H, H)-harmonic map.

e Proof. We have already shown that f is (H, H)-harmonic. From (2.17), (5.2), (5.3), one may find that f0α = f0α = 0,

that is, f is foliated. 37

e J) e ), θ, e between two pseudoDefinition 5.2. A map f : (M, H(M ), θ, J) → (N, H(N e e hermitian manifolds is called (H, H)-holomorphic (resp. (H, H)-antiholomorphic) if it satisfies (5.4)

dfH,He ◦ J = Je ◦ dfH,He (resp. dfH,He ◦ J = −Je ◦ dfH,He )

e Furthermore, if f is foliated, then it is called a foliated (H, H)-holomorphic map (resp. e a foliated (H, H)-antiholomorphic map).

e Remark 5.1. Clearly the composition of two foliated (resp. (H, H)-holomorphic) maps e is still a foliated (resp. (H, H)-holomorphic) map. Note that the foliation of a pseudoHermitian manifold M is not transversally holomorphic in general, although there is a complex structure J on its horizontal distribution H(M ). e is a smooth map between two pseudoe ), J, e θ) Suppose f : (M, H(M ), J, θ) → (N, H(N Hermitian manifolds. The complexification of dfH,He determines various partial horizontal differentials by the compositions with the inclusions of H 1,0 (M ) and H 0,1 (M ) in e )C on H 1,0 (N ) and H 0,1 (N ) respecH(M )C respectively and the projections of H(N tively. Thus we have the following bundle morphisms (cf. [Si1], [Do]) (5.5)

e 1,0 (N ), ∂fH,He : H 1,0 (M ) → H

e 0,1 (N ), ∂fH,He : H 1,0 (M ) → H

which can be locally expressed as follows (5.6)

e 1,0 (N ), ∂fH,He : H 0,1 (M ) → H e 0,1 (N ), ∂fH,He : H 0,1 (M ) → H ∂fH,He = fjα θ j ⊗ ηeα ,

∂fH,He = fjα θ j ⊗ ηeα ,

∂fH,He = fjα θ j ⊗ ηeα ,

∂fH,He = fjα θ j ⊗ ηeα .

e e From (5.4), it is clear to see that f : M → N is (H, H)-holomorphic (resp. (H, H)antiholomorphic) if and only if ∂fH,He = 0 (resp. ∂fH,He = 0).

e e Proposition 5.2. Suppose that f : M → N is either (H, H)-holomorphic or (H, H)e antiholomorphic. Then f is (H, H)-harmonic if and only if f is foliated.

e Proof. Without loss of generality, we assume that f is (H, H)-holomorphic. Then the α α α e (H, H)-holomorphicity of f means that fk = fk = 0, and thus fkj = 0. Consequently, e the (H, H)-harmonicity equation (3.14) becomes

(5.7)

α bα f 0 f β = 0. fkk +A β k k

On the other hand, the fourth equation of (2.17) yields that (5.8)

α bα f 0 f β = mif α fkk +A 0 β k k

e Therefore we conclude from (5.7) and (5.8) that f is (H, H)-harmonic if and only if it is foliated. 38

e J) e ), θ, e be either a foliated (H, H)e Theorem 5.3. Let f : (M, H(M ), θ, J) → (N, H(N e e holomorphic map or a foliated (H, H)-antiholomorphic map. Then f is a (H, H)pluriharmonic map. e Proof. Without loss of generality, we may assume that f is (H, H)-holomorphic. Then α α α we get fj = fj = fjl = 0 as in Proposition 5.2. It follows that α bα f 0 f β = 0. +A fjl β j l

(5.9)

Since f is foliated, we have f0α = 0. Consequently, (2.17) and (5.9) give that bα fl0 f β = 0. fljα + A β j

e This proves the (H, H)-pluriharmonicity of f .

Recall that a map f : M → N is called a CR map if f is horizontal and dfH,He ◦ J = e e J ◦ dfH,He ([DTo]). Thus CR maps provide us many examples of (H, H)-holomorphic e maps. Since a CR map is not necessarily a foliated map, it is not (H, H)-harmonic in general. A map f : M → N between two pseudo-Hermitian manifolds is called a CR-holomorphic map (cf. [Dr], [IP], [Ur]) if (5.10) df ◦ J = Je ◦ df.

From [IP], we know that a CR-holomorphic map is a harmonic map between the two Riemannian manifolds (M, gθ ) and (N, e gθe) in the usual sense. Clearly (5.10) implies that e ). Hence a CR-holomorphic map is just a foliated Je ◦ df (ξ) = 0 and df (H(M )) ⊂ H(N CR map. In addition, a special kind of CR maps, called pseudo-Hermitian immersions e in [Dr], also provide us a lot of examples for foliated (H, H)-holomorphic maps. e between e θ) Definition 5.3. We call a diffeomorphism f : (M 2m+1 , J, θ) → (N 2m+1 , J, e e two pseudo-Hermitian manifolds a (H, H)-biholomorphism if f and f −1 are (H, H)e H)-holomorphic respectively. Furthermore, if f is foliated, then it holomorphic and (H, e is called a foliated (H, H)-biholomorphism.

Example 5.1. Let (M, H(M ), J, θ) be a pseudo-Hermitian manifold. For any positive function u on M , we set θe = uθ. Then dθe = du ∧ θ + udθ.

Write ξe = λξ + Teα ηα + Teα ηα . By requiring that iξeθe = 1 and iξedθe = 0, one gets λ = u−1 ,

Obviously

Teα = iu−1 ηα (log u),

Teα = −iu−1 ηα (log u).

e idM : (M, H(M ), J, θ) → (M, H(M ), J, θ) e e is a (H, H)-biholomorphism. Note that H(M ) = H(M ) in this example. If u is not e constant, then ξ ∦ ξ, and thus idM is not foliated. This provides us an example of e non-foliated (H, H)-biholomorphisms. In this circumstance, we know from Proposition e is not (H, H)-harmonic. e 5.2 that idM : (M, H(M ), J, θ) → (M, H(M ), J, θ) When e e u is constant, idM : (M, H(M ), J, θ) → (M, H(M ), J, θ) is clearly a foliated (H, H)biholomorphism. 39

Example 5.2. Let S = (ξ, θ, J, gθ ) be a Sasakian structure on M . Then we have its Reeb foliation Fξ and the associated quotient vector bundle ν(Fξ ). Let πν : T M → ν(Fξ ) (X 7→ [X] for any X ∈ T M ) be the natural projection. The background structure (ξ, θ, J, gθ) induces a transversal complex structure Jν on ν(Fξ ) by Jν [X] = [JX]. According to [BG], [BGS], we consider the following space of Sasakian structures with fixed Reeb vector field ξ and fixed transversal complex structure Jν : e θ, e J, e g e) on M | ξe = ξ, Jeν = Jν } S(ξ, Jν ) = {Sasakian structure Se = (ξ, θ

e For any where Jeν denotes the transversal complex structure on ν(Fξ ) induced by J. e is a foliated (H, H)-biholomorphism. e Se ∈ S(ξ, Jν ), we assert that idM : (M, S) → (M, S) To prove this assertion, we may decompose any X ∈ T M as X = aξ+XH = bξ+XHe with e ) = ker θe for some a, b ∈ R. Then πHe (XH ) = XHe , XH ∈ H(M ) = ker θ and XHe ∈ H(M e = JX e e . By definition, Jν [X] = [JXH ] and and (1.7) implies that JX = JXH and JX H e e ]. Since πν : H(M ) → ν(Fξ ) and πν : H(M e Jeν [X] = [JX ) → ν(Fξ ) are both vector H e e e = πν JXH for any bundle isomorphisms, we see that Jν = Jν if and only if πν JX H e e e (XH ) X ∈ T M . On the other hand, diH,He ◦JX = Jdi X if and only if πHe JXH = Jπ e H H,H e that is, πHe JXH = JXHe . Taking projection πν on both sides of the previous equality, we get the result. Recall that a function u on a foliated manifold M is called basic if it is constant along ∞ the leaves. Denote by CB (M ) the space of smooth basic functions on M . In terms of ∞ (M )/R) × [BG1,2], we know that the space S(ξ, Jν ) is an affine space modeled on (CB ∞ 1 (CB (M )/R) × H (M, Z). Indeed, if S = (ξ, θ, J, gθ ) is a given Sasakian structure in e θ, e J, e g e) in it is determined by real valued S(ξ, Jν ), any other Sasakian structure Se = (ξ, θ ∞ (M ) up to a constant and α ∈ H 1 (M, Z) a harmonic 1-form basic functions ϕ, ψ ∈ CB such that θe = θ + dc ϕ + α + dψ

(5.11) √

where dc = 2−1 (∂ − ∂). Thus one may denote the Sasakian structure Se by Seϕ,ψ,α if S is fixed. In order to use the notations in §2, we write f = idM and express (5.11) as √ −1 ∗e ϕk θ k − ϕk θ k + αk θ k + αk θ k + ψk θ k + ψk θ k (5.12) f θ=θ+ 2 where ϕk = ηk (ϕ), ψk = ηk (ψ) and αk = α(ηk ). Then √ √ −1 −1 0 0 fk = − ϕk + αk + ψk , fk = ϕk + αk + ψk . 2 2 It follows that

(5.13)

0 fkk

+

0 fkk

√

−1 ϕkk − ϕkk + αkk + αkk + ψkk + ψkk 2√ m −1 =− ξ(ϕ) + αkk + αkk + ψkk + ψkk 2 40

=

√ where we use the known formula ϕkl − ϕlk = −1δkl ξ(ϕ) for a function in the second equality (see for example [Le], [CDRY]). Since ϕ is basic and α is harmonic, we find from (5.13) that 0 0 + fkk = △H (ψ). fkk e It follows that idM : (M, S) → (M, Seϕ,0,α ) is a special foliated (H, H)-biholomorphism. In general, if ψ is not constant, then idM : (M, S) → (M, Seϕ,ψ,α ) is not special in the sense of Definition 3.2.

Theorem 5.4. Suppose that M , N are Sasakian manifolds and f : M → N is a e e foliated (H, H)-harmonic map. Let U be a nonempty open subset of M . If f is (H, H)e e holomorphic (resp. (H, H)-antiholomorphic) on U , then f is (H, H)-holomorphic (resp. e (H, H)-antiholomorphic) on M .

e Proof. Without loss of generality, we assume that f is (H, H)-holomorphic on U . Although f satisfies a PDE system of ‘subelliptic type’, to the author’s knowledge, the unique continuation theorem is still open for such kind of PDE systems. We will try to show this theorem by using the Aroszajin’s continuation theorem for elliptic PDE systems and the moving frame method. Let Ω be the largest connected open subset of M containing U such that ∂fH,He vanishes identically on Ω. Suppose Ω has a boundary point q. Let W be a connected open neighborhood of q in M such that i) there exists a frame field {ξ, η1 , ...ηm , η1 , ..., ηm } of T M C on some open neighborhood of the closure of W and e ηe1 , ..., ηen, ηe , ..., ηen} of T N C on some open neighborii) there exists a frame field {ξ, 1 hood of the closure of f (W ) . The assumption that f is foliated means that (5.11)

f0α = f0α = 0.

e Since N is Sasakian, the (H, H)-harmonic equation for f becomes

(5.12)

α α fkk + fkk = 0.

