International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 140259, 11 pagesdoi:10.1155/2011/140259

Research Article

Characteristic Lightlike Submanifolds of an Indefinite -Manifold

Jae Won Lee

Department of Mathematics, Sogang University, Sinsu-Dong, Mapo-Gu, Seoul 121-742, Republic of Korea

Received 12 April 2011; Revised 12 August 2011; Accepted 23 August 2011

Academic Editor: Christian Corda

Copyright © 2011 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.

We study characteristic -lightlike submanifolds tangent to the characteristic vector fields in an indefinite metric -manifold, and we also discuss the existence of characteristic lightlike submanifolds of an indefinite -space form under suitable hypotheses: (1) is totally umbilical or (2) its screen distribution is totally umbilical in .

1. Introduction

In the theory of submanifolds of semi-Riemannian manifolds, it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is nontrivial, making it interesting and remarkably different from the study of nondegenerate submanifolds. In particular, many authors study lightlike submanifolds on indefinite Sasakian manifolds (e.g., [1–4]).

Similar to Riemannian geometry, it is natural that indefinite -manifolds are generalizations of indefinite Sasakian manifolds. Brunetti and Pastore analyzed some properties of indefinite -manifolds and gave some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields [5]. After then, they studied the geometry of lightlike hypersurfaces of indefinite -manifold [6]. As Jin's generalizations of lightlike submanifolds of the Sasakian manifolds with the general codimension [3, 4, 7], Lee and Jin recently extended lightlike hypersurfaces on indefinite -manifold to lightlike submanifolds with codimension 2 on an indefinite -manifold, called characteristic half lightlike submanifolds [8]. However, a general notion of characteristic lightlike submanifolds of an indefinite -manifold have not been introduced as yet.

The objective of this paper is to study characteristic -lightlike submanifolds of an indefinite -manifold subject to the conditions: (1) is totally umbilcial, or (2) is totally umbilcal in . In Section 2, we begin with some fundamental formulae in the theory of -lightlike submanifolds. In Section 3, for an indefinite metric -manifold we consider a lightlike submanifold tangent to the characteristic vector fields, we recall some basic information about indefinite -manifolds and deal with the existence of irrotational characteristic submanifolds of an indefinite -space form. Afterwards, we study characteristic -lightlike submanifolds of in Sections 4 and 5.

2. Preliminaries

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 nondegenerate 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 submanifolds of are characterized as follows:(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 on : 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-Weingartan 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 s 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 and is the sign of but it is ±1 related to the causal character of . From (2.18), we know that each is shape operator related to the local second fundamental form on . Replacing by in (2.14), we have for all . It follows Also, replacing by in (2.14) and using (2.20), we have For an -lightlike submanifold, replace by in (2.16), we have

Note 1. Using (2.14) and the fact that are symmetric, we have From this, (2.20) and (2.21), we show that are self-adjoint on with respect to if and only if for all and if and only if for all . We call self-adjoint the lightlike shape operators of . It follows from the above equivalence and (2.10) that the radical distribution of a lightlike submanifold , with the lightlike shape operators , is always an integrable distribution.

3. Characteristic Lightlike Submanifolds

A manifold is called a globally framed f-manifold (or -manifold) if it is endowed with a nonnull -tensor field of constant rank, such that is parallelizable, that is, there exist global vector fields , , with their dual 1-forms , satisfying and .

The -manifold , , is said to be an indefinite metric -manifold if is a semi-Riemannian metric, with index , , satisfying the following compatibility condtion for any , being according to whether is spacelike or timelike. Then, for any , one has . An indefinite metric -manifold is called an indefinite -manifold if it is normal and , for any , where for any . The normality condition is expressed by the vanishing of the tensor field , being the Nijenhuis torsion of . Furthermore, as proved in [5], the Levi-Civita connection of an indefinite -manifold satisfies: where and . We recall that and is an integrable flat distribution since (more details in [5]).

Following the notations in [9], we adopt the curvature tensor , and thus we have , and , for any , , , .

An indefinite -manifold is called an indefinite -space form, denoted by , if it has the constant -sectional curvature [5]. The curvature tensor of this space form is given by for any vector fields .

Note 2. Although is not unique, it is canonically isomorphic to the factor vector bundle considered by Kupeli [10]. Thus all screen distributions are mutually isomorphic. For this reason, we newly define generic lightlike submanifolds of as follows.

Definition 3.1. Let be a -lightlike submanifold of such that all the characteristic vector fields are tangent to . A screen distribution is said to be characteristic if and .

Definition 3.2. A -lightlike submanifold of is said to be characteristic if and a characteristic screen distribution is chosen.

