ISRN Geometry

Volume 2012 (2012), Article ID 251213, 16 pages

doi:10.5402/2012/251213

Research Article

Hemi-Slant Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

S. M. Khursheed Haider, Mamta Thakur, and Advin

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

Received 25 April 2012; Accepted 25 June 2012

Academic Editors: T. Friedrich and M. Khalkhali

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.

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.

Now, we prove a lemma which will be helpful for the study of minimal hemi-slant lightlike submanifolds.

Lemma 4.3. Let be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold such that . If is a local orthonormal basis of , then is an orthonormal basis of .

Proof. Suppose that is a local orthonormal basis of . From (3.25) together with the fact that is Riemannian, we conclude that

where . Hence our assertion follows from (4.6).

The existence of minimal hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold is given by the following two theorems.

Theorem 4.4. Let be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold . Then is minimal if and only if

for any and , where is a basis of and is a basis of .

Proof. Using and (2.2), we obtain . It is known that on [14, Proposition 4.1]. If we take as an orthonormal basis of , then is minimal if and only if

and on . From (2.10) and (2.14), we have

On the other hand, from (2.10), we infer that on if

for and . Thus, our assertion follows from (4.9) and (4.10).

Theorem 4.5. Let be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold such that . Then is minimal if and only if

for , where is a basis of and is a basis of .

Proof. From and (2.2), we obtain . Recall that on [14, Proposition 4.1]. In view of Lemma 4.3, we observe that is an orthonormal basis of . Then, for any and for some function , , we can write

from which we have

for any . Thus our assertion follows from the above equation and Theorem 4.4.

References

D. N. Kupeli, Singular Semi-Riemannian Geometry, Kluwer Academic, Dodrecht, The Netherlands, 1996.

K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.

B. Sahin, “Transversal lightlike submanifolds of indefinite kaehler manifolds,” Analele Universitatii de Vest din Timisoara, Seria Matematica-Informatica, vol. 44, no. 1, pp. 119–145, 2006.

B. Sahin, “Screen transversal lightlike submanifolds of indefinite Kaehler manifolds,” Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1439–1448, 2008.

C. Yildirim and B. Sahin, “Transversal lightlike submanifolds of indefinite Sasakian manifolds,” Turkish Journal of Mathematics, vol. 34, no. 4, pp. 561–583, 2010.

C. Yildirim and B. Sahin, “Screen transversal lightlike submanifolds of indefinite Sasakian manifolds,” Journal of the Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 18, no. 2, pp. 315–336, 2010.

S. M. K. Haider, M. Thakur, and Advin, “Screen transversal lightlike submanifolds of indefinite Sasakian manifolds,” International Journal of Mathematical Analysis, vol. 4, no. 39, pp. 1899–1911, 2010.

K. L. Duggal, “Space time manifolds and contact structures,” International Journal of Mathematics & Mathematical Sciences, vol. 13, no. 3, pp. 545–553, 1990.

N. Aktan, “On non-existence of lightlike hypersurfaces of indefinite Kenmotsu space form,” Turkish Journal of Mathematics, vol. 32, no. 2, pp. 127–139, 2008.

A. Carriazo, New Developments in Slant Submanifolds Theoty, Narosa Publishing, New Delhi, India, 2002.

B. Sahin, “Warped product submanifolds of Kaehler manifolds with a slant factor,” Annales Polonici Mathematici, vol. 95, no. 3, pp. 207–226, 2009.

B. Sahin, “Slant lightlike submanifolds of indefinite Hermitian manifolds,” Balkan Journal of Geometry and its Applications, vol. 13, no. 1, pp. 107–119, 2008.

M. Atceken and E. Kilic, “Semi-invariant lightlike submanifolds of a semi-Riemannian product manifold,” Kodai Mathematical Journal, vol. 30, no. 3, pp. 361–378, 2007.

C. L. Bejan and K. L. Duggal, “Global lightlike manifolds and harmonicity,” Kodai Mathematical Journal, vol. 28, no. 1, pp. 131–145, 2005.