By definition of covariant derivatives, we have (5.13) and

(5.14)

e ηα D∂fH,He = dfjα ⊗ θ j ⊗ ηeα + fjα ∇θ j ⊗ ηeα + fjα θ j ⊗ ∇e e ηα D2 ∂fH,He =∇dfjα ⊗ θ j ⊗ ηeα + dfjα ⊗ ∇θ j ⊗ ηeα + dfjα ⊗ θ j ⊗ ∇e e ηα + dfjα ⊗ ∇θ j ⊗ ηeα + fjα ∇2 θ j ⊗ ηeα + fjα ∇θ j ⊗ ∇e

e ηα + f α ∇θ j ⊗ ∇e e ηα + f α θ j ⊗ ∇ e 2 ηeα . + dfjα ⊗ θ j ⊗ ∇e j j 41

Now we compute the trace Laplacian of the section ∂fH,He as follows: △∂fH,He =trgθ D2 ∂fH,He α α α )θ j ⊗ ηeα =(fjkk + fjkk + fj00

e ηα =(△M fjα )θ j ⊗ ηeα + trgθ {dfjα ⊗ ∇θ j ⊗ ηeα + dfjα ⊗ θ j ⊗ ∇e

(5.15)

e ηα + dfjα ⊗ ∇θ j ⊗ ηeα + fjα ∇2 θ j ⊗ ηeα + fjα ∇θ j ⊗ ∇e

e ηα + f α ∇θ j ⊗ ∇e e ηα + f α θ j ⊗ ∇ e 2 ηeα } + dfjα ⊗ θ j ⊗ ∇e j j

where △M denotes the Laplace-Beltrami operator acting on functions. Since M and N are Sasakian, we derive from the second equation of (2.17) and (5.11) that α fj0 =0

(5.16) which yields

α = 0. fj00

(5.17)

Using (2.17), (2.38), (2.44) and (5.12), we discover (5.18)

α α α α + fjkk =fkjk + fkjk fjkk

Consequently (5.19)

bα (f γ f δ − f γ f δ ) − f α Rt + f β R b α (f γ f δ − f γ f δ ). =fkβ R t βγδ k j kkj j k j k k βγδ k j α α α + fjkk + fj00 |≤C |fjkk

X l,β

|flβ |

on W for some positive number C. From (5.15) and (5.19), we find that there is a positive number C ′ such that X X (5.20) | △M (fjα )| ≤ C ′ ( |flα |) |∇flβ | + l,β

l,α

where ∇ denotes the gradient of the functions {fjα }. By applying the Aronszajn’s unique continuation theorem (cf. [Ar], [PRS]) to the system of functions Re{fjα }, Im{fjα } (1 ≤ j ≤ m, 1 ≤ α ≤ n) and to the elliptic operator △M , we conclude from the identical vanishing of Re{fjα }, Im{fjα } on W ∩ Ω that Re{fjα }, Im{fjα } vanish identically on W . This contradicts the fact that q is a boundary point of Ω. Hence Ω = M , which implies that ∂fH,He = 0 on the whole domain manifold. Remark 5.2. Note that we verify the structural assumptions of Aronszajn-Cordes in the proof of Theorem 5.4 by adopting the moving frame method, whose advantage is its operability. This method will be used again in the appendix. e e Note that f is both (H, H)-holomorphic and (H, H)-antiholomorphic if and only if dfH,He = 0, that is, f is horizontally constant. 42

Corollary 5.5. Suppose that M , N are Sasakian manifolds and f : M → N is a foliated e (H, H)-harmonic map. Let U be a nonempty open subset of M . If f is horizontally constant on U , then f is horizontally constant on M .

e e Proof. Since f is horizontally constant on U , it is both (H, H)–holomorphic and (H, H)e antiholomorphic on U . It follows from Theorem 5.4 that f is both (H, H)-holomorphic e and (H, H)-anitholomorphic on the whole M . Hence we conclude that f is horizontally constant on M .

It would be interesting to know whether the Sasakian and foliated conditions in Theorem 5.4 or Corollary 5.5 can be removed or not. In terms of Proposition 5.2, one way to give an answer is to establish a unique continuation result for the foliated property. We would like to propose the following question: e Question. Suppose f : M → N is a (H, H)-harmonic map between two pseudoHermitian manifolds or even Sasakian manifolds . If f is foliated on a nonempty open subset U of M , can we deduce that f is foliated on the whole M ? Though a general Sasakian manifold is not a global Riemannian submersion over a Kähler manifold, the following result will help us to understand the general local picture e and properties about (H, H)-holomorphic maps between Sasakian manifolds.

e are compact e ), J, e θ) Proposition 5.6. Suppose (M 2m+1 , H(M ), J, θ) and (N 2n+1 , H(N Sasakian manifolds which are the total spaces of Riemannian submersions π : M → B e over compact K¨ e respectively. Suppose f : M → and π e:N →B ahler manifolds B and B e between the base manifolds. Then N is a foliated map which induces a map h : B → B e e f is a foliated (H, H)-holomorphic (resp. (H, H)-antiholomorphic) map if and only if h is a holomorphic (resp. anti-holomorphic) map. Proof. Since f is foliated, we have h ◦ π = π e ◦ f , and thus dh ◦ dπ = de π ◦ df . Denote e by J1 and J2 the complex structures of B and B respectively. Since dπ ◦ J = J1 ◦ dπ e e and de π ◦ Je = J2 ◦ de π , we immediately have that f is (H, H)-holomorphic (resp. (H, H)antiholomorphic) if and only if h is holomorphic (resp. antiholomorphic).

Remark 5.2. Suppose now that M and N are two general Sasakian manifolds, which are not necessarily total spaces of Riemannian submersions, and f : M → N is a foliated map. Let p be any point in M and q = f (p). We have foliated neighborhoods U1 and e2 of p and q respectively, together with Riemannian submersions π1 : U1 → W1 and U e2 → W f2 over two Kähler manifolds W1 and W f2 . Assuming that f (U1 ) ⊂ U e2 , then π e2 : U f2 . According to Proposition 5.6, we find that f induces locally a map hp : W1 → W e e f : M → N is (H, H)-holomorphic (resp. (H, H)-antiholomorphic) if and only if the locally induced map hp for each p ∈ M is holomorphic (resp. anti-holomorphic). e between two e ), J, e θ) Definition 5.3. A foliated map f : (M, H(M ), θ, J) → (N, H(N pseudo-Hermtian manifolds is called a horizontally one-to-one map if it induces a oneto-one map h : M/Fξ → N/Feξe between the spaces of leaves. 43

Proposition 5.7. Let M and N be Sasakian manifolds and let f : M → N be a foliated e (H, H)-holomorphic map. If f is both one-to-one and horizontally one-to-one, then f −1 e H)-holomorphic map, that is, f is a foliated (H, H)-biholomorphism. e is a (H,

e2 → W f2 and hp : W1 → W f2 be as in Proof. For any p ∈ M , we let π1 : U1 → W1 , π2 : U f2 is holomorphic. Since f is horizontally one-to-one, we Remark 5.2. Then hp : W1 → W see that hp : W1 → hp (W1 ) is one-to-one. By a known result for holomorphic maps (cf. Proposition 1.1.13 in [Hu]), we know that hp is a local biholomorphic, that is, hp (W1 ) f2 and hp : W1 → hp (W1 ) is biholomorphic. is an open set of W f2 = f (U1 ) ⊂ U e2 . Clearly f −1 : N → M is also a foliated map with f −1 (V f2 ) = U1 , Set V −1 −1 e and f induces the holomorphic map hp : hp (W1 ) → W1 . Using Proposition 5.6, we e H)-holomorphic at q = f (p). Since p is arbitrary, we conclude that find that f −1 is (H, e f is a foliated (H, H)-biholomorphism. 6. Lichnerowicz type results

By definition, one gets from (5.6) that (cf. [Do]) |∂fH,He |2 =h∂fH,H e (ηj )i e (ηj ), ∂fH,H m

(6.1)

and

1X = {hdfH,He (ej ), dfH,He (ej )i + hdfH,He (Jej ), dfH,He (Jej )i 4 j=1 e + 2hdfH,He (Jej ), Jdf e (ej )i} H,H

|∂fH,He |2 =h∂fH,He (ηj ), ∂fH,He (ηj )i m

(6.2)

Thus

1X = {hdfH,He (ej ), dfH,He (ej )i + hdfH,He (Jej ), dfH,He (Jej )i 4 j=1 e − 2hdfH,He (Jej ), Jdf e (ej )i}. H,H

1 |df e |2 = |∂fH,He |2 + |∂fH,He |2 . 2 H,H

(6.3) Define (6.4)

′ EH, e (f ) H

=

Z

2

M

′′ EH, e (f ) H

|∂fH,He | dvθ ,

=

Z

M

|∂fH,He |2 dvθ .

′ ′′ Then EH,He (f ) = EH, e (f ) + EH,H e (f ). Set H

(6.5)

kH,He (f ) = |∂fH,He |2 − |∂fH,He |2 ,

′ ′′ KH,He (f ) = EH, e (f ) − EH,H e (f ). H

44

e Then Lemma 6.2. Set ω M = dθ and ω N = dθ.

kH,He (f ) = hω M , f ∗ ω N i.

Proof. Choose an orthonormal frame {e1 , ..., em , Je1 , ..., Jem} of H(M ). Using (1.9), we deduce that X hω M , f ∗ ω N i = {(f ∗ ω N )(ei , ej )ω M (ei , ej ) + (f ∗ ω N )(Jei , Jej )ω M (Jei , Jej )} i 0, let n o CPk,α (Ω) = u : Ω → R : kukC k,α < ∞ , P X kukC k,α (Ω) = kXA1 · · · XAl ukCPα (Ω) ,

∂ ∂t .

Let

P

w(A1 ,...,Al )≤k

kukCPα (Ω) = |u|CPα (Ω) + kukL∞ (Ω) , ( ) |u(x, t) − u(y, s)| α : (x, t), (y, s) ∈ Ω, (x, t) 6= (y, s) , |u|CPα (Ω) = sup dP (x, t), (y, s)

where 0 ≤ As ≤ 2m, 1 ≤ s ≤ l. Similarly one may use a C ∞ partition of unity to define the function space CPk,α (M × [T1 , T2 ]) for any [T1 , T2 ] ⊂ (0, ∞). Let d(x, y) be the Riemannian distance of x and y in (M, gθ ). For the relatively compact open set U ⊂ M , there exist positive constants c1 , c2 depending on U such that (cf. [NSW]) c1 d(x, y) ≤ dC (x, y) ≤ c2 d(x, y)1/2 50

for any x, y ∈ U . We have a natural Riemannian distance b d((x, t), (y, s)) =

p

d(x, y) + |t − s|2

b one may define the usual Hölder space C k,α (Ω) for on M × (0, ∞). Using the distance d, any open subset Ω ⊂ UT . Clearly there exist two positive constants C1 , C2 such that b b C1 d((x, t), (y, s)) ≤ dP ((x, t), (y, s)) ≤ C2 d((x, t), (y, s)) 2

1

for any x, y ∈ U and 0 ≤ |t − s| 0) and let Ω ⋐ UT be a relatively compact open subset of UT . Suppose u is locally ∂ in Lp (UT ), and (△H − ∂t )u = v. p p a) If v ∈ Sk (UT ), then χu ∈ Sk+2 (UT ) for any χ ∈ C0∞ (UT ). In particular, there exists a constant c > 0 such that kuk

p (Ω) Sk+2

p ≤ c kukLp (UT ) + kvkSk (UT ) .

b) If v ∈ CPk,α (UT ), then there exists a constant c such that kukC k+2,α (Ω) P

o n ≤ c kvkC k,α (UT ) + kukL∞ (UT ) . P

Remark 7.1. (i) It is known that if kp is large enough, then the Sobolev type space Skp is contained in some Hölder space (cf. [RS], [FS], [DT], [FGN]). For example, let k = 2 and p > 2n + 4. 2n+4 If u ∈ S2p (UT ) , then for any χ ∈ C0∞ (UT ), we have χu ∈ Ω1,α . P (UT ) with α = 1 − p In particular, for any relatively compact open subset Ω of UT , there exists a positive constant c such that kukC 1,α (Ω) ≤ ckukS2p (UT ). P (ii) Combining (7.7) and Proposition 7.1(b), we have kukC l+1, α2 (Ω) ≤ C{kvkC 2l,α (UT ) + kukL∞ (UT ) }. P

Since the linearization of (7.6) is a linear subelliptic parabolic system, the short time existence and uniqueness of solution to (7.6) follow from a standard argument. By Proposition 7.1 and a bootstrapping argument, one can always assume that the shorttime solution u of (7.6) (or (7.5)) is smooth on M × [0, T ) for some T > 0. 51

