Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 610986, 16 pagesdoi:10.1155/2011/610986

Research Article

Generic Lightlike Submanifolds of an Indefinite Cosymplectic Manifold

Dae Ho Jin¹ and Jae Won Lee²

¹Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea²Department of Mathematics, Sogang University, Sinsu-dong, Mapo-gu 121-742, Republic of Korea

Received 7 June 2011; Revised 12 August 2011; Accepted 2 September 2011

Academic Editor: Gerhard-Wilhelm Weber

Copyright © 2011 Dae Ho Jin and Jae Won Lee. 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.

Lightlike geometry has its applications in general relativity, particularly in black hole theory. Indeed, it is known that lightlike hypersurfaces are examples of physical models of Killing horizons in general relativity (Galloway, 2007). In this paper, we introduce the definition of generic lightlike submanifolds of an indefinite cosymplectic manifold. We investigate new results on a class of generic lightlike submanifolds of an indefinite cosymplectic manifold .

1. Introduction

In the generalization from Riemannian to semi-Riemannian manifolds, the induced metric may be degenerate (lightlike) therefore there is a natural existence of lightlike submanifolds and for which the local and global geometry is completely different than nondengerate case. In lightlike case, the standard textbook definitions do not make sense- and one fails to use the theory of non-degenerate geometry in the usual way. The primary difference between the lightlike submanifolds and non-degenerate submanifolds is that in the first case, the normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult and different from the study of non-degenerate submanifolds. Moreover, the geometry of lightlike submanifolds is used in mathematical physics, in particular, in general relativity since lightlike submanifolds produce models of different types of horizons (event horizons, Cauchy’s horizons, and Kruskal’s horizons). The universe can be represented as a four-dimensional submanifold embedded in a -dimensional spacetime manifold. Lightlike hypersurfaces are also studied in the thoery of electromagnetism [1]. Thus, large number of applications but limited information available, motivated us to do research on this subject matter. Kupeli [2] and Bejancu and Duggal [1] developed the general theory of degenerate (lightlike) submanifolds. They constructed a transversal vector bundle of lightlike submanifold and investigated various properties of these manifolds. The geometry of both lightlike hypersurfaces and half lightlike submanifolds of indefinite cosymplectic manifolds was studied by Jin ([3, 4]). However, a general notion of generic lightlike submanifolds of an indefinite cosymplectic manifold has not been introduced as yet.

The objective of this paper is to study generic -lightlike submanifolds of an indefinite cosymplectic manifold subject to the conditions: (1) is totally umbilical, or (2) is totally umbilical in . In Section 1, we first of all recall some of fundamental formulas in the theory of -lightlike submanifolds. In Section 2, we newly define generic lightlike submanifolds. After that, we prove some basic theorems which will be used in the rest of this paper. In Section 3, we study generic -lightlike submanifolds of .

2. Lightlike Submanifolds

Let be an -dimensional lightlike submanifold of an -dimensional semi-Riemannian manifold . Then the radical distribution is a vector subbundle of the tangent bundle and the normal bundle , of rank . In general, there exist two complementary non-degenerate distributions and of in and , respectively, called the screen and coscreen distributions on , such that where the symbol denotes the orthogonal direct sum. We denote such a lightlike submanifold by . Denote by the algebra of smooth functions on and by the module of smooth sections of a vector bundle over . We use the same notation for any other vector bundle. We use the following range of indices:

Let and be complementary (but not orthogonal) vector bundles to in and in , respectively, and let be a lightlike basis of consisting of smooth sections of , where is a coordinate neighborhood of , such that where is a lightlike basis of . Then we have

We say that a lightlike submanifold of is(1)-lightlike if ;(2)coisotropic if ;(3)isotropic if ;(4)totally lightlike if .

The above three classes (2)–(4) are particular cases of the class (1) as follows: , , and , respectively. The geometry of -lightlike submanifolds is more general form than that of the other three type submanifolds. For this reason, in this paper we consider only -lightlike submanifolds , with the following local quasiorthonormal field of frames of : where the sets and are orthonormal basis of and , respectively.

Let be the Levi-Civita connection of and the projection morphism of on with respect to (2.1). For an -lightlike submanifold, the local Gauss-Weingarten formulas are given by for any , where and are induced linear connections on and , respectively, the bilinear forms and on are called the local lightlike and screen second fundamental forms on , respectively, are called the local radical second fundamental forms on . , , and are linear operators on and , , and are 1-forms on . Since is torsion-free, is also torsion-free and both and are symmetric. From the fact , we know that are independent of the choice of a screen distribution. We say that is the second fundamental tensor of .

The induced connection on is not metric and satisfies for all , where are the 1-forms such that But the connection on is metric. The above three local second fundamental forms are related to their shape operators by where is the sign of the vector field . From (2.18), we know that each is shape operator related to the local second fundamental form on . Replace by in (2.14), we have for all . It follows Also, replace by in (2.14) and use (2.21), we have For an -lightlike submanifold, replace by in (2.16), we have From (2.6), (2.10), and (2.23), for all , we have

Definition 2.1. A lightlike submanifold of a semi-Riemannian manifold is said to be irrotational if for any and for all .

