Abstract

We mainly investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. It is proved that a tangential half-lightlike submanifold of an indefinite Kenmotsu space form with semiparallel second fundamental form either satisfies or is -mixed geodesic.

1. Introduction

As the intersection of normal bundle and tangent bundle of a submanifold of a semi-Riemannian manifold may not be trivial, it is more difficult and interesting to study the geometry of lightlike submanifolds than nondegenerate submanifolds. The two standard methods to deal with the above difficulties were developed by Kupeli [1], Duggal and Bejancu [2], Duggal and Jin [3], and Duggal and Sahin [4], respectively. Let be a lightlike submanifold immersed in a semi-Riemannian manifold, it is obvious to see that there are two cases of codimension 2 lightlike submanifolds, since for this type the dimension of their radical distributions is either 1 or 2. A codimension 2 lightlike submanifold of a semi-Riemannian manifold is called a half-lightlike submanifold [5] if , where denotes the degenerate radical distribution of . For more results about half-lightlike submanifolds, we refer the reader to [4, 6, 7].

In the theory of submanifolds of Riemannian manifolds, the parallel and semiparallel immersions were studied by Ferus [8] and Deprez [9], respectively. Recently, Massamba [10–13] and Upadhyay and Gupta [14] studied the parallel and semiparallel lightlike hypersurfaces of an indefinite Sasakian, Kenmotsu, and cosymplectic manifolds, respectively. However, the parallel and semiparallel half-lightlike submanifolds of an indefinite Kenmotsu manifolds have not yet been considered. The aim of this paper is to investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds.

This paper is organized in the following way. In Section 2, we provide some well-known basic formulas and properties of indefinite Kenmotsu manifolds and half-lightlike submanifolds. Section 3 is devoted to presenting some main results on semiparallel half-lightlike submanifolds of indefinite Kenmotsu space form. Finally, in Section 4, some properties of parallel half-lightlike submanifolds of indefinite Kenmotsu manifolds are investigated.

2. Preliminaries

In this section, we follow Duggal and Sahin [4] for the notation and fundamental equations for half-lightlike submanifolds of indefinite Kenmotsu manifolds. A -dimensional semi-Riemannian is said to be an indefinite Kenmotsu manifold if it admits a normal almost contact metric structure , where is a tensor field of type (1,1), is a vector field which is called characteristic vector field, is a 1-form, and is the semi-Riemannian metric on such that for any , where denotes the Levi-Civita connection of a semi-Riemannian metric .

A plane section of an indefinite Kenmotsu manifold is called a -section if it is spanned by a unit vector field orthogonal to and , where is a nonnull vector field on . The sectional curvature of a -section is called a -sectional curvature. If has a constant -sectional curvature which does not depend on the -section at each point, then is a constant and is called a Kenmotsu space form, denoted by . The curvature tensor of an indefinite Kenmotsu space form is given in [4] as follows:

A submanifold of a semi-Riemannian manifold of codimension 2 is called a half-lightlike submanifold if the radical distribution is a vector subbundle of the tangent bundle and the normal bundle is of rank 1, where the metric induced from ambient space is degenerate. Thus there exist nondegenerate complementary distributions and of in and respectively, which are called the screen and screen transversal distributions on , respectively. Thus we have where denotes the orthogonal direct sum. Consider the orthogonal complementary distribution to in ; it is easy to see that is a subbundle of . As is a nondegenerate subbundle of , then the orthogonal complementary distribution to in is also a nondegenerate distribution. Clearly is a subbundle of . Choose as a unit vector field with . In this paper, we may assume that without loss of generality. For any null section , there exists a uniquely defined null vector field satisfying Denote by the vector subbundle of locally spanned by . Then we show that . We put . We call , , and the lightlike transversal vector field, lightlike transversal vector bundle, and transversal vector bundle of with respect to the chosen screen distribution , respectively. Then is decomposed as follows:

Let be the projection morphism of on with respect to the decomposition (7). For any , , , and , the Gauss and Weingarten formulas of and are given by where and are induced connection on and , respectively; and are called locally second fundamental forms of ; and is called the locally second fundamental form on . , , and are linear operators on and , , and are -forms on . We put for any , where is called the second fundamental form of . Note that the connection is torsion free but is not metric tensor and the connection is metric. We also know that both and are symmetric tensors on and independent of the choice of a screen distribution. Using (8)–(12) we obtain the following equations: for any , where . Denote by and the curvature tensor of semi-Riemannian connection of ; then we have

3. Semiparallel Half-Lightlike Submanifolds

In this section, we mainly investigate semiparallel half-lightlike submanifolds of indefinite Kenmotsu manifolds. First of all, we need the following two lemmas.