Lemma 7.2. Let M be a compact pseudo-Hermitian manifold and let N be a Sasakian manifold. For any 0 < T ≤ ∞, if f ∈ C ∞ (M × [0, T ); N ) solves (7.5), then EH,He (ft ) +

Z tZ 0

M

|τH,He (fs )|2 dvθ ds = EH,He (h)

for any t ∈ [0, T ). In particular, the energy EH,He decays along the flow. Proof. By Proposition 3.1, we get dEH,He (fs ) ds

=−

Z

M

h

∂f , τ e (fs )idvθ ∂s H,H

Therefore (7.4) and (7.5) imply that dEH,He (fs ) ds

=−

Z

M

|τH,He (fs )|2 dvθ

Integrating the above equality over [0, t] then proves this lemma. Let f : M × [0, T ) → N be a C ∞ solution of (7.5). In terms of Lemma 2.1, (1.10) and the assumption that N is Sasakian, we have

e e h∇τ e , dfH,H e i = ∇(τH,H e + τH,L e ), dfH,H e H,H =

2m X

e e df ( ∂ ), df e (eA ) ∇ A H,H ∂t

A=1

2m X

e ∂ df (eA ) + Tee df (eA ), df ( ∂ ) , df e (eA ) = ∇ ∇ H,H ∂t ∂t A=1

=

∂ e e. ∂t H,H

Using (2.17), we get from (4.16) that (7.8) ∂ (△H − )eH,He ∂t α α α α = |βH,He |2 − 2i(fjα f0j + fjα f0j − fjα f0j ) + (2m − 4)i(fjα fkα Ajk − fjα fkα Ajk ) − fjα f0j

e + 2 dfH,He (Ric(ηj )), dfH,H e (ηj ) − 2R dfH,H e (ηj ), dfH,H e (ηk ), dfH,H e (ηj ), dfH,H e (ηk ) e df e (ηj ), df e (η ), df e (η ), df e (ηk ) − 2R H,H H,H j H,H H,H k = |βH,He |2 + βL×H,He ∗ dfH,He + A ∗ (dfH,He ) ∗ (dfH,He )

e df e (ηj ), df e (ηk ), df e (η ), df e (η ) + 2 dfH,He (Ric(ηj ), dfH,He (ηj ) − 2R H,H H,H H,H j H,H k e df e (ηj ), df e (η ), df e (η ), df e (ηk ) , − 2R H,H H,H j H,H H,H k

where the notation Φ ∗ Ψ denotes some contraction of two tensors Φ and Ψ. 52

∂ )eL,He . In terms of (2.17), (2.35), (2.41), we deduce Now we want to compute (△H − ∂t that △H eL,He = (f0α f0α )kk + (f0α f0α )kk α α α α α α )k + (f0k f0 + f0α f0k )k = (f0k f0 + f0α f0k

α α α α α α α α f0k ) + f0α f0kk + f0α f0kk + f0α f0kk + f0α f0kk = 2(f0k f0k + f0k α α α α α α α α = 2(f0k f0k + f0k f0k ) + f0α fkk0 + f0α fkk0 + f0α fkk0 + f0α fkk0 γ δ γ δ bα bα (f γ f δ − f γ f δ ) + f α f β R + f0α fkβ R 0 k βγδ (fk f0 − f0 fk ) 0 k βγδ k 0 γ δ γ δ bα (f γ f δ − f γ f δ ) + f α f β R bα + f0α fkβ R 0 k βγδ (fk f0 − f0 fk ) 0 k βγδ k 0

α t α + Akj f0α fkj f0α ftα + Ajk f0α fjk + Ajk,k f0α fjα + Wkk α t α + Ajk,k f0α fjα + Wkk f0α ftα + Akj f0α fjk + Ajk f0α fkj α t α + Ajk f0α fkj + Ajk,k f0α fjα + Wkk f0α ftα + Akj f0α fjk

α t α + Ajk f0α fkj + Wkk f0α ftα + Ajk f0α fjk + Ajk,k f0α fjα .

Consequently (7.9) △H eL,He

e ξ τH , df e (ξ) − 2R e df e (ξ), df e (ηk ), df e (ξ), df e (η ) = |βL×H,He |2 + ∇ L,H L,H H,H L,H H,H k

+ A ∗ (dfL,He ) ∗ βH,He + ∇A ∗ (dfL,He ) ∗ (dfH,He ). Using Lemma 2.1 and (7.4), we get e ξ τH = ∇ e ξ df ( ∂ ) ∇ ∂t

(7.10)

e ∂ df (ξ) + T e df (ξ), df ( ∂ ) . =∇ ∇ ∂t ∂t From (7.9), (7.10), (1.10) and the assumption that N is Sasakian, we obtain (7.11)

(△H −

∂ e df e (ξ), df e (ηk ), df e (ξ), df e (η ) )eL,He = |βL×H,He |2 − 2R L,H H,H L,H H,H k ∂t + A ∗ (dfL,He ) ∗ βH,He + ∇A ∗ (dfL,He ) ∗ (dfH,He ).

Lemma 7.3. If N has non-positive horizontal sectional curvature, then e df e (ξ), df e (ηk ), df e (ξ), df e (η ) ≤ 0. R L,H H,H L,H H,H k

e H ≤ 0, we find Proof. Write dfH,He (ηk ) = Xk + iYk (k = 1, ..., m). Since K e df e (ξ), Xk + iYk , df e (ξ), Xk − iYk R L,H L,H

e df e (ξ), Xk , df e (ξ), Xk ) + R(df e =R e (ξ), Yk , dfL,H e (ξ), Yk L,H L,H L,H ≤ 0.

53

e H ≤ 0 and let f ∈ C ∞ (M ×[0, T ), N ) Lemma 7.4. Let N be a Sasakian manifold with K be a solution of (7.5). Set eHe = eH,He + eL,He . Then (△H −

∂ )e e ≥ −CeHe . ∂t H

Here C is a positive constant depending only on the pseudo-Hermitian Ricci curvature and the torsion of (M, H(M ), J, θ). Proof. Utilizing (7.8), (7.11), Lemma 7.3 and Cauchy-Schwarz inequality, we deduce that ∂ )e e ∂t H ≥ |βH,He |2 + |βL×H,He |2 + βL×H,He ∗ dfH,He + A ∗ (dfH,He ) ∗ (dfH,He )

+ 2 dfH,He (Ric(ηj ), dfH,He (ηj ) + A ∗ (dfL,He ) ∗ βH,He + ∇A ∗ (dfL,He ) ∗ (dfH,He )

(△H −

1 ≥ (1 − ε1 )|βH,He |2 + (1 − 2 1 ≥ (1 − ε1 )|βH,He |2 + (1 − 2

1 ε1 )|βL×H,He |2 − 2 1 ε1 )|βL×H,He |2 − 2

C1 (e e + eL,He ) ε1 H,H C1 ee ε1 H

for any ε1 > 0, where C1 is a positive constant depending only on Ric, A and ∇A. Taking ε1 = 2 in the above inequality, we prove this lemma. In order to estimate eHe , let us recall Moser’s Harnack inequality ([Mo]). For any z0 = (x0 , t0 ) ∈ M × (0, T ), let 0 < δ < inj(M ) (the injectivity radius), 0 < σ < t0 and let R(z0 , δ, σ) be the following cylinder R(z0 , δ, σ) = (x, t) ∈ M × [0, ∞) : d(x, x0 ) < δ, t0 − σ < t < t0 where d denotes the distance function of the Webster metric gθ . Lemma 7.5. Let u be a non-negative smooth solution of ∂ )u ≥ 0 ∂t

(△H − on M . Then u(z0 ) ≤ C(m, δ, σ)

Z

u(x, t)dvθ dt,

R(z0 ,δ,σ)

where C is a positive constant depending only on m, δ and σ. Since M is compact, it follows from Lemma 7.5 that (7.12)

u(z0 ) ≤ C(m, σ)

Z

t0

t0 −σ

Z

u(x, t)dvθ dt

M

for any z0 = (x0 , t0 ) ∈ M × [σ, T ). Henceforth in this section, we assume that M is also Sasakian. 54

Lemma 7.6. Let M be a compact Sasakian manifold and let N be a Sasakian manifold e H ≤ 0. Suppose f ∈ C ∞ (M × [0, T ), N ) is a solution of (7.5). Then the energy with K EL,He (f (t)) is decreasing in t. In particular, if the initial map h is foliated, then f (t) is foliated for each t ∈ [0, T ). Proof. From (7.11), the divergence theorem and Lemma 7.3, we have Z ∂ d eL,He (f ) dvθ EL,He (f (t)) = dt ∂t ZM e df e (ξ), df e (ηk ), df e (ξ), df e (η ) dvθ ≤ − |βL×H,He |2 + 2R L,H H,H L,H H,H k M

≤ 0.

Lemma 7.7. Let M be a compact Sasakian manifold and let N be a Sasakian manifold e H ≤ 0. Suppose f ∈ C ∞ (M × [0, T ), N ) (0 < T ≤ ∞) is a solution of (7.5). with K Then eHe (f ) is uniformly bounded.

Proof. From Lemma 7.4, we know that eHe satisfies

∂ )e e ≥ −CeHe ∂t H

(△H − for some constant C. Let

F (x, t) := e−Ct eHe (f ),

(x, t) ∈ M × [0, T ).

It follows that

∂ )F (x, t) ≥ 0. ∂t Let 0 < σ < T . Then for any z0 = (x0 , t0 ) ∈ M × [σ, T ), (7.12) implies that Z t0 Z Ct0 e−Ct eHe (f )dvθ dt eHe (f )(z0 ) ≤ C1 e (△H −

≤ C1 eCσ ≤ C2

Z

Z

t0

t0 −σ M t0 Z

t0 −σ

t0 −σ

M

eHe (f )dvθ dt

EHe (f (t))dt

≤ C2 EHe (h) since EH,He (f (t)) and EL,H e (f (t)) are decreasing in t in view of Lemmas 7.2, 7.6. Next we want to derive Bochner formulas for eH,Le (ft ) and eL,Le (ft ). According to (7.1) and the definition of △H , one has (7.13)

△H eH,Le = (fj0 fj0 )kk + (fj0 fj0 )kk

0 0 0 0 0 0 0 0 = 2(fjk fjk + fjk fjk ) + fj0 fjkk + fj0 fjkk + fj0 fjkk + fj0 fjkk 55

and

1 2 (f00 )2kk + (f00 )kk 2 0 0 0 0 + f0kk ) = 2f0k f0k + f00 (f0kk

△H eL,Le =

(7.14)

Using (2.14) and (2.24), we deduce from (7.16) and (7.17) that 0 0 0 0 0 0 0 0 fjk ) + fj0 fkkj + fj0 fkkj + fj0 fkkj + fj0 fkkj △H eH,Le = 2(fjk fjk + fjk 0 0 0 0 t t + fj0 ft0 Rkjk + fj0 fj0 + fj0 fj0 + ifj0 f0j − ifj0 f0j + fj0 ft0 Rkjk + ifj0 fjα fkα − fjα fkα k − ifj0 fkα fjα − fkα fjα k + ifj0 fkα fjα − fkα fjα k − ifj0 fjα fkα − fjα fkα k

(7.15)

e = |βH×H,Le |2 + h∇τ e , dfH,L e i + βH×L,L e ∗ dfH,L e + βL×H,L e ∗ dfH,L e H,L

+ dfH,Le (RicH (ηj ), dfH,Le (ηj ) + dfH,Le (RicH (ηj ), dfH,Le (ηj )

+ βH,He ∗ (dfH,He ) ∗ (dfH,Le )

and (7.16) 0 0 0 0 △H eL,Le = 2f0k f0k + f00 fkk0 + f00 fkk0 + if00 (f0α fkα − f0α fkα )k − (f0α fkα − f0α fkα )k e ξ τ e , df e (ξ)i + β = |βL×H,Le |2 + h∇ e ∗ dfH,H e ∗ dfL,L e H,L L,L L×H,H

+ βH,He ∗ dfL,Le ∗ dfL,He .

