International Journal of Mathematics and Mathematical Sciences

Volumeย 2011ย (2011), Article IDย 140259, 11 pages

http://dx.doi.org/10.1155/2011/140259

## Characteristic Lightlike Submanifolds of an Indefinite -Manifold

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: Christianย Corda

Copyright ยฉ 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., [1โ4]).

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)โ(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 ยฑ1 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 .*

#### References

- K. L. Duggal and A. Bejancu,
*Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications*, Kluwer Academic, Dodrecht, The Netherlands, 1996. - K. L. Duggal and B. Sahin, โLightlike submanifolds of indefinite Sasakian manifolds,โ
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 57585, 21 pages, 2007. View at Google Scholar ยท View at Zentralblatt MATH - D. H. Jin, โGeometry of lightlike hypersurfaces of an indefinite Sasakian manifold,โ
*Indian Journal of Pure and Applied Mathematics*, vol. 41, no. 4, pp. 569โ581, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - D. H. Jin, โGeometry of half lightlike submanifolds of an indefinite Sasakian manifold,โ submitted to
*Bulletin of the Korean Mathematical Society*. - L. Brunetti and A. M. Pastore, โCurvature of a class of indefinite globally framed f-manifolds,โ
*Bulletin Mathรฉmatique de la Sociรฉtรฉ des Sciences Mathรฉmatiques de Roumanie*, vol. 51(99), no. 3, pp. 183โ204, 2008. View at Google Scholar - L. Brunetti and A. M. Pastore, โLightlike hypersurfaces in indefinite $\mathrm{\&\#x1D4AE;}$-manifolds,โ
*Differential Geometry*, vol. 12, pp. 18โ40, 2010. View at Google Scholar - D. H. Jin, โGeneric lightlike submanifolds of an indefinite Sasakian manifoldโ.
- J. W. Lee and D. H. Jin, โHalf lightlike submanifolds in indefinite $\mathrm{\&\#x1D4AE;}$-manifolds,โ to appear in
*Taiwanese Journal of Mathematics*. - S. Kobayashi and K. Nomizu,
*Foundations of Differential Geometry*, vol. 1-2, Interscience Publishers, New York, NY, USA, 1963. - D. N. Kupeli,
*Singular Semi-Riemannian Geometry*, Kluwer Academic, Dodrecht, The Netherlands, 1996. - C. Atindogbe and K. L. Duggal, โPseudo-Jacobi operators and Osserman lightlike hypersurfaces,โ
*Kodai Mathematical Journal*, vol. 32, no. 1, pp. 91โ108, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH