CHERN CLASSES OF INTEGRAL SUBMANIFOLDS OF SOME CONTACT MANIFOLDS

A complex subbundle of the normal bundle to an integral submanifold of the contact distribution in a Sasakian manifold is given. The geometry of this bundle is investigated and some results concerning its Chern classes are obtained.


Introduction.
Let M be a (2m + 1)-dimensional manifold endowed with the almost contact metric structure F , ξ, η, g.These tensor fields satisfy the conditions for all vector fields X, Y tangent to M. Let Ᏸ be the contact distribution of M, defined by the equation η = 0.The study of the integral submanifolds of Ᏸ is very difficult for, at least, three reasons: (a) their abundance (see, e.g., [1,5,8]), (b) the nonexistence of a natural structure induced on the submanifold M, resulting from the equalities η = 0, dη = 0, true along M, and (c) for any vector field X tangent to M, the vector field FX is normal to M and therefore, freely speaking, the geometry of an integral submanifold of Ᏸ is normal to the submanifold.However, for maximal integral submanifolds (i.e., dim M = m), we know many properties (see, e.g., [1,Chapter V]); while for nonmaximal integral submanifolds, we have so few results.
In this paper, we associate to each nonmaximal integral submanifold M of M a nontrivial vector bundle τ(M).The geometry and the topology of this vector bundle are also studied.In Section 2, we give, in an "appropriate" form, the structure equations of an integral submanifold in a Sasakian manifold.In Section 3, we study the geometry of τ(M), namely, we prove that it has a natural structure of complex symplectic vector bundle.
It is well known that integral submanifolds of an almost contact manifold are antiinvariant, [8].Thus, such a submanifold is analogous to the isotropic (or totally real) submanifolds of a Kähler manifold, investigated by Chen and Morvan in [2,4], and we can use some of their technics in order to study Chern classes of the vector bundle τ(M).In Section 4, by combining these ideas with some Vaisman's results [7] concerning the characteristic classes of quaternionic bundles, we obtain stronger results than for isotropic submanifolds.Namely, we prove that if m − n is even, then all odd Chern classes of τ(M) are zero.In absence of this supposition on the dimensions, we prove that the first Chern class of τ(M) is zero when M is a Sasakian space form.

Structure equations of an integral submanifold.
Let M be an almost contact metric manifold.Furthermore, we assume that M is Sasakian and let ᐄ( M) denote the set of all vector fields tangent to M. We have [1, page 73 where ∇ is the Levi-Civita connection associated to the metric g on M.Moreover, we have the well-known equalities Now, let M be an n-dimensional submanifold of the Sasakian manifold M and denote by h, ∇⊥ , and A its second fundamental form, normal connection, and Weingarten operator, respectively.It is well known that n ≤ m (see [8] or [ If Ꮾ * = {ω 1 ,...,ω n ,ω n+1 ,...,ω m ,ω 1 * ,...,ω n * ,ω (n+1) * ,...,ω m * ,ω (m+1) * = η} is the local field of coframes of Ꮾ, then, at the points of M, we have (locally) On the other hand, by computations we prove that if (ω β α ) is the connection form of ∇, expressed with respect to Ꮾ, then, on the submanifold M, we have ) (2.5) The curvature forms of M and M are, respectively, where Rα βγδ and R a bcd are the components (with respect to Ꮾ) of the curvature tensors of M and M, respectively.Then, at the points of M, we have ) where R λ µab are the components of the curvature tensor of ∇ ⊥ .Finally, from (2.3), (2.4), and (2.5) and from the general form of the structure equations (see, e.g., [3, page 121]), we deduce the structure equations of an integral submanifold of a Sasakian manifold under the form (2.9)

Geometry of the maximal invariant normal bundle. The normal space T ⊥
x M at each point x ∈ M has the following orthogonal decomposition where is the total space of a subbundle τ(M) of T ⊥ M and Ꮾ τ = {e λ ,e λ * } = {e n+1 ,...,e m ,e (n+1) * ,...,e m * } is a local basis in the module Γ (τ) of its sections.We also denote this bundle by τ(M) and call it the maximal invariant normal bundle of the integral submanifold M.
Theorem 3.1.Let M be an integral submanifold of the Sasakian manifold M. Its maximal invariant normal bundle τ(M) has the following properties: Proof.(a) follows easily from (3.1).(b) Denote by (n λ ,n λ * ) the components of the vector n x ∈ τ x (M), relative to the basis Ꮾ τ , and let ρ : τ(M) → M be the natural projection.Then, using the classical notations, the vector charts define on τ(M) a complex vector bundle structure.(c) From (a), we deduce that the space Γ (τ) can be considered as a complex space with the following multiplication by complex numbers: Endowed with this complex structure, Γ (τ) is an (m −n)-dimensional space, denoted by Γ c (τ).Moreover, we can define the map and it has the following properties: for n, n 1 , n 2 ∈ Γ c (τ) and λ ∈ C. Hence (see [7, Section 1]), F τ defines on τ(M) a quaternionic structure.
A natural connection can be defined on τ(M).Firstly, we remark that g(∇ ⊥ X n, ξ) = 0 for all X ∈ ᐄ(M) and n ∈ Γ (τ), hence the normal vector field ∇ ⊥ X n has the following decomposition: where B n X ∈ Γ (F T M) and ∇ τ X n ∈ Γ (τ).Moreover, the maps B : Γ (τ)×ᐄ(M) → Γ (F T M) and ∇ τ : ᐄ(M) × Γ (τ) → Γ (τ) have the following properties.Proposition 3.2.(a) ∇ τ is an almost complex connection on the maximal invariant normal bundle of the integral submanifold M, that is, The proof follows from (3.5) by computation, taking into account (2.1) and (2.2) and using the Weingarten formula for the submanifold M. Now, if we extend the scalar product g over Γ c (τ) by for λ ∈ C and n 1 , n 2 ∈ Γ c (τ), then we have hence g τ is a Hermitian scalar product on the complex vector bundle τ(M).Moreover, For any n 1 , n 2 ∈ Γ c (τ), we put and a simple computation shows that Ω τ is C-linear with respect to the first argument and From these relations and because Ꮾ c τ is an orthonormal local basis, we deduce that Ω τ is a nondegenerate skew-symmetric 2-form on the complex vector bundle τ(M).Hence, we have the following proposition.Proposition 3.3.For m − n even, the maximal invariant normal bundle τ(M) of the integral submanifold M of a Sasakian manifold has a structure of complex symplectic vector bundle with the symplectic form Ω τ .and then its curvature form is