In terms of (1.7), (1.9), (1.10), Lemma 2.1 and (7.5), we have

2m X

∂ e e e Jdf h∇τH,Le , dfH,Le i = eH,Le + e (eA ), τH,H e (f ) θ dfH,L e (eA ) H,H ∂t A=1

and

e ξ τ e , df e (ξ)i = h∇ H,L L,L

In addition, (2.14) implies Consequently

(7.17)

βH×L,Le ∗ dfH,Le = βL×H,Le ∗ dfH,Le + dfH,He ∗ dfL,He ∗ dfH,Le .

(△H −

and (7.18)

∂ e df e (ξ) . e eL,Le + Jdf (ξ), τ (f ) θ e e L,H H,H L,L ∂t

(△H −

∂ )e e = |βH×H,Le |2 + dfH,He ∗ τH,He ∗ dfH,Le + βL×H,Le ∗ dfH,Le ∂t H,L + dfH,He ∗ dfL,He ∗ dfH,Le + βH,He ∗ (dfH,He ) ∗ (dfH,Le )

+ dfH,Le (RicH (ηj ), dfH,Le (ηj ) + dfH,Le (RicH (ηj ), dfH,Le (ηj ) ∂ )eL,Le = |βL×H,Le |2 + dfL,He ∗ τH,He ∗ dfL,Le + βL×H,He ∗ dfH,He ∗ dfL,Le ∂t + βH,He ∗ dfL,Le ∗ dfL,He .

Clearly the usual energy density of the map f is given by

e(f ) = eH,He (f ) + eL,He (f ) + eH,Le (f ) + eL,Le (f ) and thus E(f ) = EH,He (f ) + EL,He (f ) + EH,Le (f ) + EL,Le (f ). 56

Lemma 7.8. Let M be a compact Sasakian manifold and N be a Sasakian manifold e H ≤ 0. Suppose f ∈ C ∞ (M × [0, T ), N ) is a solution of (7.5). Then with K e(ft ) ≤ C(σ) sup E(ft ) [t−σ,t]

for t ∈ [σ, T ). Proof. From the proof of Lemma 7.4, we have

(7.19)

(△H −

∂ 1 1 ) eH,He (ft ) + eL,He (ft ) ≥ (1 − ε1 )|βH,He |2 + (1 − ε1 )|βL×H,He |2 ∂t 2 2 C1 e e (ft ) + eL,He (ft ) − ε1 H,H

Utilizing Lemma 7.7 and Cauchy-Schwarz inequality, we derive from (7.17) and (7.18) that ∂ ) eH,Le (ft ) + eL,Le (ft ) ∂t = |βH×H,Le |2 + dfH,He ∗ τH,He ∗ dfH,Le + βL×H,Le ∗ dfH,Le + dfH,He ∗ dfL,He ∗ dfH,Le

+ |βL×H,Le |2 + dfH,Le (RicH (ηj ), dfH,Le (ηj ) + dfH,Le (RicH (ηj ), dfH,Le (ηj )

(△H −

(7.20)

+ βH,He ∗ (dfH,He ) ∗ (dfH,Le ) + dfL,He ∗ τH,He ∗ dfL,Le + βL×H,He ∗ dfH,He ∗ dfL,Le

+ βH,He ∗ dfL,Le ∗ dfL,He

1 1 1 ≥ − ε2 |βH,He |2 + (1 − ε2 )|βL×H,Le |2 − ε2 eH,He (ft ) + eL,He (ft ) 2 2 2 1 C 2 − ε2 |βL×H,He |2 − eH,Le (ft ) + eL,Le (ft ) . 2 ε2

Taking ε1 = ε2 = 1, it follows from (7.19) and (7.20) that (7.21)

(△H −

∂ e t) )e(ft ) ≥ −Ce(f ∂t

e depending only on M, N and h. Therefore this lemma follows for some positive C immediately from (7.21) and Lemma 7.5. Now we want to estimate the partial energies EH,Le (ft ) and EL,Le (ft ).

Lemma 7.9. Let M , N and f ∈ C ∞ (M × [0, T ), N ) be as in Lemma 7.8. Then Z Z

d 2 EH,Le (ft ) = − |τH,L (ft )| dvθ + JτH,He (ft ), dfH,He (eA ) θe dfH,Le (eA ) dvθ . ds M M

Furthermore, we have

EH,Le (ft ) ≤ C2 t + C3 for any t ∈ [0, T ) and some constants C2 , C3 depending on M , N and h. 57

Proof. By definition, EH,Le (ft ) is given by 1 EH,Le (ft ) = 2

Z

2m X

dfH,Le (eA ), dfH,Le (eA ) dvθ

M A=1

where {eA }2m A=1 is an orthonormal frame field of H(M ). Then, by using Lemma 2.1, (1.9), (1.10) and the divergence theorem, we derive that (7.22) d E e (ft ) ds H,L Z X 2m ∂ 1 hdf e (eA ), dfH,Le (eA )idvθ = 2 M ∂t H,L A=1 Z X 2m e ∂ df (eA ), df e (eA )idvθ = h∇ H,L =

M A=1 Z X 2m M A=1

=

Z

∂t

e e df ( ∂ ), df e (eA )i + Tee df ( ∂ ), df (eA ) , df e (eA ) dvθ h∇ A H,L ∇ H,L ∂t ∂t

2m X

e e e e τH (ft ), df e (eA )i + dθ(τ h∇ dvθ e (ft ), dfH,H e (eA ))ξ, dfH,L e (eA ) A H,L H,H

M A=1

=−

Z

M

2

|τH,L (ft )| dvθ +

Z

M

e JτH,He (ft ), dfH,He (eA ) θ(df e (eA ))dvθ . H,L

In terms of Lemma 7.7, (7.25) and Hölder’s inequality, we have Z d E e (ft ) ≤ C |τH,He (ft )||dfH,Le |dvθ dt H,L M Z Z √ 1/2 1/2 2 ≤ 2C |τH,He (ft )| dvθ eH,Le (ft )dvθ M

M

which implies

Z

0

Consequently (7.23)

t

Z t Z dEH,Le (fs ) 1/2 C q ≤√ |τH,He (fs )|2 dvθ ds. 2 0 M 2 EH,Le (fs )

Z tZ q q 1/2 √ C EH,Le (ft ) − EH,Le (h) ≤ √ t. |τH,He (fs )|2 dvθ 2 0 M

Thus Lemma 7.2 and (7.23) imply q q √ C q √ EH,He (h) t + EH,Le (h). EH,Le (ft ) ≤ 2

58

Lemma 7.10. Let M , N and f ∈ C ∞ (M × [0, T ), N ) be as in Lemma 7.7. Suppose h is foliated. Then eL,Le (ft ) is uniformly bounded, and EL,Le (ft ) is decreasing in t. Proof. Since h is foliated, we see from Lemma 7.6 that ft is foliated for each t ∈ [0, T ). As a result, (7.18) becomes (△H −

(7.24)

∂ )e e (ft ) = |βL×H,Le |2 ≥ 0. ∂t L,L

By Maximum principle, we have eL,Le (ft ) ≤ supM eL,Le (h). In terms of the divergence theorem, (7.24) implies that d dt

Z

M

eL,Le (ft )dvθ =

Z

△H eL,Le (ft ) − |βL×H,Le |2 dvθ ≤ 0.

M

Hence EL,Le (ft ) is decreasing in t. From Lemmas 7.2, 7.6, 7.8, 7.9 and 7.10, we immediately get the following global existence of (7.5). Proposition 7.11. Let M and N be compact Sasakian manifolds. Suppose N has non-positive horizontal curvature and the initial map h : M → N is foliated. Then the solution f of (7.5) exits for all t ≥ 0.

e Now we are able to establish some existence results for (H, H)-harmonic maps when the target manifold N is a compact regular Sasakian manifold, that is, N can be realized as a Riemannian submersion π : (N, gθe) → (B, gB ) over a compact Kähler manifold. Let i(B) be the injectivity radius of B. We denote by Br (y) the geodesic ball centered at y with radius r in B. Hence, if r ≤ i(B), then any two points in Br (y) can be joined by a unique geodesic in Br (y).

Theorem 7.12. Let M be a compact Sasakian manifold and let N be a compact Sasakian manifold with non-positive horizontal sectional curvature. Suppose π : N → B is a Riemannian submersion over a K¨ ahler manifold B and h : M → N is a given e foliated map. Then there exists a smooth foliated (H, H)-harmonic map in the same homotopy class as h. Proof. From Proposition 7.11, we know that there is a global solution f : M × [0, ∞) → N of (7.5) with the initial map h. Set ϕt = π◦ft and ψ = π◦h. Since ft is foliated for each t ∈ [0, ∞) in view of Lemma 7.6, it follows from Proposition 3.8 that ϕ : M ×[0, ∞) → N satisfies the following harmonic heat flow (7.25)

∂ϕ ∂t

= τ gB (ϕt )

ϕ|t=0 = ψ.

Observe from (1.25) that B has non-positive sectional curvature duo to the non-positive horizontal curvature condition on N . By Eells-Sampson theorem, the solution ϕ of (7.25) converges in C ∞ (M, B) to a harmonic map ϕ∞ as t → ∞. Therefore there is a sufficiently large T > 0 such that if t ≥ T , then ϕt (x) ∈ Bi(B) (ϕ∞ (x)). In particular, 59

ϕT (x) ∈ Bi(B) (ϕ∞ (x))) for each x ∈ M . Set v(x) = exp−1 ϕT (x) (ϕ∞ (x)). Clearly v ∈ Γ(ϕ−1 (T B)). We may define a horizontal vector field ve along fT such that ve(x) ∈ HfT (x) (N ) and dπ() = v(x) for x ∈ M . Set fb(x) = exp⊥ v (x)), fT (x) (e

x ∈ M,

where exp⊥ denotes the normal exponential map along the Reeb leaf π −1 (ϕT (x)). Note that ϕT (x) and ϕ∞ (x) can be joined by a unique geodesic γx lying in Bi(B) (ϕ∞ (x)). Actually fb(x) is the image of fT (x) under the honolomy diffeomorphism hγx : π −1 ( ϕT (x)) → π −1 (ϕ∞ (x)) associated to the geodesic γx from ϕT (x) to ϕ∞ (x). Thus fb is a foliated map. Obviously the map fb : M → N lies in the same homotopy class as h and satisfies π ◦ fb = ϕ∞ . e Thus Proposition 3.8 implies that fb is a (H, H)-harmonic map.

Remark 7.2. Let V be any vertical vector field on N and let ζs denote the one parameter e transformation group generated by V . Then ζs ◦ fb is also a (H, H)-harmonic map in the e same homotopy class as h. Hence we do not have the uniqueness for (H, H)-harmonic maps in a fixed homotopy class in general.

Lemma 7.13. Let M, N and B be as in Theorem 7.12. Let f : M × [0, ∞) → N be a e solution of (7.5) with initial map h. Suppose h : M → N is a foliated (H, H)-harmonic e map. Then ft : M → N is a foliated (H, H)-harmonic map for each t ∈ [0, ∞). Proof. Set ϕt = π ◦ ft and ψ = π ◦ h, where π : N → B is the Riemannian submersion. From the proof of Theorem 7.12, we know that ϕt satisfies the harmonic map heat e flow (7.25). Since h is assumed to be a foliated (H, H)-harmonic map, Proposition 3.8 implies that ψ : M → B is a harmonic map, which may be also regarded as a solution of (7.25) independent of the time t. By the uniqueness for solutions of (7.25), we find that ϕt is harmonic for each t. It follows from Proposition 3.8 again that ft : M → N e is (H, H)-harmonic.