Note 1. For an -lightlike , the above definition is equivalent to

Denote by and the curvature tensors of and , respectively. Using the local Gauss-Weingarten formulas for , we obtain for all . Assume that is irrotational. Replace by in (2.26) and use (2.10), (2.15), (2.17), and (2.25), then we have Using (2.27) and the fact for , we get

3. Indefinite Cosymplectic Manifolds

An odd dimensional smooth manifold is called a contact metric manifold [5, 6] if there exists a -type tensor field , a vector field , called the characteristic vector field, and its 1-form satisfying for any vector fields on . Then the set is called a contact metric structure on . Note that we may assume that without loss of generality [7]. We say that has a normal contact structure [5, 8] if , where is the Nijenhuis tensor field of . A normal contact metric manifold is called a cosymplectic [9, 10] for which we have for any vector field on . A cosymplectic manifold is called an indefinite cosymplectic manifold [3, 4] if is a semi-Riemannian manifold of index . For any indefinite cosymplectic manifold, apply to for any vector field on and use (3.2), then we have . Apply to this and use (3.1) and , we get

An indefinite cosymplectic manifold is called an indefinite cosymplectic space form, denoted by , if it has the constant -sectional curvature [3, 9, 10]. The curvature tensor of this space form is given by for any vector fields , and in .

Let be an -dimensional -lightlike submanifold of an -dimensional indefinite cosymplectic manifold and the projection morphism of on with respect to (2.4). The characteristic vector field of from (2.4) is decomposed by where ,  and are smooth functions on .

Note 2. Although is not unique, it is canonically isomorphic to the factor vector bundle considered by Kupeli [2]. Thus all screen distributions are mutually isomorphic. For this reason, the following definition is well defined.

Definition 3.1 (see [6]). One says that is generic lightlike submanifold of if there exists a screen distribution of such that

Proposition 3.2 (see [3]). Let be a lightlike hypersurface of an indefinite cosymplectic manifold . Then is a generic lightlike submanifold of .

Proposition 3.3 (see [4]). Let be a 1-lightlike submanifold of codimension 2 of an indefinite cosymplectic manifold such that the coscreen distribution is spacelike. Then is a generic lightlike submanifold of .

Theorem 3.4. Let be an irrotational generic -lightlike submanifold of an indefinite cosymplectic space form . Then one has .

Proof. Assume that in (3.5). Note that (3.1) implies for all . Then, taking the scalar product with to (3.4), and using (2.28), we get Replace by and by in the equation and use (3.1), then we have because by (3.6). Replacing by in this equation, we obtain . Since , we have .Assume that . Then, taking the scalar product with to both sides of (3.4) and using (2.28) and (3.1), we obtain Replace by and by in this equation and use (3.1), then we have because by (3.6) and . Replace by in this equation, we get .

Corollary 3.5. There exist no irrotational generic -lightlike submanifolds of an indefinite cosymplectic space form with .

Proposition 3.6. Let be an lightlike submanifold of an indefinite cosymplectic manifold . Then the characteristic vector field does not belong to and .

Proof. Assume that belongs to (or ). Then (3.1) deduces to [or ]. From this, we have It is a contradiction. Thus does not belong to and .

4. Generic Lightlike Submanifolds

If the characteristic vector field is tangent to , then, by Proposition 3.6, does not belong to . This enables one to choose a screen distribution which contains . This implies that if is tangent to , then it belongs to . Călin also proved this result in his book [11] which Kang et al. [12] and Duggal and Sahin [5, 8] assumed in their papers. We also assumed this result in this paper. In this case, all of the functions , , and on , defined by (3.5), vanish identically.

Theorem 4.1. Let be a generic -lightlike submanifold of an indefinite cosymplectic manifold . Then is a parallel vector field on and . Furthermore is conjugate to any vector field on with respect to and . In particular, is an asymptotic vector field on .

Proof. Replace by to (2.6) and use (3.3) and , we get Taking the scalar product with and in this equation by turns, we have Thus is parallel on and conjugate to any vector field on with respect to . Replace by to (2.9) and use (4.2) and , we have Taking the scalar product with to this equation we have Thus is also parallel on and conjugate to any vector field on with respect to . Thus we have our assertions.

Definition 4.2. An -lightlike submanifold of is said to be totally umbilical [13] if there is a smooth vector field such that In case , we say that is totally geodesic.It is easy to see that is totally umbilical if and only if, on each coordinate neighborhood , there exist smooth functions and such that

Theorem 4.3. Let be a totally umbilical generic lightlike submanifold of an indefinite cosymplectic manifold . Then is totally geodesic.

Proof. From (4.2) and (4.6), we obtain Replace by to this equations and use , we have for all and for all . Thus is totally geodesic.

Definition 4.4. A screen distribution of is said to be totally umbilical [13] in if, for each locally second fundamental form , there exist smooth functions on any coordinate neighborhood in such that In case for all , we say that is totally geodesic in .Due to (2.18) and (4.8), we know that is totally umbilical in if and only if each shape operators of satisfies for some smooth functions on .

