Abstract

We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold.

1. Introduction

Differential geometry of submanifolds of Sasakian manifold is an interesting branch and has been subject of investigations of many mathematicians. There are four important types of submanifolds of a Sasakian manifold, namely, invariant submanifolds, anti-invariant submanifolds, CR submanifolds, and slant submanifolds. All these types of submanifolds have been studied quite extensively; for invariant and anti-invariant submanifolds one can refer to [1–5] and for slant submanifolds one can refer to [6–8] and references therein. A submanifold of a Sasakian manifold is said to be a CR-submanifold if there is a pair of orthogonal complementary distributions and defined on of which is invariant under and is totally real distribution so that is subbundle of the normal bundle. CR submanifolds of a Sasakian manifold have been studied in [9–13] getting various geometric properties of the CR-submanifold. A submanifold of a Sasakian manifold naturally carries four operators, , , , and , defined on this submanifold. In [14], it has been shown that a submanifold of a Sasakian manifold with parallel is essentially a CR-submanifold. In this paper, we are interested in finding conditions under which a submanifold of a Sasakian manifold with parallel operator is an invariant submanifold. First we show the existence of a symmetric tensor field on any submanifold of a Sasakian manifold and study its properties and use these properties to obtain conditions under which a submanifold is an invariant submanifold (cf. Theorem 5).

2. Preliminaries

Let be a -dimensional almost contact metric manifold. Then the structure tensor satisfies (cf. [15]) where and are smooth vectors fields on . An almost contact metric manifold is said to be a Sasakian manifold if it satisfies for smooth vector fields and , where is the Riemannian connection on .

Let be an -dimensional submanifold of the Sasakian manifold . Denote by the same the Riemannian metric induced on the submanifold and by the space of smooth sections of the normal bundle of . Then we define operators , , , and as follows: where is the Lie algebra of smooth vector fields on and ; (resp., ) denotes the tangent part (resp., normal part) of ; and (resp., ) denotes the tangent part (resp., normal part) of .

Also for the structure vector field of the Sasakian manifold , we set where and . Define a smooth one form on the submanifold and by , , and , . It can be easily checked that the operators , , , and satisfied the relations

Also for the submanifold , we have the following Gauss and Weingarten formulas (cf. [16]): where is the normal connection, is the second fundamental form and is the Weingarten map with respect to the normal bundle of and we have

For the operators and on the submanifold of the Sasakian manifold , we define the following covariant derivatives: Then using (2)–(4) and (11) we immediately get the following.

Lemma 1. Let be a submanifold of the Sasakian manifold . Then one has

3. Submanifolds with Parallel

In this section we will study a submanifold of the Sasakian manifold with structure vector field tangent to and the operator is parallel. Note that it is customary to require that is tangent to (cf. [13]), rather than normal as it is too restrictive, as in this case must be totally real [13, Proposition 1.1, page 43] or leads to too complicated embedding equations.

If we define the operators by using (7) it is easy to see that and are symmetric tensor fields.

In [17], Yano and Kon have studied the submanifolds of a Sasakian manifold with operator is parallel and have proved that these submanifolds are contact CR submanifolds [17, Proposition 3.3, page 241]. Thus all our results for submanifolds with parallel operator of a Sasakian manifold are automatically the results of contact CR submanifolds of the Sasakian manifold. First we will prove the following.

Theorem 2. Let be an -dimensional submanifold of a -dimensional Sasakian manifold with structure vector field tangent to . If the operator is parallel, then is also parallel.

Proof. For and we have and consequently Using Lemma 1 in above equation and noting that , we get Now since is parallel then by Lemma 1 with , we get Also, using (6) we get by using the above equation and (12) we get Using (21) and (19) in (18), we conclude that and this proves the theorem.

For , let , ; that is, is the eigenvector of corresponding to an eigenvalue .

Lemma 3. Let be an -dimensional submanifold of the Sasakian manifold with structure vector field tangent to . If the operator is parallel, then the eigenvalues of are constant.

Proof. Let , , . Without loss of generality we can assume that is a unit normal vector field. As is parallel then, by Theorem 2, we have being parallel and consequently Taking inner product with , we get which proves that is a constant.

For a local orthonormal frame (where ) on , we define smooth function by then using (6), (7), and (9), we obtain where . Also, we define since is symmetric we can choose an orthonormal frame that diagonalizes .

Lemma 4. Let be -dimensional submanifold of the Sasakian manifold with structure vector field tangent to . If the operator is parallel, then is a constant and holds; consequently is also a constant.

Proof. Suppose is parallel; then by (25), we have
By (7) we have since the operator is symmetric we get since by Theorem 2 since is parallel implies also parallel and the local orthonormal frame on normal vector fields that diagonalizes with the equation , thus we get that which proves that is a constant. Finally as is a constant, we get that is also a constant.

Note that the operator plays an important role in the geometry of submanifolds of a Sasakian manifold; for instance the following theorem shows that implies that the submanifolds are invariant submanifolds.

Theorem 5. Let be an -dimensional connected submanifold of the Sasakian manifold with structure vector field tangent to . If  , then is an invariant submanifold.

Proof. Suppose ; then we have and that gives . This also gives and consequently that is, , . That is, is an invariant submanifold.

Theorem 6. Let be an -dimensional submanifold of the Sasakian manifold with structure vector field tangent to . The operator is parallel if and only if is parallel.

Proof. If is parallel then by Lemma 1 with we get Taking inner product with and using (6) we get and using (12) and (5) we get which implies that Now using the above equation in Lemma 1 and the fact that we get that and which means that is parallel.
Conversely, if we suppose that the operator is parallel then by Lemma 1 with , we get Also using (5) to get which together with (12) and (6) gives we get that and using the above equation in Lemma 1 with we get which proves that is parallel.

As a direct consequence of the above theorem and Proposition 3.3 in [17], we have the following.

Corollary 7. Let be a submanifold of a Sasakian manifold . If is parallel, then is a contact CR-submanifold of .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.