Theorem 7.14. Let M and N be compact Sasakian manifolds and let h : M → N be a foliated map. Suppose N is regular with non-positive horizontal sectional curvature. e Then there exists a foliated special (H, H)-harmonic map in the same foliated homotopy class as h. e Proof. Without loss of generality, we may assume that h is a foliated (H, H)-harmonic map in view of Theorem 7.12. Suppose f : M × [0, ∞) is a solution of (7.25) with e the initial map h. By Lemma 7.13 , each ft is a foliated (H, H)-harmonic map. Then Lemma 7.9 gives Z d EH,Le (ft ) = − |τH,Le (ft )|2 dvθ . dt M

This yields that EH,Le (ft ) is decreasing in t and Z ∞Z (7.26) |τH,Le (ft )|2 dvθ dt < ∞. 0

M

60

Consequently Lemmas 7.2, 7.6, 7.8, 7.10 imply that e(ft ) is uniformly bounded. Applying Proposition 7.1 to the solution of (7.6), we find that all higher derivatives of f are uniformly bounded. On the other hand, (7.26) implies that there exists a sequence {tk } such that |τH,Le (ftk )|L2 (M ) → 0

(7.27)

as tk → ∞.

In terms of the Arzela-Ascoli theorem, by passing a subsequence {tkl } of {tk }, we conclude that f (·, tkl ) converges in C ∞ (M, N ) to a limit f∞ (as tkl → ∞), which satisfies both τH,He (f∞ ) = 0 and τH,Le (f∞ ) = 0. Clearly f∞ lies in the same foliated homotopy class as h. 8. Foliated rigidity and Siu-type strong rigidity results First, we introduce the following e e ), J, e θ) Definition 8.1. We say that a map f : (M 2m+1 , H(M ), J, θ) → (N 2n+1 , H(N has split horizontal second fundamental form if πHe (β(T, X)) = 0 for any X ∈ H(M ), α α that is, f0k = f0k = 0 for k = 1, ..., m. e Lemma 8.1. Let f : M 2m+1 → N 2n+1 be a (H, H)-harmonic map with split horizontal second fundamental form. Suppose that M is compact and N is Sasakian. Then f is foliated. Proof. By (2.17), we have α α = fkk + mif0α fkk

(8.1)

It follows from Corollary 3.3 and (8.1) that (8.2)

α fkk =

mi α f , 2 0

α fkk =−

mi α f . 2 0

Let us define a global 1-form on M by ρe = −i(f0α fkα θ k − f0α fkα θ k ). Using Lemma 1.3, α α (8.2) and the assumption that f0k = f0k = 0, we find that Z 0= δe ρ M Z α α =i (f0 fk )k − (f0α fkα )k dvθ ZM α α α α α α − f0k dvθ fk − f0α fkk =i f0k fk + f0α fkk M Z mi α α mi α α =i f f + f f dvθ 2 0 0 2 0 0 M Z X = −m |f0α |2 dvθ , M

α

which implies f0α = 0, that is, f is foliated. 61

e be compact Sasakian e ), J, e θ) Theorem 8.2. Let (M 2m+1 , H(M ), J, θ) and (N 2n+1 , H(N e e has nonmanifolds and let f : M → N be a (H, H)-harmonic map. If (N 2n+1 , ∇) positive horizontal sectional curvature, then f is foliated. Proof. First we define a global 1-form ρe1 by

α α k α α α α k α α f0 + f0k f0 )θ }. ρe1 = −{(f0k f0 + f0k f0 )θ + (f0k

(8.3)

Using (1.20), (1.22), (2.17), (2.35) and (2.41), we deduce that α α α α α α α α δ ρe1 =(f0k f0 + f0k f0 )k + (f0k f0 + f0k f0 )k

α α α α α 2 α 2 | + f0α (f0kk + f0kk ) + f0α (f0kk + f0kk ) =2|f0k | + 2|f0k α α 2 α 2 α α α | + f0α (fk0k + fk0k ) + f0α (fk0k + fk0k ) =2|f0k | + 2|f0k

α 2 α 2 α α α α =2|f0k | + 2|f0k | + f0α (fkk0 + fkk0 ) + f0α (fkk0 + fkk0 )

γ δ γ δ bα (f γ f δ − f γ f δ ) + f α f β R bα + f0α fkβ R 0 k βγδ (fk f0 − f0 fk ) 0 k βγδ k 0

(8.4)

γ δ γ δ bα (f γ f δ − f γ f δ ) + f α f β R bα + f0α fkβ R 0 k βγδ (fk f0 − f0 fk ) 0 k βγδ k 0

α α 2 α 2 α α α | + f0α (fkk0 + fkk0 ) + f0α (fkk0 + fkk0 ) =2|f0k | + 2|f0k

e − Q(df e (ξ) ∧ dfH,H e (ηk ), dfL,H e (ξ) ∧ dfH,H e (ηk )) L,H

e − Q(df e (ξ) ∧ dfH,H e (ηk ), dfL,H e (ξ) ∧ dfH,H e (ηk )) L,H

where dfL,He (ξ) = πHe (df (ξ)). It follows from Corollary 3.3 and (8.4) that (8.5)

α 2 α 2 e | − Q(df δ ρe1 =2|f0k | + 2|f0k e (ξ) ∧ dfH,H e (ηk ), dfL,H e (ξ) ∧ dfH,H e (ηk )) L,H

e − Q(df e (ξ) ∧ dfH,H e (ηk )) e (ξ) ∧ dfH,H e (ηk ), dfL,H L,H

Note that

e Q(df e (ξ) ∧ dfH,H e (ηk )) e (ξ) ∧ dfH,H e (ηk ), dfL,H L,H

e +Q(df e (ξ) ∧ dfH,H e (ηk ), dfL,H e (ξ) ∧ dfH,H e (ηk )) L,H

(8.6)

1e = Q(df e (ξ) ∧ dfH,H e (ek − iJek ), dfL,H e (ξ) ∧ dfH,H e (ek + iJek )) L,H 2 1e + Q(df e (ξ) ∧ dfH,H e (ek + iJek ), dfL,H e (ξ) ∧ dfH,H e (ek − iJek )) L,H 2 e =Q(df e (ξ) ∧ dfH,H e (ek ), dfL,H e (ξ) ∧ dfH,H e (ek )) L,H e +Q(df e (ξ) ∧ dfH,H e (Jek ), dfL,H e (ξ) ∧ dfH,H e (Jek )) L,H

≤0

in view of the curvature condition on N . Consequently α 2 α 2 δ ρe1 ≥ 2|f0k | + 2|f0k | 62

α α which gives that f0k = f0k = 0 by the divergence theorem. In terms of Lemma 8.1, we see that f is foliated.

e In terms of Remark 3.2 and Theorem 8.2, we find that (H, H)-harmonic maps seem to be more sensitive to the foliated structures than to the horizontal distributions, at least when the target manifolds are Sasakian manifolds with non-positive horizontal sectional curvature. However, we will see that under some further conditions on the maps and the target manifolds, these critical maps may be related to the CR structures in a horizontally projective way. e be compact Sasakian e ), J, e θ) Theorem 8.3. Let (M 2m+1 , H(M ), J, θ) and (N 2n+1 , H(N e e has strongly manifolds and f : M → N be a (H, H)-harmonic map. If (N 2n+1 , ∇) e seminegative horizontal curvature, then f is (H, H)-pluriharmonic and (8.7)

e Q(df e (ηj ) ∧ dfH,H e (ηk ), dfH,H e (ηj ) ∧ dfH,H e (ηk )) = 0 H,H

for any unitary frame {ηj } of H 1,0 (M ). Proof. Let us define a global 1-form by (8.8)

α α k α α k ρe2 = −(fjk fj θ + fjk fj θ ).

According to (1.20), (1.22), (2.17) and (2.38), we compute that α α α α fj )k fj )k + (fjk δ ρe2 =(fjk

α α α α α α fjk + fjk fjk + fjkk fjα + fjkk fjα =fjk α 2 α α =2|fjk | + (fkj − if0α δkj )k fjα + (fkj + if0α δkj )k fjα

α 2 α α j α j α =2|fjk δk )fjα − if0k δk )fjα + (fkjk + if0k | + (fkjk

(8.9)

α α α α α α α 2 | + i(f0j fj ) + fkjk fj − f0j fjα + fkjk fjα =2|fjk α 2 α α α α α α =2|fjk | + i(f0j fj ) + fkkj fj − f0j fjα + fkkj fjα

bα (f γ f δ − f γ f δ ) + f α f β R bα (f γ f δ − f γ fkδ ) + fjα fkβ R j j k βγδ k j βγδ k j j k

α α α α α α α 2 | + i(f0j fj ) + fkkj fj − f0j fjα + fkkj fjα =2|fjk

e − Q(df e (ηj ) ∧ dfH,H e (ηk ), dfH,H e (ηj ) ∧ dfH,H e (ηk )). H,H

Note that a Sasakian manifold with strongly semi-negative horizontal curvature has automatically non-positive horizontal sectional curvature. Then Theorem 8.2 yields that f0α = f0α = 0, and thus (8.10)

α α = 0. f0j = f0j

From the fourth equation of (2.17), we derive that α α α 2fkk = fkk + fkk + mif0α

(8.11)

= mif0α = 0. 63

It follows from (8.9), (8.10) and 8.11) that (8.12)

α 2 e δ ρe2 = 2|fjk | − Q(df e (ηj ) ∧ dfH,H e (ηk ), dfH,H e (ηj ) ∧ dfH,H e (ηk )). H,H

e and the divergence theorem, we get from In terms of the curvature condition of (N, ∇) (8.12) that α =0 fjk and e Q(df e (ηj ) ∧ dfH,H e (ηk ), dfH,H e (ηj ) ∧ dfH,H e (ηk )) = 0 H,H

α α for any 1 ≤ j, k ≤ m. Since f0α = 0, (2.17) implies that fkj = fjk = 0. According to e (5.2) and (5.3), f is (H, H)-pluriharmonic. Hence we complete the proof.

e are e ), J, e θ) Theorem 8.4. Suppose (M 2m+1 , H(M ), J, θ)(m ≥ 2) and (N 2n+1 , H(N compact Sasakian manifolds and N has strongly negative horizontal curvature. Supe pose f : M → N is a (H, H)-harmonic map with maxM rankR {dfH,He } ≥ 3. Then f is e e either a foliated (H, H)-holomorphic map or a foliated (H, H)-antiholomorphic map.

Proof. From Theorem 8.2, we know that f is foliated. The rank condition for f means that there exists a point p ∈ M such that rankR {dfH,He (p)} ≥ 3. Consequently rankR {dfH,He } ≥ 3 in some connected open neighborhood U of p. Write dfH,He (ηj ) = fjα ηeα + fjα ηeα . Then (8.13)

dfH,He (ηj ) ∧ dfH,He (ηk )

(1,1)

ηα ∧ ηeβ . = (fjα fkβ − fkα fjβ )e

Since N has strongly negative horizontal curvature, it follows from (8.7) and (8.13) that (8.14)

fjα fkβ − fkα fjβ = 0

for 1 ≤ j, k ≤ m and 1 ≤ α, β ≤ n. We want to show that for every point q ∈ U , either ∂fH,He (q) = 0 or ∂fH,He (q) = 0. Without loss of generality, we assume that ∂fH,He (q) 6= 0. This means that fkγ (q) 6= 0 for some 1 ≤ k ≤ m and some 1 ≤ γ ≤ n. Therefore (8.14) yields that (8.15)

γ α {f1α (q), · · · , fm (q)} (q)} = cα {f1γ (q), · · · , fm

for each 1 ≤ α ≤ m, where cα = fkα (q)/fkγ (q). If ∂fH,He (q) 6= 0 too, then fjδ (q) 6= 0 for some 1 ≤ j ≤ m and some 1 ≤ δ ≤ n. Using (8.14) again, we get (8.16)

β δ (q)} (q)} = dβ {f1δ (q), · · · , fm {f1β (q), · · · , fm 64

for each 1 ≤ β ≤ m, where dβ = fjβ (q)/fjδ (q). In terms of (8.15) and (8.16), we find dfH,He (ηi ) = fiα ηα + fiα ηα = fiγ (cα ηα ) + fiδ (dα ηα )

(8.17)

= fiγ (cα ηα ) + cδ fiγ (dα ηα ) = fiγ (cα ηα + cδ dα ηα )

at q ∈ U . From (8.17), we have spanC {dfH,He (ηi )}q = spanC {cα ηα + cδ dα ηα } ≤1

which implies that rankR {dfH,He (q)} ≤ 2, contradicting q ∈ U . Thus, for each q ∈ U , either ∂fH,He (q) = 0 or ∂fH,He (q) = 0. Since rankR {dfH,He > 0} in U , the two sets U ∩ {∂fH,He = 0} and U ∩ {∂fH,He = 0} are disjoint closed subsets of U , and their union is the connected set U . It follows that either ∂fH,He ≡ 0 on U or ∂fH,He ≡ 0 on U . From Theorem 5.4 and Theorem 8.2, we conclude that either ∂fH,He ≡ 0 on M or ∂fH,He ≡ 0 on M . e are e ), J, e θ) Theorem 8.5. Let k ≥ 2. Suppose (M 2m+1 , H(M ), J, θ) and (N 2n+1 , H(N compact Sasakian manifolds and the horizontal curvature of N is negative of order k. e Suppose f : M → N is a (H, H)-harmonic map and maxM {dfH,He } ≥ 2k. Then f is e e either a foliated (H, H)-holomorphic map or a foliated (H, H)-antiholomorphic map.