Theorem 4.5. Let be a generic -lightlike submanifold of an indefinite cosymplectic manifold such that is totally umbilical in . Then is totally geodesic in .

Proof. As is totally umbilical in . Replace by to (4.8) and use (4.4), we have for all . Replace by to this equation and use the fact , we obtain for all .From (3.6), the screen distribution splits as follows: where is a non-degenerate almost complex distribution on with respect to , that is, . Thus the general decompositions of and in (2.1) and (2.4) reduce, respectively, to where and are - and -lightlike distributions on such that In this case, is an almost complex distribution of with respect to . Consider the local null vector fields and on and the local nonnull vector field on defined respectively by Denote by the projection morphism of on with respect to the decomposition (4.11). Then any vector field on is expressed as follows: where , , and are 1-forms locally defined on by and is a tensor field of (1,1)-type globally defined on by Apply to (2.6), (2.7), (2.8), and (2.24) and use (4.13) and (4.14), we have

Theorem 4.6. Let be a generic -lightlike submanifold of an indefinite cosymplectic manifold . Then is integrable if and only if Moreover, if is totally umbilical, then is a parallel distribution on .

Proof. Take . Then we have . Apply to and use (2.6), (3.2), (4.13), and (4.14), we have By directed calculations from two equations of (4.25), we have If is an integrable distribution on , then we have for any . This implies for all and . Therefore we obtain for all . Conversely if for all , then we have for all and . This implies for all . Thus is an integrable distribution of .If is totally umbilical, from Theorem 4.3 and (4.25), we have This implies for all . Thus is a parallel distribution on .

Theorem 4.7. Let be a generic lightlike submanifold of an indefinite cosymplectic manifold . Then is parallel on with respect to if and only if is a parallel distribution on.

Proof. Assume that is parallel on with respect to . For any , we have . Taking the scalar product with and to (4.26) with , we have and for all and for each and , respectively. From (4.25), we have and . This implies for all . Thus is a parallel distribution on .Conversely, if is parallel on , from (4.25) we have For any , we show that . Replace by to the equations and use (4.2), we have and for any . Thus is parallel on with respect to by (4.25).

Theorem 4.8. Let be a generic lightlike submanifold of an indefinite cosymplectic manifold . If is parallel on with respect to , then is a parallel distribution on and is locally a product manifold , where , , and are leafs of , and , respectively.

Proof. Assume that is parallel on with respect to . Then is parallel on with respect to . By Theorem 4.7, is a parallel distribution on . Apply the operator to (4.23) with , we have due to for all and . Replace by and to this equation by turns and use (4.15), we have and . Taking the scalar product with and to (4.23) with by turns, we have for all . Replace by to (4.31), we get due to (2.23). Also replace by to (4.32), we have due to (2.17). From this results, (4.11) and (4.14), we get Thus and are also parallel distributions on . By the decomposition theorem of de Rham [14], we show that , where , and are some leafs of , and , respectively.

Theorem 4.9. Let be a generic lightlike submanifold of an indefinite cosymplectic manifold . One has the following assertions. (i)If each is parallel with respect to , then . In this case is irrotational. Moreover, one has (ii)If each is parallel with respect to , then and (iii)If each is parallel with respect to , then and Moreover, if all of , , and are parallel on with respect to , then is totally geodesic in and on . In this case, each null transversal vector fields of is a constant on .

Proof. If is parallel with respect to , then, taking the scalar product with , , and to (4.21) by turns, we have , and , respectively. Thus is irrotational. We have for all . From this result and (4.14), we obtain Apply to this equation and use , we obtain (i). In a similar way, by using (4.13), (4.14), (4.20), and (4.22), we have (ii) and (iii).Assume that all of , and are parallel on with respect to . Substituting the equation of (i) into (4.17)-1, we have Also, substituting the equation of (iii) into (4.17)-2, we have From the last two equations and the equation of (ii), we see that for all . From this and (2.18) we see that is totally geodesic in and all 1-forms , , and , defined by (2.7) and (2.8), vanish identically. Using the results and (2.7), we show that is a constant on .

Theorem 4.10. Let be a totally umbilical generic -lightlike submanifold of an indefinite cosymplectic manifold such that is totally umbilical. Then is locally a product manifold , where , , and are some leafs of , and , respectively, and .

Proof. By Theorem 4.6, is a parallel distribution . Thus, for all , we have . From (2.9) and (4.26), we have due to . If is totally umbilical in , then we have due to Theorem 4.5. By (2.9) and (4.40), we get These results and (4.25) imply for all . Thus is a parallel distribution on and , where . By Theorems 4.3 and 4.5, we have and . Thus (2.10) and (4.20)~(4.22) deduce, respectively, to Thus is also a parallel distribution on . Thus we have , where , , and are some leafs of , and , respectively, and .

Acknowledgment

The authors are thankful to the referee for making various constructive suggestions and corrections towards improving the final version of this paper.

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