International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 605816, 12 pages

doi:10.1155/2012/605816

Research Article

On GCR-Lightlike Product of Indefinite Cosymplectic Manifolds

Varun Jain,¹ Rakesh Kumar,² and R. K. Nagaich³

¹Department of Mathematics, Multani Mal Modi College, Patiala 147001, India

²Department of Basic & Applied Sciences, University College of Engineering, Punjabi University Patiala, Patiala 147002, India

³Department of Mathematics, Punjabi University Patiala, Patiala 147002, India

Received 26 March 2012; Revised 15 May 2012; Accepted 10 July 2012

Academic Editor: Hernando Quevedo

Copyright © 2012 Varun Jain 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 define GCR-lightlike submanifolds of indefinite cosymplectic manifolds and give an example. Then, we study mixed geodesic GCR-lightlike submanifolds of indefinite cosymplectic manifolds and obtain some characterization theorems for a GCR-lightlike submanifold to be a GCR-lightlike product.

1. Introduction

To fill the gaps in the general theory of submanifolds, Duggal and Bejancu [1] introduced lightlike (degenerate) geometry of submanifolds. Since the geometry of -submanifolds has potential for applications in mathematical physics, particularly in general relativity, and the geometry of lightlike submanifolds has extensive uses in mathematical physics and relativity, Duggal and Bejancu [1] clubbed these two topics and introduced the theory of -lightlike submanifolds of indefinite Kaehler manifolds and then Duggal and Sahin [2], introduced the theory of -lightlike submanifolds of indefinite Sasakian manifolds, which were further studied by Kumar et al. [3]. But -lightlike submanifolds do not include the complex and real subcases contrary to the classical theory of -submanifolds [4]. Thus, later on, Duggal and Sahin [5] introduced a new class of submanifolds, generalized-Cauchy-Riemann- (GCR-) lightlike submanifolds of indefinite Kaehler manifolds and then of indefinite Sasakian manifolds in [6]. This class of submanifolds acts as an umbrella of invariant, screen real, contact -lightlike subcases and real hypersurfaces. Therefore, the study of -lightlike submanifolds is the topic of main discussion in the present scenario. In [7], the present authors studied totally contact umbilical -lightlike submanifolds of indefinite Sasakian manifolds.

In present paper, after defining -lightlike submanifolds of indefinite cosymplectic manifolds, we study mixed geodesic -lightlike submanifolds of indefinite cosymplectic manifolds. In [8, 9], Kumar et al. obtained some necessary and sufficient conditions for a -lightlike submanifold of indefinite Kaehler and Sasakian manifolds to be a -lightlike product, respectively. Thus, in this paper, we obtain some characterization theorems for a -lightlike submanifold of indefinite cosymplectic manifold to be a -lightlike product.

2. Lightlike Submanifolds

Let be a real -dimensional vector space with a symmetric bilinear mapping . The mapping is called degenerate on if there exists a vector of such that

otherwise is called nondegenerate. It is important to note that a non-degenerate symmetric bilinear form on may induce either a non-degenerate or a degenerate symmetric bilinear form on a subspace of . Let be a subspace of and    degenerate; then is called a degenerate (lightlike) subspace of .

Let be a real -dimensional semi-Riemannian manifold of constant index such that , , and let be an -dimensional submanifold of and the induced metric of on . Thus, if is degenerate on the tangent bundle of , then is called a lightlike (degenerate) submanifold of (for detail see [1]). For a degenerate metric on , is also a degenerate -dimensional subspace of . Thus, both and are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace , which is known as radical (null) subspace. If the mapping defines a smooth distribution on of rank , then the submanifold of is called an -lightlike submanifold and is called the radical distribution on . Then, there exists a non-degenerate screen distribution which is a complementary vector subbundle to in . Therefore,

where denotes orthogonal direct sum. Let , called screen transversal vector bundle, be a non-degenerate complementary vector subbundle to in . Let and be complementary (but not orthogonal) vector bundles to in and to in , called transversal vector bundle and lightlike transversal vector bundle of , respectively. Then, we have

Let be a local coordinate neighborhood of and consider the local quasiorthonormal fields of frames of along on as , where are local lightlike bases of and and are local orthonormal bases of and , respectively. For these quasiorthonormal fields of frames, we have the following theorem.

Theorem 2.1 (see [1]). Let be an -lightlike submanifold of a semi-Riemannian manifold . Then there, exist a complementary vector bundle ltr(TM) of Rad TM in and a basis of consisting of smooth section of , where is a coordinate neighborhood of , such that

where is a lightlike basis of .

Let be the Levi-Civita connection on . Then, according to decomposition (2.4), the Gauss and Weingarten formulas are given by

for any and , where and belong to and , respectively. Here is a torsion-free linear connection on , is a symmetric bilinear form on that is called second fundamental form, and is a linear operator on , known as shape operator.

