Abstract
We study the geometry of lightlike submanifolds of a semi-Riemannian manifold of quasiconstant curvature subject to the following conditions: (1) the curvature vector field ζ of is tangent to , (2) the screen distribution of is totally geodesic in , and (3) the coscreen distribution of is a conformal Killing distribution.
1. Introduction
In the generalization from the theory of submanifolds in Riemannian to the theory of submanifolds in semi-Riemannian manifolds, the induced metric on submanifolds 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 nondegenerate case. In lightlike case, the standard text book definitions do not make sense, and one fails to use the theory of nondegenerate geometry in the usual way. The primary difference between the lightlike submanifolds and nondegenerate 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 nondegenerate 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 theory of electromagnetism [1]. Thus, large number of applications but limited information available motivated us to do research on this subject matter. Kupeli [2] and Duggal and Bejancu [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.
In the study of Riemannian geometry, Chen and Yano [3] introduced the notion of a Riemannian manifold of a quasiconstant curvature as a Riemannian manifold with the curvature tensor satisfying the conditionfor any vector fields , and on , where are scalar functions and is a 1-form defined by where is a unit vector field on which called the curvature vector field. It is well known that if the curvature tensor is of the form (1.1), then the manifold is conformally flat. If , then the manifold reduces to a space of constant curvature.
A nonflat Riemannian manifold of dimension is defined to be a quasi-Einstein manifold [4] if its Ricci tensor satisfies the condition
where are scalar functions such that , and is a nonvanishing 1-form such that for any vector field , where is a unit vector field. If , then the manifold reduces to an Einstein manifold. It can be easily seen that every Riemannian manifold of quasiconstant curvature is a quasi-Einstein manifold.
The subject of this paper is to study the geometry of lightlike submanifolds of a semi-Riemannian manifold of quasiconstant curvature. We prove two characterization theorems for such a lightlike submanifold as follows.
Theorem 1.1. Let be an -lightlike submanifold of a semi-Riemannian manifold of quasiconstant curvature. If the curvature vector field of is tangent to and is totally geodesic in , then one has the following results: (1)if is a Killing distribution, then the functions and , defined by (1.1), vanish identically. Furthermore, , , and the leaf of are flat manifolds;(2)if is a conformal Killing distribution, then the function vanishes identically. Furthermore, and are space of constant curvatures, and is an Einstein manifold such that , where is the induced scalar curvature of .
Theorem 1.2. Let be an irrotational -lightlike submanifold of a semi-Riemannian manifold of quasiconstant curvature. If is tangent to , is totally umbilical in , and is a conformal Killing distribution with a nonconstant conformal factor, then the function vanishes identically. Moreover, and are space of constant curvatures, and is a totally umbilical Einstein manifold such that , where is the scalar quantity of .
2. Lightlike Submanifolds
Let be an -dimensional lightlike submanifold of an -dimensional semi-Riemannian manifold . We follow Duggal and Bejancu [1] for notations and results used in this paper. The radical distribution is a vector subbundle of the tangent bundle and the normal bundle , of rank . Then, in general, there exist two complementary nondegenerate distributions and of in and , respectively, called the screen and coscreen distribution on , and we have the following decompositions: where the symbol denotes the orthogonal direct sum. We denote such a lightlike submanifold by . 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 say that a lightlike submanifold of is
(1) -lightlike submanifold if ,
(2) coisotropic submanifold if ,
(3) isotropic submanifold if ,
(4) totally lightlike submanifold if .
The above three classes (2)~(4) are particular cases of the class (1) as follows: , and , respectively.
Example 2.1. Consider in the 1-lightlike submanifold given by equations
then we have and , where we set
It follows that is a distribution on of rank 1 spanned by . Choose and spanned by and where are timelike and spacelike, respectively. Finally, the lightlike transversal vector bundle
and the transversal vector bundle
are obtained.
Let be the Levi-Civita connection of and the projection morphism of on with respect to the decomposition (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 second fundamental form and local screen second fundamental form on , respectively, and is called the local radical second fundamental form 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 that , we know that are independent of the choice of a screen distribution. Note that , and depend on the section . Indeed, take , then we have [5].
