Advances in Mathematical Physics

Volume 2011, Article ID 983069, 12 pages

http://dx.doi.org/10.1155/2011/983069

*GCR*-Lightlike Product of Indefinite Sasakian Manifolds

^{1}University College of Engineering, Punjabi University, Patiala 147002, India^{2}Department of Applied Sciences, Bhai Gurdas Institute of Engineering & Technology, Sangrur, Punjab 148001, India^{3}Department of Mathematics, Punjabi University, Patiala 147002, India

Received 21 March 2011; Accepted 12 April 2011

Academic Editor: Klaus Kirsten

Copyright © 2011 Rakesh Kumar 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 study mixed geodesic *GCR*-lightlike submanifolds of indefinite Sasakian manifolds and obtain some necessary and sufficient conditions for a *GCR*-lightlike submanifold to be a *GCR*-lightlike product.

#### 1. Introduction

The geometry of submanifolds is one of the most important topics of differential geometry. It is well known that the geometry of semi-Riemannian submanifolds have many similarities with their Riemannian case but the geometry of lightlike submanifolds is different since their normal vector bundle intersect with the tangent bundle making it more difficult and interesting to study. The lightlike submanifolds of semi-Riemannian manifolds were introduced and studied by Duggal and Bejancu [1]. Since contact geometry has vital role in the theory of differential equations, optics, and phase spaces of a dynamical system, therefore contact geometry with definite and indefinite metric becomes the topic of main discussion. In the process of establishment of theory of lightlike submanifolds, Duggal and Sahin [2] introduced the theory of contact -lightlike submanifold of indefinite Sasakian manifold. Later on, Duggal and Sahin [3] introduced generalized Cauchy-Riemann -lightlike submanifold of indefinite Sasakian manifolds to find an umbrella of invariant, screen, real, contact -lightlike subcases, and real hypersurfaces. In the present paper we study -lightlike submanifolds extensively and study mixed geodesic -lightlike submanifolds of indefinite Sasakian manifolds. We also obtain some necessary and sufficient conditions for a -lightlike submanifold to be a -lightlike product.

#### 2. Lightlike Submanifolds

We recall notations and fundamental equations for lightlike submanifolds, which are due to the book [1] by Duggal and Bejancu.

Let be a real -dimensional semi-Riemannian manifold of constant index such that , and be an -dimensional submanifold of and be the induced metric of on . If is degenerate on the tangent bundle of then is called a lightlike submanifold of . For a degenerate metric on is 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 .

Let be a screen distribution which is a semi-Riemannian complementary distribution of in , that is and is a complementary vector subbundle to in . Let and be complementary (but not orthogonal) vector bundles to in and to in , 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 are local orthonormal bases of and , respectively. For this quasi-orthonormal fields of frames, we have the following.

Theorem 2.1 (see [1]). *Let be an -lightlike submanifold of a semi-Riemannian manifold . Then there exists a complementary vector bundle of 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 the decomposition (2.5), the Gauss and Weingarten formulas are given by

where and belong to and , respectively. Here is a torsion-free linear connection on , is a symmetric bilinear form on which is called the second fundamental form, and is linear a operator on and called a shape operator.

According to (2.4), considering the projection morphisms and of on and , respectively, (2.11) and (2.12) give where one puts , .

As and are -valued and -valued, respectively, therefore, these are called as the lightlike second fundamental form and the screen second fundamental form on . In particular where and . Using (2.4)-(2.5) and (2.8)–(2.12), one obtains for any , and .

Let be the projection morphism of on . Then using (2.3), one 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. and are -valued and -valued bilinear forms and called as the second fundamental forms of distributions and , respectively, and one has the following equations:

Next, one recalls some basic definition and results of indefinite Sasakian manifolds [4]. An odd dimensional semi-Riemannian manifolds is called an contact metric manifold, if there is a tensor field , a vector field called characteristic vector field, and a 1-form such that where . Therefore it follows that Then is called contact metric structure of . one says that has a normal contact structure if , where is the Nijenhuis tensor field then is called an indefinite Sasakian manifold and for which one has

#### 3. Generalized Cauchy-Riemann- (GCR-) Lightlike Submanifold

Calin [5] proved that if the characteristic vector field is tangent to then it belongs to . We assume characteristic vector is tangent to , throughout this paper.