Proof. By a similar argument as that in Theorem 8.4, we may deduce the conclusion of this theorem from (8.7), (8.13) and Definition 8.2. For a manifold M , we use Hl (M, R) to denote its usual singular homology.

e is a folie ), J, e θ) Corollary 8.6. Suppose f : (M 2m+1 , H(M ), J, θ) → (N 2n+1 , H(N e ated (H, H)-harmonic map between compact Sasakian manifolds. Suppose the horizontal curvature of N is negative of order k and k ≥ 2. If the induced map f∗ : Hl (M, R) → e e Hl (N, R) is nonzero for some l ≥ 2k +1, then f is either (H, H)-holomorphic or (H, H)anti-holomorphic. Proof. The assumption that f∗ : Hl (M, R) → Hl (N, R) is nonzero for some l ≥ 2k + 1 implies that rankR {df } ≥ 2k + 1 at some point p of M . Since f is foliated, rankR {πLe ◦ df } + rankR {dfH,He } ≥ rankR {df } at each point of M . Hence rankR {dfH,He }p ≥ 2k. By Theorem 8.5, we find that f is e e either a foliated (H, H)-holomorphic or a foliated (H, H)-anti-holomorphic map.

Before investigating the strong rigidity of Sasakian manifolds with some kind of negative horizontal curvature, let us recall some basic notions and results for foliations, 65

especially for Riemannian foliations. Given a foliation F on a manifold M , a differential form ω ∈ Ω∗ (M ) is called basic if for every vector field X tangent to the leaves, iX ω = 0 and iX dω = 0, where iX denotes the interior product with respect to X. In particular, a basic form of degree 0 is just a basic function. Clearly the exterior derivative of a basic form is again basic, so all basic forms (Ω∗B (M ), dB ) constitutes a subcomplex of the de Rham complex (Ω∗ (M ), d), where dB = d|Ω∗B (M ) . The cohomol∗ ogy HB (M/F ) = kerdB /ImdB of this subcomplex is called the basic cohomology of (M, F ). In general, the basic cohomology groups are not always finite dimensional. It is known that Riemannian foliations on compact manifolds form a large class of foliations for which the basic cohomology groups are finite-dimensional (cf. [KHS], [KT], [PR]). Let (M, F ) and (N, Fe) be two manifolds endowed with complete Riemannian foliations F and Fe respectively. Similar to Definition 3.3, we have the notion of continuous foliated map between (M, F ) and (N, Fe ), that is, a continuous map from M to N mapping leaves of F into leaves of Fe. A homotopy between M and N consisting of continuous (resp. smooth) foliated maps is called a continuous (resp. smooth) foliated homotopy, and the corresponding homotopy equivalence can be defined in a natural way. Actually we may work in smooth category due to the following result. Lemma 8.7. ([LM]) Any continuous foliated map between complete Riemannian foliations is foliatedly homotopic to a C ∞ foliated map. In view of Lemma 8.7, two complete Riemannian foliated manifolds (M, F ) and (N, Fe) have the same foliated homotopy type in the C ∞ sense if and only if they have the same foliated homotopy type in the usual continuous sense. Another important property of basic cohomologies of Riemannian foliations is the homotopy invariance. Lemma 8.8. (cf. [Vi]) Let (M, F ) and (N, Fe) be two compact Riemannian foliated manifolds. Suppose f, g : (M, F ) → (N, Fe) are foliated homotopic. Then f ∗ = g ∗ : ∗ ∗ HB (N/Fe) → HB (M/F ). In particular, if f : (M, F ) → (N, Fe ) is a foliated homotopy ∗ ∗ equivalence, then f ∗ : HB (N/Fe ) → HB (M/F ) is an isomorphism.

As mentioned in §1, the Reeb foliation of a Sasakian manifold is a Riemannian foliation of dimension 1. Suppose (M 2m+1 , H(M ), J, θ) is a compact Sasakian manifold with k the Reeb foliation Fξ . Then each basic cohomology group HB (M/Fξ ) (k = 0, 1, ..., 2m) is finite dimensional. Using the second property in (1.4), it is easy to verify that dθ is a basic form, and thus so is (dθ)k for 2 ≤ k ≤ m.

Lemma 8.9. Suppose (M 2m+1 , H(M ), J, θ) is a compact Sasakian manifold. Then 2k 0 6= [(dθ)k ]B ∈ HB (M/F ) (1 ≤ k ≤ m). Proof. We prove this lemma by contradiction. Suppose (dθ)k = dα for some α ∈ (M ). Then Ω2k−1 B (8.18)

(dθ)m = dα ∧ (dθ)m−k = d α ∧ (dθ)m−k .

Using (8.18) and the Stokes formula, the volume of M is given, up to a positive constant, 66

by

Z

M

θ ∧ (dθ)

m

= =

Z

ZM

M

θ ∧ d α ∧ (dθ)m−k dθ ∧ α ∧ (dθ)m−k

= 0,

where the last equality follows from the fact that dθ ∧ α ∧ (dθ)m−k = 0, since it is a basic form of degree 2m + 1. Hence we get a contradiction. Remark 8.1. The above result is actually true for a general compact pseudo-Hermitian manifold. Recall that a smooth complex-valued function f : M → C on a CR manifold M is called a CR function if Z(f ) = 0 for any Z ∈ H 0,1 (M ). Definition 8.2. Let M 2m+1 be a Sasakian manifold. We say that a subset V of M is a foliated analytic subvariety if, for any point p ∈ V , there exists a foliated coordinate chart (U, Φ; ϕ) of p such that V ∩U is the common zero locus of a finite collection of basic CR functions f1 , ..., fk on U . In particular, V is called a foliated analytic hypersurface if V is locally the zero locus of a single nonzero basic CR function f . More explicitly, let ϕ : U → W ⊂ C m be the submersion associated with the foliated coordinate chart (U, Φ; ϕ), that is, ϕ = π ◦ Φ (see (1.23) in §1). Since the CR functions f1 , .., fk are constant along the leaves, there are holomorphic function fe1 , ..., fek on W such that fi = fei ◦ π (i = 1, ..., k). Set Veϕ = ϕ(V ∩ U ). Thus Veϕ is a complex analytic subvariety in W defined by the common zero locus of fe1 , .., fek . A point p ∈ V is called a smooth point of V if V is a submanifold of M near p. The locus of smooth points of V is denoted by V ∗ . A point p ∈ V − V ∗ is called a singular point of V ; the singular locus V − V ∗ of V is denoted by V s . Similarly we have the notions of smooth points and singular points for Veϕ . Let Veϕ∗ (resp. Veϕs ) denote the locus of smooth points (resp. singular points) of Veϕ . Clearly Veϕ∗ = ϕ(V ∗ ∩ U ) and Veϕs = ϕ(V s ∩U ) of Ve . By the proposition on page 32 in [GH], we know that Veϕ∗ has finite volume in bounded regions. Consequently V ∗ has finite volume in bounded regions too. Therefore we may define the integral of a differential form ω on M over V to be the integral of ω over the smooth locus V ∗ of V . We need the following Stokes’ formula for foliated analytic subvarieties, which is a generalization of the usual Stokes’ formula for analytic subvarieties. Proposition 8.10. Let M be a Sasakian manifold and let V ⊂ M be a foliated analytic subvarieties of real dimension 2k + 1. Suppose α is a differential form of degree 2k with compact support in M . Then Z dα = 0.

V

Proof. The question is local, it will be sufficient to show that for every point p ∈ V , there exists a neighborhood U of p such that for every α ∈ A2k c (U ) (the space of differential 67

forms of degree 2k with compact support in U ) Z dα = 0. V

Suppose (U, Φ; ϕ) is a bounded foliated coordinate chart around p. Let ϕ : U → W be the submersion associated with (U, Φ; ϕ) and let pe = ϕ(p) ∈ Veϕ ⊂ W . By the local structure of an analytic subvariety in C m (cf. [GH]), we may find a coordinate system z = (z1 , ..., zm ) and a polycylinder △ around pe such that the projection π e : ′ e (z1 , ..., zm) → (z1 , ..., zk ) expresses Vϕ ∩ △ as a branched cover of △ = π e(△), branched ′ ε over an analytic hypersurface Σ ⊂ △ . Let T be the ε-neighborhood of Σ in △′ and Veϕε = (Veϕ ∩ △) − π e−1 (T ε ) ⊂ Veϕ∗ .

Set V ε = ϕ−1 (Veϕε ). Clearly ϕ−1 (△) ⊂ U is a foliated neighborhood of p. For α ∈ −1 A2k (△)), we have c (ϕ Z Z dα =

V

dα

V ∩ϕ−1 (△)

= lim

ε→0

= lim

ε→0

= lim

ε→0

Z

dα

Vε

Z

α

∂V ε

Z

α.

ϕ−1 (∂e π −1 (T ε ))

Thus to prove the result, we simply have to prove that vol(ϕ−1 (∂e π −1 (T ε ))) → 0 as −1 ε ε → 0 or equivalently vol(∂e π (T )) → 0 as ε → 0. However, the latter one has already be shown on page 33 in [GH] . e We will use special (H, H)-harmonic maps to establish strong rigidity results for Sasakian manifolds. e is a folie ), J, e θ) Lemma 8.11. Suppose f : (M 2m+1 , H(M ), J, θ) → (N 2n+1 , H(N e ated special (H, H)-harmonic map between Sasakian manifolds. If M is compact, then e df (ξ) = λξ with λ constant.

Proof. Since f is foliated, df (ξ) = λξe for some smooth function λ on M, that is, f00 = λ and f0α = f0α = 0. Then (2.14) yields that (8.19)

0 0 = fj0 . f0j

0 0 f0j = fj0 ,

In terms of (2.24) and (8.19), we get (8.20)

0 0 0 f0jj = fj0j = fjj0 ,

0 0 0 f0jj = fj0j = fjj0 .

Since f is special, we derive from (8.19) and (8.20) that 0 0 0 0 + f0jj = fjj0 + fjj0 △b λ =f0jj

=0.

Due to the compactness of M , it follows that λ is constant. 68

e be a foliated special (H, H)e ), J, e θ) e Theorem 8.12. Let f : (M, H(M ), J, θ) → (N, H(N harmonic map of Sasakian manifolds, both of CR dimension m ≥ 2. Suppose the horizontal curvature of N is either strongly negative or adequately negative. Suppose f 2m−2 2m−2 is of degree 1 and the map f ∗ : HB (N ) → HB (M ) induced by f is surjective. e e Then f : M → N is either (H, H)-biholomorphic or (H, H)-anti-biholomorphic.

