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ISRN Geometry
Volume 2012 (2012), Article ID 804520, 17 pages
http://dx.doi.org/10.5402/2012/804520
Research Article

Geodesic Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India

Received 16 September 2012; Accepted 3 October 2012

Academic Editors: A. Ferrandez, A. Fino, T. Friedrich, and A. Morozov

Copyright © 2012 S. M. Khursheed Haider et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of the present paper is to study geodesic contact screen Cauchy Riemannian (SCR-) lightlike submanifolds, geodesic screen transversal lightlike, and geodesic transversal lightlike submanifolds of indefinite Kenmotsu manifolds.

1. Introduction

The study of the geometry of submanifolds of a Riemannian or semi-Riemannian manifold is one of the interesting topics of differential geometry. Despite of some similarities between semi-Riemannian manifolds and Riemannian manifolds, the lightlike submanifolds are different since their normal vector bundle intersect with the tangent bundle making it more interesting and difficult to study. These submanifolds were introduced and studied by Duggal and Bejancu [1]. On the other hand, geodesic CR-lightlike submanifolds in Kähler manifolds were studied by Sahin and Gunes [2], and geodesic lightlike submanifolds of indefinite Sasakian manifolds were investigated by Dong and Liu [3]. In 2006, Sahin [4] initiated the study of transversal lightlike submanifolds of an indefinite Kähler manifold which are different from CR-lightlike [1] and screen CR-lightlike submanifolds [5]. Recently, Sahin [6] introduced the notion of screen transversal lightlike submanifolds of indefinite Kähler manifolds and obtained some useful results. In this paper, we study geometric conditions under which some lightlike submanifolds of an indefinite Kenmotsu manifold are totally geodesic.

2. Preliminaries

We follow [1] for the notation and fundamental equations for lightlike submanifolds used in this paper. A submanifold immersed in a semi-Riemannian manifold is called a lightlike submanifold if it admits a degenerate metric induced from whose radical distribution is of rank , where and 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 complementary vector bundle of in . Since for any local basis of , there exists a local null frame of sections with values in the orthogonal complement of in such that and , it follows that there exists a lightlike transversal vector bundle locally spanned by [1, page 144]. Let be the complementary (but not orthogonal) vector bundle to in .

Then, Let be the Levi-Civita connection on . Then, in view of the decomposition (2.3), the Gauss and Weingarten formulas are given by where and belong to and , respectively, and are linear connection on and on the vector bundle , respectively. Moreover, we have for all, , and . If we denote the projection of on by , then by using (2.6)–(2.8) and the fact that is a metric connection, we obtain From the decomposition of the tangent bundle of a lightlike submanifold, we have for any , and . By using above equation we obtain

An odd dimensional semi-Riemannian manifold is said to be an indefinite contact metric manifold [7] if there exists a (1,1) tensor field , a vector field , called the characteristic vector field, and its 1-form satisfying where . It is not difficult to show that .

An indefinite almost contact metric manifold is said to be an indefinite Kenmotsu manifold [8] if for any .

Without loss of generality, we take . For any vector field tangent to , we put where and are the tangential and transversal parts of , respectively.

For any , we have where and are the tangential and transversal parts of , respectively.

From now on, we denote by in this paper.

3. Geodesic Contact -Lightlike Submanifolds

In this section, we study the geometric conditions under which the distributions involved in the definition of a contact -lightlike submanifold and the submanifold itself are totally geodesic. We recall the following definition of contact -lightlike submanifold of an indefinite Kenmotsu manifold given by Duggal and Sahin [5].

Definition 3.1. A lightlike submanifold , tangent to structure vector field , immersed in an indefinite Kenmotsu manifold is said to be contact -lightlike submanifold of if the following conditions are satisfied. (a) There exists real nonnull distribution and such that

, where is the orthogonal complementary to in .(b) .

The tangent bundle of a contact -lightlike submanifold is decomposed as We will use the symbol to denote the orthogonal complement of in .