According to (2.3), considering the projection morphisms and of on and , respectively, then (2.6) gives

where we put , .

As and are -valued and -valued, respectively, they are called the lightlike second fundamental form and the screen second fundamental form on . In particular,

where , and . By using (2.3)-(2.4) and (2.7)-(2.8), we obtain

for any , , and .

Let be the projection morphism of on . Then, using (2.2), we can induce some new geometric objects on the screen distribution on as

for any and , where and belong to and , respectively. and are linear connections on complementary distributions and , respectively. Then, using (2.7), (2.8), and (2.11), we have

Next, an odd-dimensional semi-Riemannian manifold is said to be an indefinite almost contact metric manifold if there exist structure tensors , where is a tensor field, is a vector field called structure vector field, is a -form, and is the semi-Riemannian metric on satisfying (see [10])

for any .

An indefinite almost contact metric manifold is called an indefinite cosymplectic manifold if (see [11])

3. Generalized Cauchy-Riemann Lightlike Submanifolds

Calin [12] proved that if the characteristic vector field is tangent to , then it belongs to . We assume that the characteristic vector is tangent to throughout this paper. Thus, we define the generalized Cauchy-Riemann lightlike submanifolds of an indefinite cosymplectic manifold as follows.

Definition 3.1. Let be a real lightlike submanifold of an indefinite cosymplectic manifold such that the structure vector field is tangent to ; then is called a generalized-Cauchy-Riemann- (GCR-) lightlike submanifold if the following conditions are satisfied:

(A)there exist two subbundles and of such that

(B)there exist two subbundles and of such that

where is invariant nondegenerate distribution on , is one-dimensional distribution spanned by , and and are vector subbundles of and , respectively.

Therefore, the tangent bundle of is decomposed as

A contact -lightlike submanifold is said to be proper if , and . Hence, from the definition of -lightlike submanifolds, we have that

(a)condition (A) implies that ,(b)condition (B) implies that and , and thus and . (c)any proper -dimensional contact -lightlike submanifold is -lightlike, (d)(a) and contact distribution imply that index .

The following proposition shows that the class of -lightlike submanifolds is an umbrella of invariant, contact and contact -lightlike submanifolds.

Proposition 3.2. A -lightlike submanifold of an indefinite cosymplectic manifold is contact -submanifold (resp., contact -lightlike submanifold) if and only if (resp., ).

Proof. Let be a contact -lightlike submanifold; then is a distribution on such that . Therefore, and . Since , this implies that . Conversely, suppose that is a -lightlike submanifold of an indefinite Cosymplectic manifold such that . Then, from (3.1), we have , and therefore . Hence, is a vector subbundle of . This implies that is a contact -lightlike submanifold of an indefinite cosymplectic manifold. Similarly the other assertion follows.

The following construction helps in understanding the example of -lightlike submanifold. Let be with its usual Cosymplectic structure and given by

where are the Cartesian coordinates.

Example 3.3. Let be a semi-Euclidean space and a -dimensional submanifold of that is given by

where is of signature with respect to the canonical basis . Then, the local frame of is given by

Hence, is a -lightlike as . Also, and ; these imply that and , respectively. Since ,  . By straightforward calculations, we obtain

where ; this implies that . Moreover, the lightlike transversal bundle is spanned by

where and . Hence, . Therefore, . Thus, is a -lightlike submanifold of .

Let , , be the projection morphism on , , , respectively; therefore

for . Applying to (3.9), we obtain

where , , and , or, we can write (3.10) as

where and are the tangential and transversal components of , respectively.

Similarly,

where and are the sections of and , respectively. Differentiating (3.10) and using (2.8)–(2.10) and (3.12), we have

for all . By using, cosymplectic property of with (2.7), we have the following lemmas.

Lemma 3.4. Let be a -lightlike submanifold of an indefinite cosymplectic manifold ; then one has

where and

Lemma 3.5. Let be a -lightlike submanifold of an indefinite cosymplectic manifold ; then one has

where and and

4. Mixed Geodesic GCR-Lightlike Submanifolds

Definition 4.1. A -lightlike submanifold of an indefinite cosymplectic manifold is called mixed geodesic -lightlike submanifold if its second fundamental form satisfies , for any and .

Definition 4.2. A -lightlike submanifold of an indefinite cosymplectic manifold is called geodesic -lightlike submanifold if its second fundamental form satisfies , for any .

Theorem 4.3. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, is mixed geodesic if and only if and , for any and .

Proof. Using, definition of -lightlike submanifolds, is mixed geodesic if and only if , for , and . Using (2.8) and (2.11), we get

Therefore, from (4.1), the proof is complete.