Lemma 1 (see [7]). Let be a half-lightlike submanifold of an indefinite almost contact metric manifolds . Then there exists a screen distribution such that

Using the similar method shown in [15], it is easy to see the following result.

Lemma 2. Let be a half-lightlike submanifold of an indefinite Kenmotsu manifolds . Then the structure vector field does not belong to and .

If the structure vector field is tangent to , then belongs to . We say that a half-lightlike submanifold of an indefinite Kenmotsu manifold is tangential if is tangent to the characteristic vector field of . Thus, by using the above two lemmas we see that there exists a nondegenerate almost complex distribution on with respect to ; that is, , such that Thus, the general decompositions (5) and (7) reduce the following: where is an almost complex distribution of with respect to . Consider a pair of local null vectors and a local nonnull vector field on defined by Denote by the projection morphism of on with respect to the decomposition (20); then for any vector field on we have where , , and are 1-forms locally defined on by and is a tensor of type globally defined on by . From the second term of (3) and (8) we have the following identities: for any .

Definition 3. Let be a half-lightlike submanifold of an indefinite Kenmotsu manifold , tangent to the characteristic vector . Then is said to be semiparallel if its second fundamental form satisfies for any .

Definition 4 (see [1]). A half-lightlike submanifold of a semi-Riemannian manifold is said to be irrotational if for any .

From (8) and (12) we see that a necessary and sufficient condition for a half-lightlike submanifold to be irrotational is for any .

Lemma 5. Let be a half-lightlike submanifold of an indefinite Kenmotsu space from ; then the curvature tensor of is given by for any .

Proof. The proof follows from (4) and (17).

Theorem 6. Let be a semiparallel half-lightlike submanifold of an indefinite Kenmotsu space from , tangent to the characteristic vector . If is irrotational, then either or is -mixed geodesic.

Proof. Putting (27) and into (26), and using the decomposition (7) and (25) we obtain
Replacing by in (28) and (29), respectively, and noting that is irrotational, then we obtain Substituting into (30) and (31), respectively, then we get
Finally, substituting into (32) and (33) implies that for any . Thus, it follows that either or for any , which completes the proof.

4. Parallel Half-Lightlike Submanifolds

In this section, we mainly prove some properties of parallel half-lightlike submanifolds of indefinite Kenmotsu manifolds.

Lemma 7. Let be a half-lightlike submanifold of an indefinite Kenmotsu space from ; then the local second fundamental forms and are given, respectively, by for any .

Proof. The proof follows from (4) and (17).

Definition 8. Let be a half-lightlike submanifold of an indefinite Kenmotsu manifold , tangent to the characteristic vector . Then is said to be parallel (see [12]) with respect to if its second fundamental form satisfies for any .
Using , then a straightforward calculation gives that is said to be with the parallel second fundamental form if and only if for any .

Theorem 9. Let be a parallel half-lightlike submanifold of an indefinite Kenmotsu space from , tangent to the characteristic vector . Then and hence is totally geodesic.

Proof. Substituting the second term of (37) into (35) gives for any . Replacing and by and , respectively, in (38), we obtain Thus, substituting into (39) gives . From (25) and (37) we have for any . On the other hand, it follows from the first term of (25) and (40) that for any . Thus, the proof follows from (41).

Let be a half-lightlike submanifold of an indefinite Kenmotsu manifold. We say that the local second fundamental forms of are parallel if and only if   and for any .

Lemma 10. Let be a half-lightlike submanifold of an indefinite Kenmotsu space from , tangent to the characteristic vector . If the local second fundamental form is parallel with respect to , then . Moreover, in this case either or .

Proof. Suppose that the local second fundamental form is parallel with respect to ; that is, for any . Then it follows from (35) that for any . Substituting into (42) and using and , we obtain for any . Putting into (43) and using (1) give that for any ; finally, replacing by in this equation gives .
Using in (35) and noting that is parallel with respect to , then we have for any . Replacing by in (44) and using give for , which completes the proof.

Theorem 11. Let be a half-lightlike submanifold of an indefinite Kenmotsu space from , tangent to the characteristic vector . If the local second fundamental forms and are parallel with respective to , then . Moreover, if and , then is -totally geodesic if and only if for any .

Proof. Using the parallelism of two local second fundamental forms and and Lemma 10, it follows from (34) that for any . If , then from Lemma 10 we know ; thus for any . Replacing by in (46) and using (13), we obtain for any . Then the proof follows from (47) and Lemma 10.

Acknowledgments

This work is supported by NSFC (no. 10931005) and Natural Science Foundation of Guangdong Province of China (no. S2011010000471).