Generic Submanifolds of Nearly Kaehler Manifolds with Certain Parallel Canonical Structure

The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds, and CR-submanifolds. In this paper we initiate the study of generic submanifolds in a nearly Kaehler manifold from differential geometric point of view. Some fundamental results in this paper will be obtained.


Introduction
Nearly Kaehler manifolds have been studied intensively in the 1970's by Gray [1]. These nearly Kaehler manifolds are almost Hermitian manifolds with almost complex structure for which the tensor field ∇ is skew-symmetric. In particular, the complex structure is nonintegrable if the manifold is non-Kaehler. As we all know, there are two natual types of submanifolds of nearly Kaehler (or more generally, almost Hermitian) manifold, namely, almost complex and totally real submanifolds. Almost complex submanifolds are submanifolds whose tangent spaces are invariant under and totally real submanifolds are opposite. A well known example is the nearly Kaehler 6-dimensional sphere which has been studied by many authors (see, e.g., [2][3][4][5][6][7]).
In 1981, Chen introduced preliminary the differential geometry of real submanifolds in a Kaehler manifold ( [8]) and gave some basic formulas and definitions. Inspired by that paper, we will generalize some important formulas and proprerties in a Kaehler manifold to a nearly Kaehler manifold. The paper is organized as follows: the basic on nearly Kaehler manifolds and submanifold theory will be recapitulated in Section 2. In Section 3, we give the integrability conditions of the two natural distributions H and H ⊥ associated with a generic submanifold of nearly Kaehler manifold. Finally, we consider generic submanifolds with one of its canonical structures to be parallel. These results enable us to prove the following theorem.

Theorem 1.
Let be a generic submanifold in a nearly Kaehler manifold . If (or ) is parallel, then the holomorphic distribution H is intergrable.
The operator (or ) is a canonical structure as the following paper introduced.

Preliminaries
An almost Hermitian manifold ( , , ) is a manifold endowed with an almost complex structure , that is, compatible with the metric , that is, an endomorphism : → such that 2 = − for every ∈ and ( , ) = ( , ). A nearly Kaehler manifold is an almost Hermitian manifold with the extra condition that the (1,2)-tensor field = ∇ is skew-symmetric: for every , ∈ . Here ∇ stands for the Levi-Civita connection of the metric . The tensor field on satisfies the following properties ( [1,2]): where , , and are arbitrary vector fields on . We denote the metrics of and its submanifold by the same letter , is the tangent bundle of , and ⊥ is the normal bundle of . If ∇ and ∇ ⊥ denote the Riemannian connection induced on and the connection in the normal bundle ⊥ , respectively, then the Gauss and Weingarten formulas are where , ∈ and ∈ ⊥ . The second fundamental form and the shape operator are related to each other by For any vector field tangent to , we put where and are the tangential and normal components of , respectively. Then is an endomorphism of the tangent bundle and is a normal-bundle-valued 1-form on . For any vector field normal to , we put where and are the tangential and normal components of , respectively. Then is an endomorphism of the normal bundle ⊥ and is a tangent-bundle-valued 1-form on ⊥ . For a submanifold in a nearly Kaehler manifold we define the holomorphic tangent space of at . H is the maximal complex subspace of which is contained in . Similar to [8], we will give several definitions as follows.

Definition 2. A submanifold
in a nearly Kahler manifold (or in an almost complex manifold in general) is called a generic submanifold if dim H is constant along and H defines a differentiable distribution on , called the holomorphic distribution.
For a generic submanifold in a nearly Kaehler manifold , the orthogonal complementary distribution H ⊥ , called the purely real distribution, satisfies From (9) it is clear that the normal-bundle-valued 1-form induces an isomorphism from H ⊥ onto H ⊥ . Let V be the vector space of holomorphic normal vectors to at , or simply the holomorphic normal space of at ; that is, Then V defines a differentiable vector subbundle of ⊥ . we have that

Integrability
In this section we study the integrability of the holomorphic distribution H and the purely real distribution H ⊥ . First we give the following.
for any vector , ∈ H and ∈ V.
Proof. From (2) and (6), we obtain where ∈ H, ∈ , and ∈ V. This implies that where , ∈ H and ∈ V. Since the second fundamental form is symmetric, we have From the equations above, we prove the lemma. Proof. Since is nearly Kaehlerian, using formulas (2) and (6), we have  Proof. For any vector fields , ∈ H ⊥ , (6) gives and it follows immediately from (6), (7), and (9) that Comparing the tangential parts, we have Thus we get

Theorem 8. Let be a generic submanifold in a nearly Kaehler manifold . If H ⊥ is integrable and its leaves are totally geodesic in , then
That is, From this we obtain the theorem.

Generic Submanifolds with Parallel Canonical Structure
For the endomorphism : → , we put for any vector fields , ∈ . The endomorphism is said to be parallel if ∇ = 0 for any vector ∈ . From (6), (7), and (9) we can obtain the following: That is, By comparing the tagential parts, we have the following: Therefore, for any vector fields , , ∈ , we have So, we obtain the Lemma as follows.
Lemma 9. Let be a generic submanifold in a nearly Kaehler manifold . The P is parallel, that is, ∇ = 0, if and only if for any vectors , ∈ .
Theorem 10. Let be a generic submanifold in a nearly Kaehler manifold . If is parallel, then , for any vector fields ∈ H and ∈ (ii) the holomorphic distribution H is intergrable.
Proof. From Lemma 9, for any vector fields ∈ H and ∈ , we know = 0; this implies that ( , ) ⊤ = − . On the other hand, for any vector fields , ∈ H, ( , ) ⊤ = − = 0, then ( , ) is normal to . By (i), we can get that is, for any vector fields , ∈ H and ∈ . The equations above imply that for any vector fields , ∈ H and ∈ H ⊥ . These give For the normal bundle-valued 1-form , we put For any vector fields , ∈ . The endomorphism is said to be parallel if ∇ = 0 for any vector ∈ . By comparing the normal parts of (37), we have the following: for any vectors , ∈ . Hence, for any vector field ∈ ⊥ , it follows from (4) From which we obtain the Lemma as follows.
Lemma 11. Let be a generic submanifold in a nearly Kaehler manifold . The is parallel, that is, ∇ = 0, if and only if for any vectors ∈ and ∈ ⊥ .
Theorem 12. Let be a generic submanifold in a nearly Kaehler manifold . If is parallel, then the holomorphic distribution H is intergrable.