Definition 3.2. A contact -lightlike submanifold of an indefinite Kenmostsu manifold is said to be(i)-totally geodesic contact -lightlike submanifold if for any ,(ii)mixed totally geodesic contact -lightlike submanifold if for any and .

Let be a contact -lightlike submanifold of indefinite Kenmotsu manifold and let and be the projection morphisms on and , respectively. Then for any vector field tangent to , we can write Applying to (3.3) and using (2.19), we obtain If we denote by and by , then (3.4) can be rewritten as where and .

For any , we have where and .

In view of the above arguments, we are in a position to prove the following characterization theorem for the existence of a -totally geodesic contact SCR-lightlike submanifold immersed in indefinite Kenmotsu manifolds.

Theorem 3.3. Let be a contact lightlike submanifold of an indefinite Kenmotsu manifolds . Then is -totally geodesic if and only if(i).(ii) and for any .

Proof. Suppose that the contact -lightlike submanifold is totally geodesic. Then we see that and   for all.
Also, from (2.6) and (2.15), we obtain from which we derive where we have used (2.8), (2.18), and (3.5). Using (2.9) in the above equation, we get On the other hand, making use of (2.6), (2.8), (2.15), (2.18), (3.5), and (3.6), we arrive at Hence, (i) and (ii) follows from (3.9) and (3.10) together with the fact that and .
Converse part directly follow from (3.9) and (3.10).

The necessary and sufficient conditions for contact SCR-lightlike submanifolds to be mixed totally geodesic is given by the following theorem.

Theorem 3.4. Let be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold . Then is mixed totally geodesic if and only if(i), (ii) and for any and .

Proof. Assume that is mixed totally geodesic. Then , for any and .
Moreover, using (2.6), (2.8), (2.15), (2.18), and (3.5) a direct calculation shows that From (2.9) and (3.11), we have On the other hand, using (2.6), (2.8), (2.15), (2.18), (3.5), and (3.6), we derive Thus, (i) and (ii) follow from (3.12) and (3.13) along with .

Converse part directly follows from (3.12) and (3.13).

4. Screen Transversal Lightlike Submanifolds

We begin this section by recalling the following definitions from [6].

Definition 4.1. An -lightlike submanifold of an indefinite Kenmotsu manifold is said to be screen transversal () lightlike submanifold of if there exists a screen transversal bundle such that

Definition 4.2. An -lightlike submanifold of an indefinite Kenmotsu manifold is said to be (i)radical -lightlike submanifold if is invariant with respect to .(ii)-anti-invariant lightlike submanifold if is screen transversal with respect to , that is,

For a radical screen transversal lightlike submanifolds immersed in indefinite Kenmotsu manifold , we will denote the projection morphisms of and by and , respectively. Then for , we can write We apply to (4.3) and then using (2.19), we obtain Denoting by and by , (4.4) can be we rewritten as where and .

For , we write where , and  ( is the orthogonal complement of in ).

The geometric conditions under which the distribution is totally geodesic is given by the following theorem.

Theorem 4.3. Let be a radical screen transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then is totally geodesic if and only if has no component in .

Proof. If the distribution is totally geodesic, then for any and on [9]. On the other hand, using (2.6), (2.15), (2.18), and (4.6), we arrive at Thus, our assertion follows from (4.8) and (4.7).
Converse part directly follows from (4.8) and (4.7).

For the screen distribution to be totally geodesic in , we have the following.

Theorem 4.4. Let be a radical screen transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then, is totally geodesic if and only if   for all    and  .

Proof. Suppose that the distribution is totally geodesic. Then for any , and . Using (2.6), (2.8), (2.15), (2.18), and (4.5), a direct calculation shows that On the other hand, from (2.6), (2.7), (2.12), (2.15), (2.18), (4.5), and (4.6), we obtain Thus, our assertion follows from (4.9),(4.10),  and (4.11).
Converse part directly follows from (4.10) and (4.11).