The induced connection on is not metric and satisfies
where is the 1-form such that
But the connection on is metric. The above three local second fundamental forms of and are related to their shape operators by
and , where . From (2.19), we know that the operators are shape operators related to for each , called the radical shape operators on . From (2.16), we know that the operators are valued. Replace by in (2.15), then we have for all . It follows that
Also, replace by in (2.15) and use (2.20), then we have
Thus is an eigenvector field of corresponding to the eigenvalue 0. For an -lightlike submanifold, replace by in (2.17), then we have
From (2.15)~(2.18), we show that the operators and are not self-adjoint on but self-adjoint on .
Theorem 2.2. Let be an -lightlike submanifold of a semi-Riemannian manifold , then the following assertions are equivalent: (i) are self-adjoint on with respect to , for all ,(ii) satisfy for all and ,(iii) for all and , that is, the image of with respect to for each is a trivial vector bundle,(iv) for all and , that is, is a shape operator on , related by the second fundamental form .
Proof. From (2.15) and the fact that are symmetric, we have
(i)(ii). If for all and , then we have for all , that is, are self-adjoint on with respect to . Conversely, if are self-adjoint on with respect to , then we have
for all . Replace by in this equation and use the second equation of (2.20), then we have for all and .
(ii)(iii). Since is nondegenerate, from the first equation of (2.21), we have , for all and .
(ii)(iv). From (2.16), we have for any and for all and .
Corollary 2.3. Let be a 1-lightlike submanifold of a semi-Riemannian manifold , then the operators are self-adjoint on with respect to .
Definition 2.4. An -lightlike submanifold of a semi-Riemannian manifold is said to be irrotational if for any and .
For an -lightlike submanifold of , the above definition is equivalent to and for any . In this case, are self-adjoint on with respect to , for all .
We need the following Gauss-Codazzi equations (for a full set of these equations see [1, chapter 5]) for and . Denote by , and the curvature tensors of the Levi-Civita connection of , the induced connection of , and the induced connection on , respectively:
The Ricci tensor of is given by for any ,. Let = . Locally, is given by where is an orthonormal frame field of . If and then is an Einstein manifold. If , any is Einstein, but in (2.33) is not necessarily constant. The scalar curvature is defined by Putting (2.33) in (2.34) implies that is Einstein if and only if
3. The Tangential Curvature Vector Field
Let denote the induced Ricci tensor of type on , given by Consider an induced quasiorthonormal frame field
where is a basis of on a coordinate neighborhood of such that and . By using (2.29) and (3.1), we obtain the following local expression for the Ricci tensor: Substituting (2.25) and (2.27) in (3.3) and using (2.15)~(2.18) and (3.4), we obtain
for any , . This shows that is not symmetric. A tensor field of , given by (3.1), is called its induced Ricci tensor if it is symmetric. From now and in the sequel, a symmetric tensor will be denoted by .
Using (2.28), (3.5), and the first Bianchi identity, we obtain
From this equation and (2.28), we have
Theorem 3.1 (see[5]). Let be a lightlike submanifold of a semi-Riemannian manifold , then the tensor field is a symmetric Ricci tensor Ric if and only if each 1-form is closed, that is, , on any .
Note 1. Suppose that the tensor is symmetric Ricci tensor , then the 1-form is closed by Theorem 3.1. Thus, there exist a smooth function on such that . Consequently, we get . If we take , it follows that . Setting in this equation, we get for any . We call the pair on such that the corresponding 1-form vanishes the canonical null pair of .
For the rest of this paper, let be a lightlike submanifold of a semi-Riemannian manifold of quasiconstant curvature. We may assume that the curvature vector field of is a unit spacelike tangent vector field of and , for all . Substituting (3.8)~(3.10) into (3.5), we have
Definition 3.2. We say that the screen distribution of is totally umbilical [1] in if, on any coordinate neighborhood , there is a smooth function such that for any , or equivalently,
In case on , we say that is totally geodesic in .
A vector field on is said to be a conformal Killing vector field [6] if for any smooth function , where denotes the Lie derivative with respect to , that is,
In particular, if , then is called a Killing vector field [7]. A distribution on is called a conformal Killing (resp., Killing) distribution on if each vector field belonging to is a conformal Killing (resp., Killing) vector field on . If the coscreen distribution is a Killing distribution, using (2.10) and (2.17), we have
Therefore, since are symmetric, we obtain
Theorem 3.3. Let be an -lightlike submanifold of a semi-Riemannian manifold , then the coscreen distribution is a conformal Killing (resp., Killing) distribution if and only if there exists a smooth function such that
Theorem 3.4. Let be an irrotational -lightlike submanifold of a semi-Riemannian manifold of quasiconstant curvature. If the curvature vector field of is tangent to , is totally umbilical in , and is a conformal Killing distribution, then the tensor field is an induced symmetric Ricci tensor of .