Proof. Let V be the set of points of M where f is not locally diffeomorphic. Suppose V is nonempty. We will derive a contradiction in the following steps. Step 1. Since deg(f ) = 1, V 6= M and f maps M − f −1 (f (V )) bijectively onto e N − f (V ). By either Theorem 8.4 or Theorem 8.5, f is either (H, H)-holomorphic or e (H, H)-antiholomorphic. Without loss of generality, we assume now that f is a foliated e special (H, H)-holomorphic map. In terms of Lemma 8.11 and V 6= M , we see that e df (ξ) = λξ for some nonzero constant λ. For any point p ∈ M , let q = f (p) ∈ N . Let (U1 , Φ1 ; ϕ1 ) and (U2 , Φ2 ; ϕ2 ) be foliated coordinate charts around p and q respectively, and let ϕi : Ui → Wi ⊂ C m be the submersion associated with (Ui , Φi ; ϕi ) (i = 1, 2). Suppose f (U1 ) ⊂ U2 . Then f induces a holomorphic map fe : W1 → W2 such that ϕ2 ◦ f = fe ◦ ϕ1 . Clearly p ∈ V if and only if fe is not diffeomorphic at ϕ1 (p). Hence V is defined locally by the zero locus of det(∂wα ◦ fe/∂z i ) ◦ ϕ1 , where (z i ) and (wα ) are holomorphic coordinate systems of W1 and W2 respectively. This shows that V is e a foliated analytic hypersurface in M . Since f is a foliated (H, H)-holomorphic map, f (V ) is a foliated analytic subvariety in N . It is obvious that both M − f −1 (f (V )) and N − f (V ) are foliated open submanifolds of M and N respectively. Step 2. We claim that f is a horizontally one-to-one map from M − f −1 (f (V )) to N − f (V ). Let fb : (M − f −1 (f (V )))/Fξ → (N − f (V ))/Feξe denote the induced map of f |M −f −1 (f (V )) . Suppose there are two leaves L1 , L2 ⊂ M − f −1 (f (V )) and a leaf e ⊂ N − f (V ) such that f (L1 ), f (L2 ) ⊂ L. e Let γ1 (t), γ2 (t) (t ∈ (−∞, ∞)) be the L integral curves of ξ whose images are L1 and L2 respectively. In terms of the fact that df (ξ) = λξe with constant λ 6= 0, we see that both f (γ1 ) and f (γ2 ) are integral curves e that is, of λξe with possibly different initial points. As a result, their image must be L, e The injectivity of f |M −f −1 (f (V )) implies that L1 = L2 , that is, fb is f (L1 ) = f (L2 ) = L. injective. By the surjectivity of f from M − f −1 (f (V )) to N − f (V ), we conclude that fb is surjective, and thus fb is bijective. By Proposition 5.7, f : M −f −1 (f (V )) → N −f (V ) e is a foliated (H, H)-biholomorphism. Step 3. Now we assert that f (V ) must be a foliated analytic subvariety with real codimension at least 4, that is, the transversal complex codimension is at least two. Otherwise, suppose f (V ) is also a foliated analytic hypersurface, we are going to prove that the critical points set V of f is actually removable. For any p ∈ V , let (U1 , Φ1 ; ϕ1 ) and (U2 , Φ2 ; ϕ2 ) be foliated coordinate charts around, respectively, p and q = f (p) as in Step 1, such that f (U1 ) ⊂ U2 and f (V ) ∩ U2 is defined by the zero locus of a single basic CR function. Set Vϕ1 = ϕ1 (V ∩ U1 ) and [f (V )]ϕ2 = ϕ2 (f (V ) ∩ U2 ). From Steps 1 and 2, we know that the induced holomorphic map fe : W1 → W2 is injective on fe : W1 − fe−1 ([f (V )]ϕ2 ). Clearly Vϕ1 and [f (V )]ϕ2 are analytic hypersurfaces in W1 and W2 respectively, and fe(Vϕ1 ) ⊂ [f (V )]ϕ2 . Then there exists v ∈ Vϕ1 such that v is an 69

isolated point of fe−1 (fe(v)) and, by using a local coordinate chart (z i ) of W1 at v and by applying the Riemann removable singularity theorem to z i ◦ fe−1 on W2 − [f (V )]ϕ2 for some open neighborhood of fe(v) in W2 , we find that fe is locally diffeomorphic at v. This implies that f is locally diffeomorphic at any point in ϕ−1 1 (v), contradicting −1 ϕ1 (v) ⊂ V . Step 4. From Lemma 8.9, we know that [(dθ)m−1 ]B 6= 0. By the assumption that 2m−2 2m−2 2m−2 (N ) (M ) is surjective, there exists an element [β]B ∈ HB (N ) → HB f ∗ : HB ∗ m−1 such that f [β]B = [(dθ) ]B , that is, f ∗ β = (dθ)m−1 + dα

(8.21)

2m−3 e we may write f ∗ θe = λθ + f 0 θ j + f 0 θ j . Note (M ). Since df (ξ) = λξ, for some α ∈ ΩB j j

that fj0 θ j +fj0 θ j is a global 1-form on M . Since V is foliated and iξ {(fj0 θ j +fj0 θ j )∧f ∗ β} = 0, we have Z (8.22) (fj0 θ j + fj0 θ j ) ∧ f ∗ β = 0. V

From (8.21), (8.22) and Proposition 8.12, we get Z Z Z ∗e ∗ ∗ 0 j j f θ∧f β =λ θ∧f β+ fj θ + fj θ ∧ f ∗ β V V ZV Z m−1 =λ θ ∧ (dθ) + θ ∧ dα (8.23) V V Z Z m−1 =λ θ ∧ (dθ) + dθ ∧ α. V

V

Clearly we have iξ (dθ ∧ α) = 0, which implies that Z (8.24) dθ ∧ α = 0. V

It follows from (8.23) and (8.24) that Z Z ∗e ∗ (8.25) f θ∧f β =λ θ ∧ (dθ)m−1 > 0. V

V

On the other hand, since f (V ) is of dimension less than 2m − 3, one has Z Z ∗e ∗ f θ∧f β = θe ∧ β = 0 V

f (V )

which contradicts to (8.25). It follows from the above discussion that V must be empty. In terms of step 2, we e may conclude that f : M → N is a foliated (H, H)-biholomorphism. Remark 8.2. The argument for removing the critical points of fe in Step 3 is inspired by the related argument in Theorem 8 of [Si1]. 70

e (n ≥ 2) be a compact regular Sasakian mane ), J, e θ) Corollary 8.13. Let (N 2n+1 , H(N ifold with either strongly negative or adequately negative horizontal curvature. Suppose (M, H(M ), J, θ) is a compact Sasakian manifold with the same foliated homoe e ), J) e topy type as N . Then (M, H(M ), J) is (H, H)-biholomorphic to either (N, H(N e ), −J). e or (N, H(N

Proof. Since M is foliatedly homotopic to N , we have a foliated smooth homotopy equiv2m−2 2m−2 alent map h : M → N . Consequently deg(h) = 1 and h∗ : HB (N ) → HB (M ) is e an isomorphism. By Theorem 7.14, there exists a foliated special (H, H)-harmonic map f : M → N which is foliatedly homotopic to h. In terms of Lemma 8.8, deg(f ) = 1 2m−2 2m−2 (M ) is an isomorphism too. This corollary then follows (N ) → HB and f ∗ : HB immediately from Theorem 8.12. e Remark 8.3. Note that the (H, H)-biholomorphism f between M and N in Corollary 8.13 is actually a vertically homothetic map, that is, df (ξ) = λξe with λ constant. In Example 1.1, we give some Sasakian manifolds with either strongly negative or adequately negative horizontal curvature. These Sasakian manifolds, which may be regarded as model spaces, appear also in the classification of contact sub-symmetric spaces by [BFG]. As applications, Corollary 8.13 exhibits the foliated strong rigidity of these model spaces. Appendix A. Pseudo-Hermitian harmonic maps In this subsection, we introduce another natural generalized harmonic map between pseudo-Hermitian manifolds. e is called a e ), J, e θ) Definition A1. A map f : (M 2m+1 , H(M ), J, θ) → (N 2n+1 , H(N pseudo-Hermitian harmonic map if it satisfies (A1)

τ (f ) = trgθ β = 0,

that is (A2)

α α α + fkk =0 + fkk f00 0 0 0 f00 + fkk + fkk = 0.

Remark A1. Similar ideas for introducing generalized harmonic maps as Definition A1 were also mentioned in [DT] and [Kok]. Clearly if f is a foliated pseudo-Hermitian harmonic map, then f automatically satisα α e fies the equation fkk + fkk = 0, that is, f is (H, H)-harmonic. Concerning the Question proposed in section 5, we establish a continuation result about the foliated property for pseudo-Hermitian harmonic maps. Theorem A1. Let M and N be Sasakian manifolds and let f : M → N be a pseudoHermitian harmonic map. Assume that U is a nonempty open subset of M . If f is foliated on U , then f is foliated on M . 71

Proof. The argument is similar to that for Theorem 5.4. Choose any point q ∈ ∂U . Let W be a connected open neighborhood of q such that there exist a frame field e ηe1 , ..., ηen, ηe , ..., ηen } on some open {ξ, η1 , ..., ηm , η1 , ..., ηm} on W and a frame field {ξ, 1 neighborhood of f (W ) respectively. Write dfL,He 1,0 = f0α θ ⊗ ηeα .

e 1,0 ). By definition, it is easy to derive the following Clearly dfL,He 1,0 ∈ Hom(L, f −1 H △dfT,He 1,0 = trgθ D2 dfT,He 1,0

(A3)

α α α + f0kk )θ ⊗ ηeα = (f000 + f0kk

e ηα + f α θ ⊗ ∇ e 2 ηeα } = (△M f0α )θ ⊗ ηeα + trgθ {df0α ⊗ θ ⊗ ∇e 0

where the property that ∇θ = 0 is used. From (2.17) and the Sasakian conditions of both M and N , we have α α = fk0 . f0k

α α f0k = fk0 ,

(A4)

In terms of (A4), (2.35) and (2.41), we derive that α α α α + f0kk = fk0k + fk0k f0kk

(A5)

α bα (f γ f0δ − f γ f δ ) + f α + f β R bα (f γ f δ − f γ fkδ ) + fkβ R = fkk0 0 k 0 βγδ k kk0 k βγδ k 0

α α bα (f γ f0δ − f γ f δ ) + f β R bα (f γ f δ − f γ fkδ ). + fkk0 + fkβ R = fkk0 0 k 0 βγδ k k βγδ k 0

It is easy to see from (A2) and (A5) that there exists a constant C such that X α α α (A6) |f000 + f0kk + f0kk |≤C |f0α | α

on a fixed open subset W of M . Consequently X (A7) | △M f0α | ≤ C ′ |f0α | + |∇f0α | , α

that is, {f0α } satisfies the structural assumptions of Aronszajn-Cordes. Since f0α = 0 on W ∩ U , then f0α = 0 on W , and thus we may conclude that f is foliated on M . Remark A2. From the proof of Theorem A1, we see that the above continuation result about the foliated property still holds if f only satisfies trgθ (πHe β) = 0. e we introduce the e ), J, e θ), For a smooth map f : (M 2m+1 , H(M ), J, θ) → (N 2n+1 , H(N following two 1-forms: (A8)

α α α α f0 )θ + ρe1 ρe3 = −(f00 f0 + f00

where ρe1 is given by (8.3), and (A9)

0 0 0 0 k 0 0 k f0 θ ). ρe4 = −(f00 f0 θ + f0k f0 θ + +f0k

72

Lemma A2. Let f : M → N be a map between two Sasakian manifolds. Then α 2 α 2 α 2 α α α δe ρ3 = 2|f00 | + 2|f0k | + 2|f0k | + (f000 + fkk0 + fkk0 )f0α

(A10)

α α α b + fkk0 )f0α − R(df + (f000 + fkk0 e (ξ) ∧ dfH,H e (ηk ), dfL,H e (ξ) ∧ dfH,H e (ηk )) L,H

b − R(df e (T ) ∧ dfH,H e (ηk ), dfL,H e (T ) ∧ dfH,H e (ηk )). L,H

If f is foliated, then (A11)

0 2 0 2 0 2 0 0 0 δe ρ4 = |f00 | + |f0k | + |f0k | + f00 (f000 + fkk0 + fkk0 ).

