ISRN Geometry

Volume 2013 (2013), Article ID 231869, 8 pages

http://dx.doi.org/10.1155/2013/231869

Research Article

Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

Rashmi Sachdeva,¹ Rakesh Kumar,² and Satvinder Singh Bhatia¹

¹School of Mathematics and Computer Applications, Thapar University, Patiala 147004, India

²University College of Engineering, Punjabi University, Patiala 147002, India

Received 1 April 2013; Accepted 12 May 2013

Academic Editors: A. Fino, C. Qu, and E. H. Saidi

Copyright © 2013 Rashmi Sachdeva 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 totally contact umbilical slant lightlike submanifolds of indefinite cosymplectic manifolds. We prove that there do not exist totally contact umbilical proper slant lightlike submanifolds in indefinite cosymplectic manifolds other than totally contact geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally we give characterization theorems on minimal slant lightlike submanifolds.

1. Introduction

The notion of slant submanifolds was initiated by Chen [1, 2], as a generalization of both holomorphic and totally real submanifolds in complex geometry. Since then such submanifolds have been studied by many authors. Lotta [3, 4] defined and studied slant submanifolds in contact geometry. Cabrerizo et al. studied slant, semislant, and bislant submanifolds in contact geometry [5, 6]. They all studied the geometry of slant submanifolds with positive definite metric. Therefore this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Thus the geometry of slant submanifolds with indefinite metric became a topic of chief discussion, and Şahin [7] played a very crucial role in this direction by introducing the notion of slant lightlike submanifolds of indefinite Hermitian manifolds. Recently Gupta et al. [8] introduced the notion of slant lightlike submanifolds of an indefinite cosymplectic manifold and obtained necessary and sufficient conditions for the existence of slant lightlike submanifolds. In [9], Jain et al. studied -lightlike submanifolds of indefinite cosymplectic manifolds and proved that every totally contact umbilical GCR-lightlike submanifold of an indefinite cosymplectic manifold is a totally contact geodesic -lightlike submanifold.

In this paper we prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic manifolds other than totally contact geodesic. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally, we give characterization theorems on minimal slant lightlike submanifolds.

2. Preliminaries

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]) the following:

for , where denotes the Lie algebra of vector fields on .

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

for any , where denote the Levi-Civita connection on .

Let be a real -dimensional semi-Riemannian manifold of constant index such that , , let be an -dimensional submanifold of , and let be the induced metric of on . If is degenerate on the tangent bundle of , then is called a lightlike submanifold of (see [12]). 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 .

Screen distribution is a semi-Riemannian complementary distribution of in ; that is, and is a complementary vector subbundle to in . Let and ltr be complementary (but not orthogonal) vector bundles to in and to in , respectively. Then we have

For a quasi-orthonormal fields of frames on , we have the following.

Theorem 1 (see [12]). 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 (5), 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 which is called the second fundamental form, and is a linear operator on and known as a shape operator.

According to (4), considering the projection morphisms and of on and , respectively, then (7) becomes

where we put , and . As and are valued, and valued, respectively, therefore they are called as the lightlike second fundamental form and the screen second fundamental form on . In particular

where and . Using (7) and (8) we obtain

Let be a projection of on then using the decomposition , we can write

for any and , where , and , belong to and , respectively. Here and are linear connections on and , respectively. By using (8), (9) and (11), we obtain

3. Slant Lightlike Submanifolds

A lightlike submanifold has two distributions, namely, the radical distribution and the screen distribution. The radical distribution is totally lightlike and it is not possible to define angle between two vector fields of radical distribution where the screen distribution is nondegenerate. There are some definitions for the angle between two vector fields in Lorentzian setup [13], but they are not appropriate for our goal. Therefore to introduce the notion of slant lightlike submanifolds, one needs a Riemannian distribution, and Gupta et al. [8] played a crucial role in the development of the theory of slant lightlike submanifolds of indefinite cosymplectic manifolds.

Definition 2 (see [8]). Let be an -lightlike submanifold of an indefinite cosymplectic manifold of index with structure vector field tangent to . Then one say that is a slant lightlike submanifold of if the following conditions are satisfied.

(a) is a distribution on such that . (b)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 complementary distribution to in screen distribution .

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

Since a submanifold is invariant (anti-invariant, resp.), if , (, for any , Therefore from the previous definition, it is clear that is invariant (anti-invariant, resp.) if , resp.).

