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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 304647, 11 pages
http://dx.doi.org/10.1155/2012/304647
Research Article

On Geometry of Submanifolds of -Manifolds

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa University, 60250 Tokat, Turkey

Received 13 September 2011; Revised 16 December 2011; Accepted 17 December 2011

Academic Editor: Attila Gilányi

Copyright © 2012 Mehmet Atceken. 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 show geometrical properties of a submanifold of a -manifold. The properties of the induced structures on such a submanifold are also studied.

1. Introduction

The geometry of manifolds endowed with geometrical structures has been intensively studied, and several important results have been published. In this paper, we deal with manifolds having a Lorentzian concircular structure (-manifold) [13] (see Section 2 for detail).

The study of the Lorentzian almost paracontact manifold was initiated by Matsumoto in [4]. Later on, several authors studied the Lorentzian almost paracontact manifolds and their different classes including [1, 4, 5]. Recently, the notion of the Lorentzian concircular structure manifolds was introduced in (briefly (LCS)-manifolds) with an example, which generalizes the notion of the LP-Sasakian manifolds introduced by Matsumoto in [4].

Papers related to this issue are very few in the literature so far. But the geometry of submanifolds of a (LCS)-manifold is rich and interesting. So, in the present paper we introduce the concept of submanifolds of a (LCS)-manifold and investigate the fundamental properties of such submanifolds. We obtain the necessary and sufficient conditions for a submanifold of (LCS)-manifold to be invariant. In this case, the induced structures on submanifold by the structure on ambient space are classified. I think that the results will contribute to geometry.

2. Preliminaries

An -dimensional Lorentzian manifold is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric , that is, admits a smooth symmetric tensor field of type such that, for each point , the tensor is a nondegenerate inner product of signature , where denotes the tangent vector space of at and is the real number space. A nonzero vector is said to be timelike (resp., non-spacelike, null, and spacelike) if it satisfies (resp., , , and ) [6].

Definition 2.1. In a Lorentzian manifold , a vector field defined by for any , is said to be a concircular vector field if where is a nonzero scalar and is a closed 1-form and denotes the operator of covariant differentiation with respect to the Lorentzian metric .

Let be an -dimensional Lorentzian manifold admitting a unit time-like concircular vector field , called the characteristic vector field of the manifold. Then, we have Since is a unit concircular vector field, it follows that there exists a nonzero 1-form such that, for the equation of the following form holds: for all vector fields , , where denotes the operator of covariant differentiation with respect to the Lorentzian metric and is a nonzero scalar function satisfying being a certain scalar function given by . If we put then from (2.5) and (2.7) we have from which, it follows that is a symmetric (1,1) tensor and called the structure tensor of the manifold. Thus, the Lorentzian manifold together with the unit time-like concircular vector field , its associated 1-form , and an (1,1) tensor field is said to be a Lorentzian concircular structure manifold (briefly, -manifold). Especially, if we take , then we can obtain the LP-Sasakian structure of Matsumoto [4]. In a -manifold , the following relations hold: for all .

3. Submanifolds of a (LCS)-Manifold

Let be an isometrically immersed submanifold of a -manifold with induced metric ; we define the isometric immersion by and denote by the differential of . The induced Riemannian metric on by satisfies , for all .

We denote the tangent and normal spaces of at point by and , respectively. Let be an orthonormal basis of the normal space , where , that is, .

For any , we can write where , and denote induced (1-1)-tensor, 1-forms, vector fields and functions on , respectively. The vector field on -manifold can be written as follows: where and are vector field and functions on and , respectively. From (3.1) and (3.2), we can derive that is, is symmetric and Here, we note that the induced (1-1)-tensor field is also symmetric because is symmetric.

Next, we will the following Lemmas for later use.

Lemma 3.1. Let be an isometrically immersed submanifold of a -manifold . Then, the following assertions are true: where denotes the induced 1-form on by on and given by .

Proof. For any , by using (2.10), (3.1), and (3.2), we have Also considering (3.3), we arrive at From the tangential components of (3.10), we conclude that which is equivalent to (3.6). On the other hand, with the normal components of (3.10), we have which implies that that is, This proves (3.7). In order to prove (3.8), taking (2.10) and (3.2), into account we have Taking the product of (3.15) with , , we reach which gives us (3.8).