In respect of a radical screen transversal lightlike submanifold to be mixed totally geodesic and totally geodesic, we have the following two theorems.

Theorem 4.5. Let be a radical screen transversal lightlike submanifold of . Then is mixed totally geodesic if and only if and .

Proof. Assume that the submanifold is mixed geodesic. Then for any , and . Also, from (2.14) we have On the other hand, by the use of (2.6), (2.8), (2.15), (2.18), and (4.6), we obtain Thus, our assertion follows from (4.12) and (4.14).
Converse part directly follows from (4.14) and the fact that .

Theorem 4.6. Let be a radical screen transversal lightlike submanifold of . Then is totally geodesic if and only if(i) and ,(ii)  and  for any and .

Proof. If the submanifold is totally geodesic, then for any , and . Also, making use of (2.6), (2.8), (2.15), (2.17), (2.18), and (4.5), a direct calculation shows that On the other hand, from (2.6), (2.8), (2.11), (2.15), (2.18), (4.5), and (4.6), we obtain Thus, (i) and (ii) follow from (4.16), (4.17), and (4.15).
Converse part directly follows from (4.16) and (4.17).
For a -anti-invariant lightlike submanifold immersed in , if we denote the projection morphism of and by and , respectively, then for any vector field tangent to we can write By applying to (4.18) and then using (2.19), we obtain
Denoting by and by . Then (4.19) can be rewritten as where and .
For , writing where , and ( is the orthogonal complement of in ).

In view of the above discussions, the conditions under which the distribution of a ST-anti-invariant lightlike submanifold immersed in indefinite Kenmotsu manifolds to be totally geodesic is given by the following result.

Theorem 4.7. Let be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold . Then the distribution is totally geodesic if and only if and , for any .

Proof. Suppose that the distribution is totally geodesic. Then we see that for any and . Recall that on [9]. On the other hand, by the use of (2.6), (2.8), (2.15), (4.20), and (4.21), we obtain Thus, our assertion follows from (4.22) and (4.23).
Converse part directly follows from (4.22) and (4.23).

For the screen distribution of a ST-anti-invariant lightlike submanifold to be totally geodesic, we have the following.

Theorem 4.8. Let be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold . Then the distribution is totally geodesic if and only if and , for any .

Proof. If is totally geodesic, then for any , and .
On the other hand, using (2.6), (2.15), (2.18), and (4.20), we obtain Also, where we have used (2.6), (2.8), (2.15), (2.18), (4.20), and (4.21). Thus, our assertion follows from (4.25), (4.26), and (4.24).
Converse part directly follows from (4.25) and (4.26).

The necessary and sufficient conditions for a ST-anti-invariant lightlike submanifold to be mixed totally geodesic, we have the following.

Theorem 4.9. Let be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold . Then is mixed totally geodesic if and only if and . for any and .

Proof. Assume that the submanifold is mixed gedesic. Then for any , and . By virtue of (2.14), we have On the other hand, using (2.6), (2.8), (2.15), (2.18), (4.20), and (4.21), we get Thus, our assertion follows from (4.27) and (4.29).
Converse part directly follows from (4.29).

Now, we prove the following.

Theorem 4.10. Let be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold . Then is totally geodesic if and only if and , for any .

Proof. The submanifold is totally geodesic if and only if for any , and . By the use of (2.6), (2.8), (2.15), (2.18), and (4.20), we obtain On the other hand, from (2.6), (2.8), (2.15), (4.20), and (4.21), we have Thus, our assertion follows from (4.31), (4.32), and (4.30).
Converse part directly follows from (4.31) and (4.32).

5. Transversal Lightlike Submanifolds

The purpose of this section is to study transversal and radical transversal lightlike submanifolds in an indefinite Kenmotsu manifold. We recall here the definitions of these submanifolds given by Yıldırım and Sahin [10].