Proposition 3.3 (see [6]). Let be a lightlike hypersurface of an indefinite -manifold such that the characteristic vector fields are tangent to . Then there exists a screen distribution such that and , where is a nonzero section of .

Proposition 3.4 (see [8]). Let be a 1-lightlike submanifold of codimension 2 of an indefinite -manifold . Then is a characteristic lightlike submanifold of .

Definition 3.5. A lightlike submanifold is said to be irrotational [10] if for any and for all .

Note 3. For an -lightlike , the above definition is equivalent to

The extrinsic geometry of lightlike hypersurfaces depends on a choice of screen distribution, or equivalently, normalization. Since the screen distribution is not uniquely determined, a well-defined concept of -manifold is not possible for an arbitrary lightlike submanifold of a semi-Riemannian manifold, then one must look for a class of normalization for which the induced Riemannian curvature has the desired symmetries. Let be a semi-Riemannian manifold, . is said to be an algebraic curvature tensor [11] on if it satisfies the following symmetries:

Definition 3.6. A screen distribution is said to be admissible if the associated induced Riemannian curvature is an algebraic curvature tensor.

Theorem 3.7. Let be an irrotational generic characteristic lightlike submanifold of an indefinite -space form with an admissible screen distribution . Then one has .

Proof. Denote by and the curvature tensors of and , respectively. Using the local Gauss-Weingarten formulas for , we obtain for all . Replace by in (3.6) and use (2.10), (2.15), (2.17), and (3.4), we have Using (3.7), the fact for , and a screen distribution is admissible, we get On the other hand, since and for any , is an indefinite -space form implies the Riemannian curvature in (3.3) is given by for any . So, replacing , , by , , in (3.9), we find Then, using (3.3), (3.8), and (3.9), we get Choosing , we obtain .

Corollary 3.8. There exist no irrotational characteristic -lightlike submanifolds of an indefinite -space form with such that the screen distribution is admissible.

4. Totally Umbilical Characteristic Lightlike Submanifolds

Definition 4.1. An -lightlike submanifold of is said to be totally umbilical [1] if there is a smooth vector field such that for all . 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 for any . From (4.2) we show that any totally umbilical -lightlike submanifold of is irrotational. Thus, by Theorem 3.7, we have the following.

Theorem 4.2. Let be a totally umbilical characteristic -lightlike submanifold of an indefinite -space form . Then one has .

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

Proof. Apply to with , for all and , and use (2.8), (2.10), (2.15), (2.17), (2.22), and (3.2), we have Assume that is totally umbilical. Then we have Replace by and by by turns, we get for all and for all . Thus we show that and is totally geodesic.

Corollary 4.4 (see [1]). Let be a totally umbilical characteristic -lightlike submanifold of an indefinite -manifold . Then there exists a unique torsion-free metric connection on induced by the connection on .

Proof. From (4.2) and Theorem 4.3, we have for all and . Thus, using (2.12), we obtain our assertion.

5. Totally Umbilical Screen Distributions

Definition 5.1. A screen distribution of is said to be totally umbilical [1] 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 (5.1), we know that is totally umbilical in if and only if each shape operators of satisfies for some smooth functions on .

In general, is not necessarily integrable. The following result gives equivalent conditions for the integrability of a screen .

Theorem 5.2 (see [1]). Let be an -lightlike submanifold of a semi-Riemannian manifold . Then the following assertions are equivalent: (1) is integrable,(2) is symmetric on , for each ,(3) is self-adjoint on with respect to , for each .

We know that, from (5.2), each shape operator is self-adjoint on with respect to , which further follows from that above theorem that any totally umbilical screen distribution of is integrable.

Theorem 5.3. Let be a characteristic -lightlike submanifold of an indefinite -manifold . If is totally umbilical in , then is totally geodesic in .

Proof. Apply the operator to for some such that , and use (2.7) and (2.18) Assume that is totally umbilical in . Then we have Replacing by in (5.4) and taking and by turns and use the above method, we have for all . Thus we have our assertion.

Theorem 5.4. Let be a characteristic -lightlike submanifold of an indefinite -manifold such that is totally umbilical in . Then is not irrotational.

Proof. Apply the operator to for all and , and use (2.6), (2.10), (2.15), (2.18), and Theorem 5.3, we have Since is totally umbilical, by Theorem 5.3, we have that is totally geodesic. Then, by (5.5), we have Apply the operator to and use (2.6), (2.10), (2.15), we have Replace by in this equation and (5.6), we have It is a contradiction. Thus is not irrotational.

Since any totally umbilical -lightlike submanifold of is irrotational, by Theorem 5.4, we have the following result.

Corollary 5.5. There exist no totally umbilical characteristic -lightlike submanifolds of an indefinite -manifold equipped with a totally umbilical screen distribution in .

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