Lemma 3.2. Let be an isometrically immersed submanifold of a -manifold . Then, the following assertions are true: for any .

Proof. Making use of and (3.3), we have From the tangential and normal components of this last equation, respectively, we get Again, taking into account that is time-like vector and (3.3), we reach Finally, we conclude that This proves our assertions.

Now, we suppose that and are two orthonormal bases of at and set by means of . So, we mean that the basis with another basis transition matrix is an orthogonal matrix. From (3.24) we have Taking (3.24) into account, (3.1), (3.2), and (3.3) are, respectively, written in the following way: where Furthermore, because is symmetric, from (3.30), we can derive that under the suitable transformation (3.24) reduce to , where are eigenvalues of matrix . So, again (3.27) and (3.8) can be, respectively, written in the following way: which implies that and for .

Now, let be an isometrically immersed submanifold of a -manifold . If for any point , then is said to be an invariant submanifold of . In this case, (3.1), (3.2), and (3.3) become, respectively, for any .

Lemma 3.3. Let be an invariant submanifold of a -manifold . Then, the following assertions are true: for any .

Proof. The proof is obvious. Therefore, we omit it.

Theorem 3.4. Let be an invariant submanifold of a (LCS)-manifold . One of the following cases occurs.(1)If is normal to , then the induced structure on is an almost product Riemannian structure whenever is nontrivial.(2)If is tangent to , then the induced structure on is a Lorentzian concircular structure.

Proof. (1) If is normal to the submanifold, then the vector field . From (3.35) and (3.37), we have , , that is, is an almost product Riemannian structure whenever is nontrivial.
(2) If is tangent to the submanifold (i.e., , ), then we have , , , , that is, is a Lorentzian concircular structure.

Theorem 3.5. Let be a submanifold of a (LCS)-manifold . The submanifold of a (LCS)-manifold is invariant if and only if the induced structure on is an almost product Riemannian structure whenever is nontrivial or the induced structure on is a Lorentzian concircular structure.

Proof. From Theorem 3.4 we know that the necessary is obvious.
Conversely, we suppose that the induced structure is an almost product Riemannian structure. Then, from (3.19), we have that is, , . So from (3.1) and (3.3) we can derive that the submanifold is invariant and is normal to .
Now, we suppose that the induced structure is a Lorentzian concircular structure. Then, from (3.6), we get which implies that , . From (3.7), by a direct calculation, we derive , . So from (3.1) and (3.3), we conclude that is invariant submanifold and is tangent to .

Theorem 3.6. Let be an isometrically immersed submanifold of (LCS)-manifold . Then, is invariant submanifold if and only if the normal space , at every point , admits an orthonormal basis consisting of the eigenvectors of the matrix .

Proof. Let us suppose that is invariant.
(1) When is normal to , at we consider an -dimensional vector space and investigate the eigenvalues of the matrix . From (3.36) and (3.37), it is easy to see that the vector of the vector space is a unit eigenvector of the matrix and its eigenvalue is equal to 0.
Now, we suppose that a vector satisfying is an eigenvector and its eigenvalue is . Then, we have which implies that from which On the other hand, from (3.36) we get that is, , which is equivalent to .
Consequently, if by a suitable transformation of the orthonormal basis of , the matrix becomes a diagonal matrix, then the diagonal components satisfy relations In this case, if we denote by another orthonormal basis of , then, from (3.31), we have , . So, , , are eigenvectors of the matrix- and .
(2) When is tangent to , since , , from (3.36), we have If we denote by an eigenvector of matrix and by its eigenvalue, then we have So, we obtain that is, , which implies that . Since the eigenvalues of are , by a suitable transformation of the orthonormal basis of , to become , then are eigenvectors of matrix-.
Conversely, if the orthonormal basis of consists of eigenvectors of matrix- and these eigenvalues satisfy and or 0, then we have and we conclude that , , and so is invariant.

Acknowledgment

The author would like to express my gratitude to the referees for valuable comments and suggestions.

References

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