Generic Submanifolds of Nearly Kaehler Manifolds with Certain Parallel Canonical Structure

Abstract

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.

1. 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., [2–7]).

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 and 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 is intergrable.

The operator (or ) is a canonical structure as the following paper introduced.

2. 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 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 . 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 is constant along and defines a differentiable distribution on , called the holomorphic distribution.

Definition 3. A generic submanifold in a nearly Kaehler manifold is a totally real (resp., complex) submanifold if (resp., ).

For a generic submanifold in a nearly Kaehler manifold , the orthogonal complementary distribution , called the purely real distribution, satisfies From (9) it is clear that the normal-bundle-valued 1-form induces an isomorphism from onto . Let be the vector space of holomorphic normal vectors to at , or simply the holomorphic normal space of at ; that is, Then defines a differentiable vector subbundle of . we have that

3. Integrability

In this section we study the integrability of the holomorphic distribution and the purely real distribution . First we give the following.

Lemma 4. Let be a generic submanifold in a nearly Kaehler manifold . Then for any vector and .

Proof. From (2) and (6), we obtain where , , and . This implies that where and . Since the second fundamental form is symmetric, we have
From the equations above, we prove the lemma.

Proposition 5. Let be a generic submanifold in a nearly Kaehler manifold . Then the holomorphic distribution is integrable if and only if for any vector fields , , and .

Proof. Since is nearly Kaehlerian, using formulas (2) and (6), we have So we get for any vector field in . If the holomorphic distribution is integrable, the right-hand-side of (22) lies in ; thus we obtain . In particular, we have (20). Conversely, if (20) holds, then by Lemma 4 and (14) we have for any vectors in . Thus by (22) we obtain . Since is tangent to , this implies that lies in . Thus we proved the proposition from the Frobenius theorem.

Proposition 6. Let be a generic submanifold in a nearly kaehler manifold . Then the purely real distribution is integrable if and only if lies in , for any vector fields .

Proof. For any vector fields , (6) gives and it follows immediately from (6), (7), and (9) that From which we obtain Comparing the tangential parts, we have Thus we get Since , this implies that lies in if and only if (23) lies in . The proposition is proved.

Theorem 7. Let be a generic submanifold in a nearly Kaehler manifold . If is integrable and its leaves are totally geodesic in , then

Proof. Since is integrable and its leaves are totally geodesic in , we have , for any . So , and we can get for any vector and . From (2), (6), (7), and (9), we get which implies that This proves the theorem.

Theorem 8. Let be a generic submanifold in a nearly Kaehler manifold . If is integrable and its leaves are totally geodesic in , then

Proof. Under the hypothesis, for any vector fields in and in , it follows from (3), (6), (7), and (9) that That is, From this we obtain the theorem.

4. Generic Submanifolds with Parallel Canonical Structure

For the endomorphism , we put for any vector fields . The endomorphism is said to be parallel if 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, , if and only if for any vectors .

Theorem 10. Let be a generic submanifold in a nearly Kaehler manifold . If is parallel, then(i), for any vector fields and (ii)the holomorphic distribution is intergrable.

Proof. From Lemma 9, for any vector fields and , we know ; this implies that . On the other hand, for any vector fields , , then is normal to . By (i), we can get that is, for any vector fields and . The equations above imply that for any vector fields and . These give From Proposition 5, the theorem holds.

For the normal bundle-valued 1-form , we put For any vector fields . The endomorphism is said to be parallel if 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), (8), and (10) that

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, , 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 is intergrable.

Proof. From (46) we have, for any vectors and . Since is parallel, then That is, This implies that for any vectors and . From Proposition 5, the theorem holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Grant no. 2011J05001 of NSF of Fujian Province, China, and the Grant no. JA11052 of the Fund of the Education Department of Fujian Province, China, and was partially supported by Grant no. 11171139 of NSFC.