Normal Chern classes of an integral submanifold. As a complex vector bundle, the basic characteristic classes of the maximal invariant normal bundle τ(M) are the Chern classes [γ k (τ)], represented by the Chern forms
On the other hand, from (2.9), it follows the complex form of the second structure equation of τ(M), namely, , and then we have and the proof is complete.
Let n be a vector field normal to the integral submanifold M of the Sasakian manifold M. For X ∈ ᐄ(M), the equality α n (X) = g(F n, X) defines a 1-form α n on M. In [6], this form is used for the study of some remarkable vector fields on M (Legendrian, Hamiltonian, and harmonic variations).Another 1-form on M is defined by θ = n a=1 ω a * a , and we can state the following proposition.
for all X, Y ∈ ᐄ(M); (c) the exterior derivative of θ is given by where S is the Ricci tensor of M.
On the other hand, using (2.1), we have and then, applying the Weingarten formula in (4.12), it follows that On the other hand, on a Sasakian manifold, the following equalities are true [1, page 93]:

Theorem 4 . 1 . 2 . ( 4 . 2 )Theorem 4 . 2 . 1 ( 3 )
curvature forms of ∇ τ and δ ••• ••• is the multiindex Kronecker symbol.We say that γ k (τ) is the kth normal Chern form of the submanifold M and the purpose of this section is to obtain some results concerning the computation of γ k (τ) and thekth normal Chern class [γ k (τ)] of M. Let M be an n-dimensional integral submanifold of a Sasakian manifold of dimension 2m + 1.If m − n is even, then γ 2k+1 (τ) = 0 for k = 0,1,..., m − n − 1 Proof.By Theorem 3.1(c), the maximal invariant normal bundle τ(M) has a quaternionic structure, and then we can apply [7, Proposition 2.1].Now, we will analyse the first normal Chern form and its associated class in absence of the supposition that m − n is even.The first normal Chern form of the n-dimensional integral submanifold M in a Sasakian manifold of dimension 2m + 1, m > n, is given by γ Proof.Using (3.5), (2.1), and the Weingarten formula, we obtain the components of the curvature tensor τ R of ∇ τ under the form τ R λ * λab = R λ * λab + g B e λ e b ,B e λ * e a − g B e λ e a ,B e λ * e b , (4.4)

Proposition 4 . 3 .
The forms α n and θ have the following properties: (a) α ξ = 0 and θ = −nα H , where H is the mean curvature vector of M; (b) α n is closed if and only if

F
Proof.(a) We have the well-known equality ∇X e α = for any X ∈ ᐄ( M), and, by using (2.4) and (2.5), we obtain θ e b = n a=1 g ∇e b e a ,e a * , b ∈ {1, 2,...,n}.(4.10) Taking into account (2.1) and the Gauss formula, we deduce ∇ea e b ,e a = − n a=1 g ∇ea e b * ,e a = n a=1 g e b * , ∇ea e a = n a=1 g e b * ,h e a ,e a = ng e b * ,H = −ng e b ,FH = −nα H e b .(4.11) (b) From the definition of the 1-form α n , we obtain .14) (c) From (2.4) and (2.5) and taking into account the second structure equation of M the Gauss formula for the submanifold M in (4.9), we have n b=1 ω b a (X)e b + ω b * a (X)e b * + m λ=n+1 ω λ a (X)e λ + ω λ * a (X)e λ * = ∇ X e a + h X, e a (4.18) for any X ∈ ᐄ(M).It follows that ω µ a (X) = g h X, e a ,e µ ac are the components of h(e a ,e c ) with respect to the basis Ꮾ τ .Therefore, we have any α = λ or α = λ * .Finally, from (4.17) and (4.20) we deduce dθ = n a,b,c=1 h λ ab h λ * ac − h λ ac h λ * ab ω b ∧ ω c + Sasakian, its curvature tensor R satisfies the following equality [1, page 75]: R(X, Y )ξ = η(Y )X − η(X)Y , X, Y ∈ ᐄ M , (4.22) hence the Ricci tensor S of M is given by S(X, Y ) = 2m α=1 e α ,X,e α ,Y − g(X, Y ), (4.23) for all X, Y ∈ ᐄ( M) orthogonal to ξ, where is the Riemann-Christoffel curvature tensor field of M. Using (2.3), from the first equality in (2.6), we deduce of the submanifold M.Moreover, using the first Bianchi identity relative to M, we have Ra * abc = e a * ,e a ,e b ,e c = e c ,e a * ,e a ,e b + e c ,e a ,e b ,e c * .(4.25)