Theorem 4.4. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, is geodesic if and only if and , for any , and .

Proof. The proof is similar to the proof of Theorem 4.3.

Lemma 4.5. Let be a mixed geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Then , for any , .

Proof. For and , using (2.7) we have

Since is mixed geodesic, we obtain . Here, using (2.11), we get , and then, by virtue of (3.11), we obtain . Comparing the transversal components, we get ; this implies that

If , then the nondegeneracy of implies that there must exist a such that . But using the hypothesis that is a mixed geodesic with (2.7) and (2.11), we get

Therefore,

Also using (2.13), and (2.15), we get

Therefore,

Hence, from (4.3), (4.5), and (4.7), the result follows.

Corollary 4.6. Let be a mixed geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Then, , for any and .

Proof. The result follows from (2.12) and Lemma 4.5.

Theorem 4.7. Let be a mixed geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Then, and , for any and .

Proof. Since is mixed geodesic -lightlike submanifold for any , and thus (2.6) implies that

Since is an anti-invariant distribution there exists a vector field such that . Thus, from (2.8), (2.14), (3.11), and (3.12), we get

Comparing the transversal components, we get . Since and , this implies that and . Hence, and .

5. GCR-Lightlike Product

Definition 5.1. -lightlike submanifold of an indefinite cosymplectic manifold is called -lightlike product if both the distributions and define totally geodesic foliation in .

Theorem 5.2. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, the distribution define a totally geodesic foliation in if and only if , for any .

Proof. Since , defines a totally geodesic foliation in if and only if , for any , , and . Using (2.7) and (2.14), we have

Hence, from (5.1) and (5.2), the assertion follows.

Theorem 5.3. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . Then, the distribution defines a totally geodesic foliation in if and only if has no component in and has no component in , for any and .

Proof. From the definition of a -lightlike submanifold, we know that defines a totally geodesic foliation in if and only if

for and . Using (2.7) and (2.8), we have

Using (2.7), (2.15), and (2.14), we obtain

Thus, from (5.4)–(5.7), the result follows.

Theorem 5.4. Let be a -lightlike submanifold of an indefinite cosymplectic manifold . If , then is a lightlike product.

Proof. Let ; therefore . Then using (3.15) with the hypothesis, we get . Therefore the distribution defines a totally geodesic foliation. Next, let ; therefore . Then using (3.14), we get . Therefore, defines a totally geodesic foliation in . Hence, is a lightlike product.

Definition 5.5. A lightlike submanifold of a semi-Riemannian manifold is said to be an irrotational submanifold if , for any and . Thus, is an irrotational lightlike submanifold if and only if and .

Theorem 5.6. Let be an irrotational -lightlike submanifold of an indefinite cosymplectic manifold . Then, is a lightlike product if the following conditions are satisfied:

(A), (B).

Proof. Let hold; then, using (2.8), we get , , and for . These equations imply that the distribution defines a totally geodesic foliation in , and, with (2.9), we get . Hence, the non degeneracy of implies that . Therefore, has no component in . Finally, from (2.10) and the hypothesis that is irrotational, we have , for and . Assume that holds; then . Therefore, has no component in . Thus, the distribution defines a totally geodesic foliation in . Hence, is a lightlike product.

Definition 5.7 (see [13]). If the second fundamental form of a submanifold, tangent to characteristic vector field , of a Sasakian manifold is of the form

for any , where is a vector field transversal to , then is called a totally contact umbilical submanifold of a Sasakian manifold.

Theorem 5.8. Let be a totally contact umbilical -lightlike submanifold of an indefinite cosymplectic manifold . Then, is a -lightlike product if , for any .

Proof. Let ; then the hypothesis that implies that the distribution defines a totally geodesic foliation in .

If we assume that , then, using (3.14), we have , and taking inner product with and using (2.6) and (2.14), we obtain

where . For any from (3.14), we have . Therefore, using the hypothesis with (5.8), we get ; this implies that , and thus (5.9) becomes . Then, the nondegeneracy of the distribution implies that the distribution defines a totally geodesic foliation in . Hence, the assertion follows.

Theorem 5.9. Let be a totally geodesic -lightlike submanifold of an indefinite cosymplectic manifold . Suppose that there exists a transversal vector bundle of which is parallel along with respect to Levi-Civita connection on , that is, , for any , . Then, is a -lightlike product.

Proof. Since is a totally geodesic -lightlike , for ; this implies defines a totally geodesic foliation in .

Next implies , and hence, by Theorem 5.3, the distribution defines a totally geodesic foliation in . Hence, the result follows.

Acknowledgment

The authors would like to thank the anonymous referee for his/her comments that helped them to improve this paper.

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