Then the tangent bundle of is decomposed as

where . For any , we write

where is the tangential component of and is the transversal component of . Similarly for any , we write

where is the tangential component of and is the transversal component of . Using the decomposition in (13), we denote by , , and the projections on the distributions , , , and , respectively. Then for any , we can write

where .

Applying to (16), we obtain

Then using (14) and (15), we get

Differentiating (17) and using (8)-(9), (14) and (15), for any , we have

By using cosymplectic property of with (7), we have the following lemmas.

Lemma 3. Let be a slant lightlike submanifold of an indefinite cosymplectic manifold , then on has

where and

Lemma 4. Let be a slant lightlike submanifold of an indefinite cosymplectic manifold then we have

where and and

Theorem 5. Let be a slant lightlike submanifold of an indefinite cosymplectic manifold ; then

(a)the distribution is integrable, if and only if, , , and , for any ,(b)the distribution is integrable, if and only if , for any .

Proof. Using (20) and (21), we have , for any . Here replacing by and then subtracting the resulting equation from this equation, we get . Next from (22) and (23), we have

for all . Then, similarly to the above, we have , and this completes the proof of .

Corollary 6 (see [8]). Let be a slant lightlike submanifold of an indefinite cosymplectic manifold with structure vector field tangent to . Then one has

for any .

4. Totally Contact Umbilical Slant Lightlike Submanifolds

Definition 7 (see [14]). 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 and totally contact geodesic if .

The above definition also holds for a lightlike submanifold . For a totally contact umbilical lightlike submanifold , we have

where and .

Lemma 8. Let be a slant lightlike submanifold of an indefinite cosymplectic manifold ; then , for any .

Proof. Using (4) and (6), it is clear that if , for any . Therefore

Hence the result follows.

Thus from Lemma 8 it follows that is a subspace of . Therefore there exists an invariant subspace of such that

Thus

Theorem 9. Let be a totally contact umbilical slant lightlike submanifold of an indefinite cosymplectic manifold . Then at least one of the following statements is true:

(i) is an anti-invariant submanifold, (ii),(iii)if is a proper slant submanifold, then .

Proof. Let be a totally contact umbilical slant lightlike submanifold of an indefinite cosymplectic manifold ; then for any with (29), we have

Therefore from (7), (27), and the above equation, we get

Using (14) and the fact that is cosymplectic manifold, we obtain

Then using (8) and (9), we get

Thus using (14), (15), (30), and (31), we have

Equating the transversal components, we get

On the other hand, (28) holds for any , and by taking the covariant derivative with respect to , we obtain

Now taking the inner product in (40) with , we obtain

Then using (28) and (41), we get

Thus from (43), it follows that; , or . This completes the proof.

Lemma 10. Let be a totally contact umbilical slant lightlike submanifold of an indefinite cosymplectic manifold ; then , for any and .

Proof. Let ; therefore , and then using (3), (8), and (9) for a totally contact umbilical slant lightlike submanifold, we have

since for , using (2), (14), and (30), we have = = . Also for , we have , and therefore by replacing by and by in (10) and using the hypothesis that is totally contact umbilical slant lightlike submanifold, we obtain

Therefore from (44) and (45), the result follows.

Theorem 11. Every totally contact umbilical proper slant lightlike submanifold of an indefinite cosymplectic manifold is totally contact geodesic.

Proof. Since is a totally contact umbilical slant lightlike submanifold, therefore for any , using (29), we have , and therefore using (27), we obtain

Using (20) for , we have

Since is totally contact umbilical slant lightlike submanifold therefore = , therefore using (46) and (47), we get

Taking the scalar product of both sides of (48) with respect to for , we obtain

Here using (28), we get

Since (28) holds for any and by taking the covariant derivative with respect to , we get

Using (51) in (50), we obtain

Since is a proper slant lightlike submanifold and is a Riemannian metric on , therefore we have . Thus using Lemma 8 and (33), we obtain

Now using the cosymplectic property of , we have , for any , and then using (9), (14), and (29), we obtain

Taking the scalar product of both sides of (54), with respect to and then using (2), (34) and (53), we obtain

Since is an invariant subspace, therefore using the cosymplectic character of , we have , and this implies that

Taking the scalar product of both sides of the previous equation with respect to , we obtain

From (15), we know that and are tangential and transversal components of , respectively, for any . Therefore if , then and . Moreover, since , therefore for any , and . Since , and therefore , and using this with (28), we get

