Abstract

We introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We give an example of hemi-slant lightlike submanifold and establish two characterization theorems for the existence of such submanifolds. We prove some theorems which ensure the existence of minimal hemi-slant lightlike submanifolds and obtain a condition under which the induced connection ∇ on M is a metric connection. An example of proper minimal hemi-slant lightlike submanifolds is also given.

1. Introduction

Given a semi-Riemannian manifold, one can consider its lightlike submanifold whose study is important from application point of view and difficult in the sense that the intersection of normal vector bundle and tangent bundle of these submanifolds is nonempty. This unique feature makes the study of lightlike submanifolds different from the study of nondegenerate submanifolds. The general theory of lightlike submanifolds was developed by Kupeli [1]; Duggal and Bejancu [2]. On the other hand, the concepts of transversal and screen transversal lightlike submanifolds of an indefinite Kaehler manifold were given by Sahin [3, 4]. In Sasakian setting, these submanifolds were investigated by Yildirim and Sahin [5–7].

In the present paper, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. The paper is arranged as follows. In Section 2, we recall definitions for indefinite Kenmotsu manifolds and give basic information on the lightlike geometry needed for this paper. In Section 3, after defining hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold, we give an example which ensures the existence of hemi-slant lightlike submanifolds. We obtain integrability conditions of distributions involved, investigate the geometry of leaves of distributions, and establish two characterization theorems for the existence of hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold. We prove that there does not exist curvature invariant hemi-slant lightlike submanifold in an indefinite Kenmotsu space form with some condition on and obtain a geometric condition under which the induced connection on is a metric connection. In Section 4, we study minimal hemi-slant lightlike submanifolds and give an example of proper minimal hemi-slant lightlike submanifolds immersed in .

2. Preliminaries

An odd dimensional semi-Riemannian manifold is said to be an indefinite contact metric manifold [8] if there exists a (1,1) tensor field , a vector field , called the characteristic vector field, and its 1-form satisfying where . It follows that .

An indefinite almost contact metric manifold is said to be an indefinite Kenmotsu manifold [9] if for any .

We follow [2] for the notation and formulae used in this paper. A submanifold immersed in a semi-Riemannian manifold is called a lightlike submanifold if the metric induced from is degenerate and the radical distribution is of rank , where . Let be a screen distribution which is a semi-Riemannian complementary distribution of in , that is, Consider a screen transversal vector bundle , which is a semi-Riemannian complementry vector bundle of in . Since for any local basis of , there exist a local null frame of sections with values in the orthogonal complement of in such that , it follows that there exist a lightlike transversal vector bundle locally spanned by [2, page 144]. Let be complementary (but not orthogonal) vector bundle to in . Then The Gauss and Weingarten equations are as follows: where and belong to and , respectively, and are linear connections on and on the vector bundle , respectively. Moreover, we have for each , and . If we denote the projection of on by , then by using (2.7)–(2.9) and the fact that is a metric connection, we obtain From the decomposition of the tangent bundle of a lightlike submanifold, we have for and . By using the above equation, we get It is important to note that the induced connection on is not a metric connection whereas is a metric connection on .

We denote curvature tensors of and by and , respectively. The Gauss equation for is given by for all .

The curvature tensor of an indefinite Kenmotsu space form is given by [9] as follows: for any .

From now on, we denote by in this paper.

3. Hemi-Slant Lightlike Submanifolds

The concept of pseudoslant (antislant) submanifolds with definite metric is given by Carriazo [10] which was renamed by Sahin as hemi-slant submanifolds and studied their warped product in Kaehler manifolds [11]. In this section, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold for which we need the following lemma from [3].

Lemma 3.1. Let be a -lightlike submanifold of an indefinite Kaehler manifold with constant index such that . Then the screen distribution of lightlike submanifold is Riemannian.

In view of the above lemma, we can define hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold as follows.

Definition 3.2. Let be a -lightlike submanifold of an indefinite Kenmotsu manifold of index with the structure vector field tangent to such that . Then, the submanifold is said to be hemi-slant lightlike submanifold of if the following conditions are satisfied: (i) is a distribution on such that (ii)for all and for each non zero vector field tangent to if and are linearly independent, then the angle between and the vector space is constant, where is the complementary distribution to in the screen distribution .

The angle is called the slant angle of the distribution . A hemi-slant lightlike submanifold is said to be proper if and .

From Definition 3.2, we have the following decomposition:

Denoting as a manifold with its usual Kenmotsu structure given by where are the cartesian coordinates.

The existence of proper hemi-slant lightlike submanifolds in indefinite Kenmotsu manifolds is ensured by the following example.

Example 3.3. Let be a submanifold of defined by where and is a semi-Euclidean space of signature .
Then, a local frame of is given by We see that is a -lightlike submanifold with and which is Riemannian and can be easily proved that is a slant distribution with slant angle . Moreover, the screen transversal bundle is spanned by and the transversal lightlike bundle is spanned by It is not difficult to see that Hence, is a hemi-slant lightlike submanifold of .

For any , we write where and are the tangential and transversal components of , respectively. Similarly, for , we write where and are the tangential and transversal components of , respectively. If the projections on the distributions and are denoted by and , respectively, then any tangent to can be written as We observe that . By applying to (3.11) and using (3.9), we conclude that

The following two theorems are the characterizations of hemi-slant lightlike submanifolds which are similar to the characterization of slant submanifolds in indefinite Hermitian manifolds given by Sahin [12].

Theorem 3.4. The necessary and sufficient condition for a -lightlike submanifold of an indefinite Kenmotsu manifold of index with structure vector field tangent to to be hemi-slant lightlike submanifold is that (i) is a distribution on ; (ii)for any vector field tangent to linearly independent of structure vector field , there exists a constant such that where and is the slant angle of .

Proof. Assume that is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold . Then , from which we have . Hence is a distribution on . In order to prove (ii), we observe that, for any linearly independent of structure vector field and , one has But can also be computed as From (3.15) and (3.16), we arrive at Taking into account that is constant on , from (3.17) we infer that where we have used . Clearly . Converse part directly follows from (i) and (ii).

Theorem 3.5. The necessary and sufficient condition for a -lightlike submanifold of an indefinite Kenmotsu manifold of index with structure vector field tangent to to be hemi-slant lightlike submanifold is that (i) is a distribution on , (ii)for any vector field X tangent to linearly independent of structure vector field , there exists a constant such that
where and is the slant angle of .

Proof. If is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold of index , then is a distribution on . For the proof of (ii), applying to (3.12) and using (3.9) and (3.10), we arrive at Comparing the components of screen distribution on both sides of the above equation, we get In view of Theorem 3.4 and the fact that is a hemi-slant lightlike submanifold, we conclude that Hence (ii) follows from (3.21) and (3.22) together with . Conversely, assume that (i) and (ii) hold. From (ii) and (3.21), we observe that If we take , then . Hence the proof of Theorem follows from Theorem 3.4.

Corollary 3.6. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold . Then for any .

Differentiating (3.12), taking into account that is an indefinite Kenmotsu manifold and then comparing tangential, lightlike transversal and screen transversal parts, we obtain

The integrability conditions of the distributions involved in the definition of hemi-slant lightlike submanifolds are given by the following theorems.

Theorem 3.7. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold of index . Then the distribution is integrable if and only if(i);(ii)for any .

Proof. For any , from (3.26) we have Interchanging and in (3.29) and subtracting (3.29) from the resulting equation, we get Taking inner product of the above equation with for , after simplification, we obtain Also, from (3.28) we have Interchanging the role of and in (3.32) and subtracting (3.32) from the resulting equation, we get Taking inner product of above equation with , we arrive at Hence our assertion follows from (3.31) and (3.34).

Theorem 3.8. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold of index . Then the distribution is integrable if and only if for any .

Proof. For any , from (3.27) we have Interchanging and in (3.36) and subtracting (3.36) from the resulting equation, we obtain which proves our assertion.

In view of the above argument, if we restrict the vector fields and to , then we see that the distribution is not integrable.

The following two theorems deals with the totally geodesic foliations of the leaves of the distributions and .

Theorem 3.9. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold . Then the distribution defines totally geodesic foliation in if and only if has no component in for any .

Proof. Using (2.7), (2.9), (3.12), and (3.14), for any and , we obtain which can be written as Thus, our assertion follows from (3.39).

Theorem 3.10. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold of index . Then the distribution defines totally geodesic foliation in if and only if has no component in for any and .

Proof. For any and , from (2.7), (2.9), (2.12), and (3.12), we obtain Hence our assertion follows from (3.40).

The necessary and sufficient conditions under which the induced connection on a hemi-slant lightlike submanifold immersed in indefinite Kenmotsu manifold to be metric connection is given by the following result.

Theorem 3.11. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifolds . Then the induced connection on is a metric connection if and only if for any and .

Proof. For and , from (3.1), we have Making use of (2.3), (2.7), and (2.9) in the above equation, we arrive at Using (3.9) and (3.10) in (3.43) and comparing the tangential components of the resulting equation, we obtain from which our assertion follows.

Theorem 3.12. Let be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold . Then is never parallel on .

Proof. For , from (2.2) and (2.7), we have Comparing the tangential Components of the above equation, we get Since , so differentiating covariantly with respect to we obtain Using (3.46) in the above equation, we get from which our assertion follows.

Similar to the notion of curvature invariant submanifolds given by Atceken and Kilic [13], we define curvature invariant lightlike submanifolds as follows.

Definition 3.13. Let be a -lightlike submanifold of an indefinite Kenmotsu manifold . Then is said to be curvature invariant lightlike submanifold of if the covariant derivatives of the second fundamental forms and of satisfy for any .

Theorem 3.14. There does not exist any curvature-invariant proper hemi-slant lightlike submanifold of an indefinite Kenmotsu space form with .

Proof. Using (2.15) and (2.16), for any and , we obtain Thus, our assertion follows from (3.50) and (3.51).

4. Minimal Hemi-Slant Submanifolds

In this section, we study minimal hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold. First we recall the following definition of minimal lightlike submanifolds of a semi-Riemannian manifold given by Bejan and Duggal [14].

Definition 4.1. A lightlike submanifold isometrically immersed in a semi-Riemannian manifold is said to be minimal if(i) on and(ii)trace ,
where trace is written with respect to restricted to .

In [14], it has been shown that the above definition is independent of and , but it depends on the choice of transversal bundle .

For the existence of proper minimal hemi-slant lightlike submanifolds, we give the following example.

Example 4.2. Let be a semi-Euclidean space of signature with respect to canonical basis .
Consider an almost contact structure on defined by , , , , , , , , , , , , , .
For some . Let be a submanifold of defined by the following equations: Then the tangent bundle of is spanned by Suppose that the distribution . Then is a 2-lightlike submanifold of . If we take , then is Riemannian vector subbundle and one can easily see that is a slant distribution with slant angle . A direct calculation shows that the screen transversal bundle is spanned by and is spanned by For a vector field tangent to , from Gauss formula and after calculations, we get Hence, is a minimal proper hemi-slant lightlike submanifold of , and the induced connection on is a metric connection. We note that is neither totally geodesic nor totally umbilical.