Proof. From (8.4) and (A8), we immediately get (A10). Now suppose f is foliated. Then (2.14) yield 0 0 f0k = fk0 ,

(A12)

0 0 f0k = fk0 .

Using (A12), (2.24), we deduce from (A9) that 0 0 0 0 2 0 2 0 2 | + f00 (f000 + f0kk + f0kk ) δe ρ4 = |f00 | + |f0k | + |f0k

0 2 0 2 0 2 0 0 0 = |f00 | + |f0k | + |f0k | + f00 (f000 + fk0k + fk0k )

0 0 0 0 2 0 2 0 2 | + f00 (f000 + fkk0 + fkk0 ). = |f00 | + |f0k | + |f0k

Remark A3. We only use the Sasakian condition on M to derive (A11), so it is still valid if N is any pseudo-Hermitian manifold. Theorem A3. Let M and N be two compact Sasakian manifolds and let f : M → N be e has non-positive horizontal sectional a pseudo-Hermitian harmonic map. If (N 2n+1 , ∇) e curvature, then f is a foliated special (H, H)-harmonic map with df (ξ) = λξe for some constant λ.

Proof. Using Lemma A2 and the divergence theorem, we get Z α 2 α 2 α 2 0≥ {2|f00 | + 2|f0k | + 2|f0k | }dvθ , M

and thus (A13)

α α α f00 = f0k = f0k = 0.

e Consequently, f is a (H, H)-harmonic map with split horizontal second fundamental form. Therefore Lemma 8.1 implies that f is foliated. Since f is both foliated and pseudo-Hermitian harmonic, we know that (A11) holds and becomes (A14)

0 2 0 2 0 2 | . δe ρ4 = |f00 | + |f0k | + |f0k 73

Applying the divergence theorem to (A14), we obtain 0 0 0 = 0. f00 = f0k = f0k

(A15)

From (A2) and (A15), we find that f is special in the sense of Definition 3.2, and df (ξ) = λξe with λ constant. e B. An explicit formulation for special (H, H)-harmonic maps

e Now we want to give the explicit formulations for both the special (H, H)-harmonic map equation and its parabolic version, which are convenient for proving the exise tence theory of special (H, H)-harmonic maps between two pseudo-Hermitian man2m+1 2n+1 e e As in the theory of harmonic e θ). ifolds (M , H(M ), J, θ) and (N , H(N ), J, maps (cf. [Li]), one can always assume, in view of the Nash embedding theorem, that I : (N, gθe) ֒→ (RK , gE ) is an isometric embedding in some Euclidean space, where gE e denote the Tanaka-Webster condenotes the standard Euclidean metric. Let ∇ and ∇ θe nections of M and N respectively, and let ∇ and D denote the Levi-Civita connections of (N, gθe) and (RK , gE ) respectively. e between the two manifolds, the second fundamental For a map f : (M, ∇) → (N, ∇) e is defined by form of f with respect to (∇, ∇) e e Y df (X) − df (∇Y X) β(f ; ∇, ∇)(X, Y)= ∇

(B1)

for any vector fields X, Y on M . Applying the composition formula for second fundae mental forms (see Proposition 2.20 on page 16 of [EL]) to the maps f : (M, ∇) → (N, ∇) K e → (R , D), we have and I : (N, ∇) (B2)

e e D) df (·), df (·) . β(I ◦ f ; ∇, D)(·, ·) = dI β(f ; ∇, ∇)(·, ·) + β(I; ∇,

Define a 2-tensor field S on N by

e e Z Z1 S(Z1 , Z2 ) = ∇θZ2 Z1 − ∇ 2

(B3)

where Z1 , Z2 are any vector fields on N . Therefore (B4) e

e D)(·, ·) = β(I; ∇θe, D)(·, ·) + dI S(·, ·) . β(I; ∇,

Note that β(I; ∇θ , D) is the usual second fundamental form of the submanifold I : (N, gθe) ֒→ (RK , gE ). For simplicity, we shall identify N with I(N ), and write I ◦ f as u, which is a map from M to RK . Set (B5)

τH (u; ∇, D) = trgθ β(u; ∇, D)|H . 74

Lemma B1. Suppose f : M → N is a map between pseudo-Hermitian manifolds and N is Sasakian. Suppose I : N ֒→ RK is an isometric embedding. Set u = I ◦ f . Then e f is a special (H, H)-harmonic map if and only if e τH (u; ∇, D) − trgθ β(I; ∇θ , D)(dfH , dfH ) − trgθ dI S(dfH , dfH ) = 0 where dfH denotes the restriction of df to H(M ).

Proof. From (B2), (B4) and (B5), we get e (B6) τH (u; ∇, D) = dI τH (f ) + trgθ β(I; ∇θ , D)(dfH , dfH ) + trgθ dI S(dfH , dfH ) e H . We know from Corollary 3.3 that if N is Sasakian, where τH (f ) = trgθ β(f ; ∇, ∇)| e then f is a special (H, H)-harmonic map if and only if τH (f ) = 0. Consequently this proposition follows immediately from (B6).

Suppose now that N is a compact Sasakian manifold. By compactness of N , there exists a tubular neighborhood B(N ) of N in RK which can be realized as a submersion Π : B(N ) → N over N . Actually the projection map Π is simply given by mapping any point in B(N ) to its closest point in N . Clearly its differential dΠy : Ty RK → Ty RK when evaluate at a point y ∈ N is given by the identity map when restricted to the tangent space T N of N and maps all the normal vectors to N to the zero vector. Since Π ◦ I = I : N ֒→ RK , we have e

e

β(I; ∇θ , D)(·, ·) = dΠ(β(I; ∇θ , D)(·, ·)) + β(Π; D, D)(dI, dI) and thus e

β(I; ∇θ , D) = β(Π; D, D)(dI, dI).

(B7)

The tensor field S may be extended to a tensor field Sb = Π∗ (dI ◦ S) on B(N ), that is, b 1 , W2 ) = dI S(dΠ(W1 ), dΠ(W2 )) (B8) S(W

for any W1 , W2 ∈ T B(N ). Let {y a }1≤a≤K be the natural Euclidean coordinate system of b ·) = Sba (·, ·) ∂ a . Choose an orthonormal RK . Set ua = y a ◦u, Πa = y a ◦Π, and write S(·, ∂y frame field {eA }A=0,1,...,2m around any point of M such that span{eA }1≤A≤2m = H(M ). By definition of the second fundamental form , we have τH (u; ∇, D) = △H ua

(B9)

∂ , ∂y a

and e

trgθ β(I; ∇θ , D)(dfH , dfH ) = trgθ β(Π; D, D)(duH , duH ) (B10)

2m X ∂ 2 Πa ∂ = eA (ub )eA (uc ) a b c ∂y ∂y ∂y A=1

= Πabc h∇H ub , ∇H uc i 75

∂ ∂y a

where Πabc =

∂ 2 Πa . ∂y b ∂y c

In terms of (B8), we derive that

∂ trgθ dI S(dfH , dfH ) = trgθ Sba (duH , duH ) a ∂y 2m X ∂ ∂ ∂ = Sba ( b , c )eA (ub )eA (uc ) a ∂y ∂y ∂y

(B11)

A=1

a = Sbbc h∇H ub , ∇H uc i

∂ ∂y a

a where Sbbc = Sba ( ∂y∂ b , ∂y∂ c ). It follows from (B6), (B9), (B10) and (B11) that

(B12)

∂ a dI(τH (f )) = (△H ua − Πabc h∇H ub , ∇H uc i − Sbbc h∇H ub , ∇H uc i) a . ∂y

In view of (B12), we obtain

e Proposition B2. Let M , N , f and u be as in Lemma B1. Then f is a special (H, H)harmonic map if and only if a △H ua − Πabc h∇H ub , ∇H uc i − Sbbc h∇H ub , ∇H uc i = 0

∂ 2 Πa ∂ ba ba ∂ where Πabc = ∂y b ∂y c and Sbc = S ( ∂y b , ∂y c ) (1 ≤ a, b, c ≤ K) are smooth functions defined on B(N ) ⊂ RK .

e In section 7, we study the existence problem of the special (H, H)-harmonic map equation τH (f ) = 0 by solving the corresponding subelliptic heat flow (7.5), that is,

∂ft ∂t

f |t=0

= τH (ft ) =h

for some map h : M → N . Inspired by the above explicit formulation, we will establish the fact that in order to solve (7.5), it suffices to solve the following system (B13)

∂ua ∂t ua |t=0

a = △H ua − Πabc h∇H ub , ∇H uc i − Sbbc h∇H ub , ∇H uc i,

= ha ,

(1 ≤ a, b, c ≤ K),

where ha = y a ◦ h. Let us define a map ρ : B(N ) → RK by ρ(y) = y − Π(y),

y ∈ B(N ).

Obviously, ρ(y) is normal to N and ρ(y) = 0 if and only if y ∈ N . 76

Lemma B3. Let u(x, t) = (ua (x, t)) ((x, t) ∈ M × [0, T0 )) be a solution of (B13) with initial condition h = (ha ) : M → RK . Then the quantity Z |ρ(u(x, t)|2dvθ M

is a nonincreasing function of t. In particular, if h(M ) ⊂ N , then u(x, t) ∈ N for all (x, t) ∈ M × [0, T0 ). Proof. Since ρ(y) = y − Π(y), we have

ρab = δba − Πab

(B14) and

ρabc = −Πabc

(B15) 2 a

a

ρ and ρabc = ∂y∂ b ∂y where ρab = ∂ρ c . By applying the composition law ([EL]) to the maps ∂y b ut : (M, ∇) → (B(N ), D) and ρ : (B(N ), D) → (RK , D), we have

△H ρ(u) = dρ(△H u) + trgθ β(ρ; D, D)(duH , duH ).

(B16)

Using (B16), (B14), (B15) and (B13), we derive that (△H ρ(u))a = ρab △H ub + ρabc h∇H ub , ∇H uc i

= △H ua − Πab △H ub − Πabc h∇H ub , ∇H uc i ∂ua a = + Sbbc h∇H ub , ∇H uc i − Πab △H ub ∂t ∂ub ∂ub a = ρab + Πab ( − △H ub ) + Sbbc h∇H ub , ∇H uc i ∂t ∂t

(B17)

b Since dΠ( ∂u ∂t − △H u) and S(duH , duH ) are tangent to N and ρ(u) is normal to N , we find from (B17) that a

ρa (u) (△H ρ(u)) = ρa (u)ρab (u)

(B18)

∂ub . ∂t

Using (B18), we deduce that Z Z ∂ub ∂ a 2 (ρ (u)) dvθ = 2 ρa (u)ρab (u) dvθ ∂t M ∂t M Z a =2 ρa (u) (△H ρ(u)) dvθ M Z = −2 |∇H ρ(u)|2 dvθ M

which implies that

R

M

≤0

|ρ(u)|2 dvθ is decreasing in t.

In terms of (B12) and Lemma B3, we conclude that 77

Theorem B4. Let h : M → N ⊂ RK be a smooth map given by h = (h1 , ..., hK ) in the Euclidean coordinates. If u : M × [0, T0 ) → N ⊂ RK is a solution of the following system ∂ua a = △H ua − Πabc h∇H ub , ∇H uc i − Sbbc h∇H ub , ∇H uc i, ∂t

1 ≤ a ≤ K,

with initial condition (ua (x, 0)) = (ha (x)) for all x ∈ M , then u solves the subelliptic heat flow ∂u = τH (u) ∂t with initial condition u(x, 0) = h(x). Acknowledgments: The author would like to thank Professors Aziz El Kacimi and E. Macas-Virgós for their useful discussions on the homotopy invariance of basic cohomology of Riemannian foliations. He would also like to thank Ping Cheng for helpful conversations. References [Ar]

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School of Mathematical Science and Laboratory of Mathematics for Nonlinear Science Fudan University, Shanghai 200433, P.R. China [email protected]

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