Since is a metric connection, therefore . This further implies that , and therefore using (58), we obtain

From (55) and (59), we have

Then using (10) in (60), we obtain

this implies that

Since is a Riemannian metric on , thus we obtain

Next, for , using the cosymplectic character of , we have , and this implies that . Since is totally contact umbilical slant lightlike manifold, therefore using in the previous equation and then comparing the tangential components, we obtain

Taking the scalar product of both sides of (64) with respect to and using Lemma 10, we get

Now using (9), we have

Since is a totally contact umbilical slant lightlike submanifold therefore using (63) in the previous equation, we obtain

Using (67) in (65), we obtain that and then using (30), we get

Since is a Riemannian metric on , therefore , then using (6), we obtain that

Thus from (63) and (69), the proof is complete.

Denote by and the curvature tensors of and respectively, and then by straightforward calculations [12], we have

where

An indefinite cosymplectic space form is a connected indefinite cosymplectic manifold of constant holomorphic sectional curvature and is denoted by . Then the curvature tensor of is given by

for vector fields on .

Theorem 12. There do not exist totally contact umbilical proper slant lightlike submanifolds of an indefinite cosymplectic space form such that .

Proof. Suppose be a totally contact umbilical proper lightlike submanifold of such that . Then using (73), for any , , and , we obtain

Then using (2), we get

On the other hand using (31) and (63) in (70), we get

Using (30) and (72), we have

Similarly

Using (77) and (78) in (76), we obtain

Now using (69), we have

Thus using (80) in (75), we have

Since is a Riemannian metric on and (81) implies that , therefore . This contradiction completes the proof.

In [12], a minimal lightlike submanifold is defined when is a hypersurface of a -dimensional Minkowski space. Then in [15], a general notion of minimal lightlike submanifold of a semi-Riemannian manifold is introduced as follows.

Definition 13. A lightlike submanifold isometrically immersed in a semi-Riemannian manifold is minimal if

(i) on ,(ii), where trace is written with respect to restricted to .

We use the quasi-orthonormal basis of given by

such that , , , and form a basis of , , , and , respectively.

Definition 14 (see [16]). A lightlike submanifold is called irrotational if and only if for all and .

Theorem 15. Let be an irrotational slant lightlike submanifold of an indefinite cosymplectic manifold . Then is minimal if and only if

where is a basis of and is a basis of .

Proof. Since , therefore using (8), we get and . Moreover, irrotational M implies for and . Thus vanishes on . Hence is minimal if and only if trace on ; that is, is minimal if and only if

Using (10) and (12), we obtain

Similarly, we have

Thus our assertion follows from (85) and (86).

References

B.-Y. Chen, “Slant immersions,” Bulletin of the Australian Mathematical Society, vol. 41, no. 1, pp. 135–147, 1990.

B.-Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Louvain, Belgium, 1990.

A. Lotta, “Slant submanifolds in contact geometry,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 39, pp. 183–198, 1996.

A. Lotta, “Three-dimensional slant submanifolds of K-contact manifolds,” Balkan Journal of Geometry and Its Applications, vol. 3, no. 1, pp. 37–51, 1998.

J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Slant submanifolds in Sasakian manifolds,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 125–138, 2000.

J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Semi-slant submanifolds of a Sasakian manifold,” Geometriae Dedicata, vol. 78, no. 2, pp. 183–199, 1999.

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

R. S. Gupta, A. Upadhyay, and A. Sharfuddin, “Slant lightlike submanifolds of indefinite cosymplectic manifolds,” Mediterranean Journal of Mathematics, vol. 8, no. 2, pp. 215–227, 2011.

V. Jain, R. Kumar, and R. K. Nagaich, “Non existence of totally contact umbilical GCR-lightlike submanifolds of indefinite cosymplectic manifolds,” Vietnam Journal of Mathematics, 2013.

R. Kumar, R. Rani, and R. K. Nagaich, “On sectional curvatures of -Sasakian manifolds,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 93562, 8 pages, 2007.

D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 203 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 2002.

K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1983.

K. Yano and M. Kon, Structures on Manifolds, vol. 3 of Series in Pure Mathematics, World Scientific Publishing, Singapore, 1984.

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

K. L. Duggal and B. Sahin, “Screen Cauchy Riemann lightlike submanifolds,” Acta Mathematica Hungarica, vol. 106, no. 1-2, pp. 137–165, 2005.