*Definition 3.1. *Let be a real lightlike submanifold of an indefinite Sasakian manifold then is called generalized Cauchy-Riemann -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 , are vector subbundles of and , respectively.

Then tangent bundle of is decomposed as Let , , be the projection morphism on , , , respectively, therefore for . Applying to (3.4), we obtain where , , and and we can write (3.5) as where and are the tangential and transversal components of , respectively. Similarly where and are the sections of and , respectively.

Differentiating (3.5) and using (2.11)–(2.13), (2.14), and (3.7), we have for all . Using Sasakian property of with (2.13) and (2.14), we have the following lemmas.

Lemma 3.2. *Let be a -lightlike submanifold of an indefinite Sasakian manifold . Then one has
**
where and
*

Lemma 3.3. *Let be a -lightlike submanifold of an indefinite Sasakian manifold . Then one has
**
where , and
*

#### 4. Mixed Geodesic GCR-Lightlike Submanifolds

*Definition 4.1. *A -lightlike submanifold of an indefinite Sasakian 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 Sasakian 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 Sasakian 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 (2.12) and (2.17) we get
Therefore from (4.1), the proof is complete.

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

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

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

*Proof. *For and , using (2.9) we have
Since is mixed geodesic therefore . Using (2.16) and (2.17) we get then using (3.6) we obtain . Comparing the transversal components we get
or
If then the nondegeneracy of implies that there must exists a such that
But from (2.9) and (2.17) we get
Therefore
Also using (2.20), (2.21), and (2.24), we get
Therefore
Hence from (4.4), (4.7), and (4.9) the result follows.

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

*Proof. *From (2.18) and above lemma, the result follows.

Lemma 4.7. *Let be a -lightlike submanifold of an indefinite Sasakian manifold . Then , for any and .*

*Proof. *Using (2.12), we have
for and .

Theorem 4.8. *Let be a mixed geodesic -lightlike submanifold of an indefinite Sasakian manifold . Then and , for any and .*

*Proof. *Since is mixed geodesic, therefore for any , therefore (2.7) gives
Since is anti-invariant there exist such that . Thus from (2.12), (3.6), and (3.7) we get
Comparing transversal components we get
since and , hence and .

#### 5. GCR-Lightlike Product

*Definition 5.1. *A -lightlike submanifold of an indefinite Sasakian manifold is called -lightlike product if both the distributions and defines totally geodesic foliation in .

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

*Proof. *Since , therefore defines a totally geodesic foliation in if and only if
for any , and . Using (2.9) and (2.25), we have
Hence, from (5.2) the assertion follows.

Theorem 5.3. *Let be a -lightlike submanifold of an indefinite Sasakian 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. *We know that defines a totally geodesic foliation in if and only if
for , , , and . Using (2.9) and (2.11) we have
Using (2.9), (2.10), and (2.25) we obtain
also
Thus from (5.4)–(5.6), the result follows.

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

*Proof. *Let therefore , then using (3.11) with the hypothesis, we get therefore the distribution defines a totally geodesic foliation.Let therefore then using (3.9), we get
Equating components along the distribution of above equation, we get , therefore define 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 .

Theorem 5.6. *Let be an irrotational -lightlike submanifold of indefinite Sasakian manifold . Then is a lightlike product if the following conditions are satisfied. *(A)*. *(B)*. *

*Proof. *Let holds, then using (2.11) and (2.12), we get , and for . These equations imply that the distribution defines a totally geodesic foliation in and with (2.13), we get . Hence non degeneracy of implies that . Therefore has no component in . Finally, from (2.14) and being irrotational, we have , for and . Assume 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 [6]). *If the second fundamental form of a submanifold tangent to characteristic vector field , of an indefinite Sasakian manifold is of the form
for any , where is a vector field transversal to , then is called a totally contact umbilical submanifold of an indefinite Sasakian manifold.

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

*Proof. *Let , then the hypothesis , implies that the distribution defines totally geodesic foliation in .Let then using (3.9), we have
let and using (2.8) and (2.25), then above equation becomes
where . For from (3.10), we get
Using the hypothesis with (5.8), we get , this implies therefore, (5.10) becomes . Then non degeneracy 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 Sasakian manifold . Suppose 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, therefore , for , this implies defines a totally geodesic foliation in . Next, implies . Hence by Theorem 5.3, the distribution defines a totally geodesic foliation in . Thus is a contact -lightlike product.

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