Proof. From (2.17)~(2.20), (2.22), (3.16), and (3.11), we have Using (3.17), we show that is symmetric.
4. Proof of Theorem 1.1
As , we get by (2.30). From (2.27) and (3.16), we have
By Theorems 3.1 and 3.4, we get on . Thus, we have due to (2.28). From the above results, we deduce the following equation: Taking and to (4.2) and then comparing with (3.9), we have
Case 1. If is a Killing distribution, that is, , then we have Substituting (4.3) into (1.1) and using (2.25) and the facts and due to (1.1), we have Thus, is a space of constant curvature . Taking to (4.3), we have . Substituting (4.3) into (3.18) with , we have On the other hand, substituting (4.5) and into (3.4), we have From the last two equations, we get as . Thus, , and and are flat manifolds by (1.1) and (4.5). From this result and (2.29), we show that is also flat.
Case 2. If is a conformal Killing distribution, assume that . Taking to (4.3), we have . From this and (4.3), we show that Substituting (4.8) into (1.1) and using (2.25) with and (3.16), we have for all . Substituting (4.8) into (3.18) with , we have by the fact that . On the other hand, from (2.27), (3.9), and (4.3), we have . Substituting this result and (4.9) into (3.4), we have The last two equations imply as . It is a contradiction. Thus, and is a space of constant curvature . From (2.29) and (4.9), we show that is a space of constant curvature . But is not a space of constant curvature by (3.17). Let , then the last two equations reduce to Thus is an Einstein manifold. The scalar quantity of [8], obtained from by the method of (2.34), is given by Since is an Einstein manifold satisfying (4.12), we obtain Thus, we have which provides a geometric interpretation of half lightlike Einstein submanifold (the same as in Riemannian case) as we have shown that the constant .
5. Proof of Theorem 1.2
Assume that is tangent to , is totally umbilical, and is a conformal Killing vector field. Using (1.1), (2.26) reduces to
for all . Replacing by to (1.1), we have
for all and where . Applying to (3.12) and using (2.13), we have
for all . Substituting this equation into (2.30), we obtain
Substituting this equation and (5.2) into (2.27) and using , we obtain
Replacing by to this and using (2.20) and the fact , we havefor all . Differentiating (3.16) and using (5.1), we have
Replacing by in the last equation and using (2.20), we obtain
As the conformal factor is nonconstant, we show that . Thus, we have where . From (3.17) and (5.9), we show that the second fundamental form tensor , given by , satisfies
Thus, is totally umbilical [5]. Substituting (5.9) into (5.6), we have
for all . Taking to this equation, we have
Assume that , then we have Substituting (5.13) into (1.1) and using (2.25), (3.12), (3.17), and (5.9), we have for all . Substituting (5.9) and (5.13) into (3.18), we have On the other hand, substituting (5.14) and the fact that
into (3.4), we have Comparing (5.15) and (5.17), we obtain . As , we have , which is a contradiction. Thus, we have . Consequently, by (1.1), (2.29), and (5.14), we show that and are spaces of constant curvatures and , respectively. Let
then (5.15) and (5.17) reduce to
Thus, is an Einstein manifold. The scalar quantity of is given by
Thus, we have
Example 5.1. Let be a lightlike hypersurface of an indefinite Kenmotsu manifold equipped with a screen distribution , then there exist an almost contact metric structure on , where is a -type tensor field, is a vector field, is a 1-form, and is the semi-Riemannian metric on such that for any vector fields on , where is the Levi-Civita connection of . Using the local second fundamental forms and of and , respectively, and the projection morphism of on , the curvature tensors , and of the connections , , and on , and , respectively, are given by (see [9]) for any . In case the ambient manifold is a space form of constant -holomorphic sectional curvature , is given by (see [10]) Assume that is almost screen conformal, that is, where is a nonvanishing function on a neighborhood in , and is tangent to , then, by the method in Section 2 of [9], we have where is a nonvanishing function on a neighborhood . Then the leaf of is a semi-Riemannian manifold of quasiconstant curvature such that , , and in (1.1).
Acknowledgment
The authors are thankful to the referee for making various constructive suggestions and corrections towards improving the final version of this paper.