Definition 5.1. A lightlike submanifold tangent to structure vector field immersed in an indefinite Kenmotsu manifold is said to be(i)radical transversal lightlike submanifold of if (ii)transversal lightlike submanifold of if
For a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold , if and are the projection morphism on and , respectively, then any vector field tangent to can be written as We apply to (5.3) and then using (2.19), we get If we denote by and by , then (5.4) can be rewritten as where and .
Moreover, if , then from which we observe that .

Using the above notations, one can prove the following.

Theorem 5.2. Let be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then the distribution is totally geodesic if and only if for any and .

Proof. Since on [9], we observe that the distribution is totally geodesic if and only if for any and . On the other hand, using (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we arrive at Thus, our assertion follows from (5.7) and (5.8).
Converse part directly follows from (5.8).

A screen distribution of a radical transversal lightlike submanifold in indefinite Kenmotsu manifolds to be totally geodesic, we have the following.

Theorem 5.3. Let be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then the distribution is totally geodesic if and only if and for any and .

Proof. We note that the distribution is totally geodesic if and only if for any , and . Making use of (2.6), (2.11), (2.15), (2.18), and (5.5), we get On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we have Thus, our assertion follows from (5.10), (5.11), and (5.9).
Converse part directly follows from (5.10) and (5.11).

The conditions under which a radical transversal lightlike submanifold immersed in indefinite Kenmotsu manifolds to be mixed totally geodesic is given by the following theorem.

Theorem 5.4. Let be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then is mixed totally geodesic if and only if for any and .

Proof. The submanifold is mixed totally geodesic if and only if for any , and . From (2.14), we have On the other hand, using (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we obtain Thus, our assertion follows from (5.12) and (5.14).
Converse part directly follows from (5.14).

Theorem 5.5. Let be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then the submanifold is totally geodesic if and only if (i).(ii)
for any .

Proof. We observe that the submanifold is totally geodesic if and only if for any , and .
By the use of (2.6), (2.8), (2.11), (2.15), (2.18), (5.5), and (5.6), we arrive at On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we have Thus, our assertion follows from (5.15), (5.16), and (5.17).
Converse part directly follows from (5.16), and (5.17).

If we denote the projections on the distributions and involved with the definition of a transversal lightlike submanifold immersed in indefinite Kenmotsu manifold by and , respectively, then any vector field tangent to can be written as Applying to (5.18) and then using (2.19), we get If we denote by and by , then (5.19) can be written as where and .

For , we have where and ( is the orthogonal complement of in ).

Theorem 5.6. Let be a transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then the distribution is totally geodesic if and only if and for any .

Proof. The distribution is totally geodesic if and only if for any . In view of on [9], we have On the other hand, making use of (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we get Thus, our assertion follows from (5.22), (5.23), and (5.24).
Converse part directly follows from (5.23) and (5.24).

Theorem 5.7. Let be a transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then is totally geodesic if and only if for any .

Proof. We note that the distribution is totally geodesic if and only if and for any , and .
Combining (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we obtain On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.20), and (5.21), we have Thus, our assertion follows from (5.25), (5.26), and (5.27).
Converse part directly follows from (5.26) and (5.27).

Theorem 5.8. Let be a transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then is mixed totally geodesic if and only if and for any and .

Proof. We observe that the submanifold is mixed totally geodesic if and only if for all and .
From (2.14), we infer that On the other hand, by the use of (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we arrive at Thus, our assertion follows from (5.28), (5.29), and (5.30).
Converse part directly follows from (5.29) and (5.30).

Theorem 5.9. Let be a transversal lightlike submanifold of an indefinite Kenmotsu manifold . Then is totally geodesic if and only and for all .

Proof. The submanifold is totally geodesic if and only if for any and .
By virtue of (2.6), (2.7), (2.8), (2.15), (2.18), (5.20) and (5.21), we have On the other hand, by the use of (2.6), (2.7), (2.8), (2.15), (2.18), (5.20), and (5.21), we get Thus, our assertion follows from (5.31), (5.32), and (5.33).
Converse part directly follows from (5.32), and (5.33).

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