Abstract
We study lightlike hypersurfaces of a semi-Riemannian product manifold. We introduce a class of lightlike hypersurfaces called screen semi-invariant lightlike hypersurfaces and radical anti-invariant lightlike hypersurfaces. We consider lightlike hypersurfaces with respect to a quarter-symmetric nonmetric connection which is determined by the product structure. We give some equivalent conditions for integrability of distributions with respect to the Levi-Civita connection of semi-Riemannian manifolds and the quarter-symmetric nonmetric connection, and we obtain some results.
1. Introduction
The theory of degenerate submanifolds of semi-Riemannian manifolds is one of important topics of differential geometry. The geometry of lightlike submanifolds of a semi-Riemannian manifold, was presented in [1] (see also [2, 3]) by Duggal and Bejancu. In [4], Atçeken and Kılıç introduced semi-invariant lightlike submanifolds of a semi-Riemannian product manifold. In [5], Kılıç and Şahin introduced radical anti-invariant lightlike submanifolds of a semi-Riemannian product manifold and gave some examples and results for lightlike submanifolds. The lightlike hypersurfaces have been studied by many authors in various spaces (for example [6, 7]).
In [8], Hayden introduced a metric connection with nonzero torsion on a Riemannian manifold. The properties of Riemannian manifolds with semisymmetric (symmetric) and nonmetric connection have been studied by many authors [9–14]. In [15], Yaşar et al. have studied lightlike hypersurfaces in semi-Riemannian manifolds with semisymmetric nonmetric connection. The idea of quarter-symmetric linear connections in a differential manifold was introduced by Golab [11]. A linear connection is said to be a quarter-symmetric connection if its torsion tensoris of the form: for any vector fields on a manifold, where is a form and is a tensor of type (1,1).
In this paper, we study lightlike hypersurfaces of a semi-Riemannian product manifold. As a first step, in Section 3, we introduce screen semi-invariant lightlike hypersurfaces and radical anti-invariant lightlike hypersurfaces of a semi-Riemannian product manifold. We give some examples and study their geometric properties. In Section 4, we consider lightlike hypersurfaces of a semi-Riemannian product manifold with quarter-symmetric nonmetric connection determined by the product structure. We compute the Riemannian curvature tensor with respect to the quarter-symmetric nonmetric connection and give some results.
2. Lightlike Hypersurfaces
Let be an dimensional semi-Riemannian manifold with and let be a hypersurface of , with . If the induced metric on is degenerate, then is called a lightlike (null or degenerate) hypersurface [1] (see also [2, 3]). Then there exists a null vector field on such that The radical or the null space of , at each point , is a subspace defined by whose dimension is called the nullity degree of . We recall that the nullity degree of for a lightlike hypersurface of is . Since is degenerate and any null vector being perpendicular to itself, is also null and Since and , we have . We call a radical distribution and it is spanned by the null vector field . The complementary vector bundle of in is called the screen bundle of . We note that any screen bundle is nondegenerate. This means that Here denotes the orthogonal-direct sum. The complementary vector bundle of in is called screen transversal bundle and it has rank . Since is a lightlike subbundle of there exists a unique local section of such that Note that is transversal to and is a local frame field of and there exists a line subbundle of , and it is called the lightlike transversal bundle, locally spanned by . Hence we have the following decomposition: where is the direct sum but not orthogonal [1, 3]. From the above decomposition of a semi-Riemannian manifold along a lightlike hypersurface , we can consider the following local quasiorthonormal field of frames of along : where is an orthonormal basis of . According to the splitting (2.6), we have the following Gauss and Weingarten formulas, respectively: for any , where and . If we set and , then (2.8) become and are called the second fundamental form and the shape operator of the lightlike hypersurface , respectively [1]. Let be the projection of on . Then, for any , we can write where is a form given by
From (2.9), we get and the induced connection is a nonmetric connection on . From (2.4), we have where and belong to . , and are called the local second fundamental form, the local shape operator and the induced connection on , respectively. Also, we have the following identities: Moreover, from the first and third equations of (2.15) we have
Now, we will denote and the curvature tensors of the Levi-Civita connection on and the induced connection on . Then the Gauss equation of is given by where . Then the Gauss-Codazzi equations of a lightlike hypersurface are given by for any .
For geometries of lightlike submanifolds, hypersurfaces and curves, we refer to [1–3].
2.1. Product Manifolds
Let be an dimensional differentiable manifold with a tensor field of type (1,1) on such that Then is called an almost product manifold with almost product structure . If we put then we have Thus and define two complementary distributions and has the eigenvalue of or . If an almost product manifold admits a semi-Riemannian metric such that for any vector fields on , then is called a semi-Riemannian almost product manifold. From (2.19) and (2.22), we have
If, for any vector fields on , then is called a semi-Riemannian product manifold, where is the Levi-Civita connection on .
3. Lightlike Hypersurfaces of Semi-Riemannian Product Manifolds
Let be a lightlike hypersurface of a semi-Riemannian product manifold . For any we can write where is a (1,1) tensor field and is a 1-form on given by .
Definition 3.1. Let be a lightlike hypersurface of a semi-Riemannian product manifold :(i)if and then we say that is a screen semi-invariant lightlike hypersurface;(ii)if then we say that is a screen invariant lightlike hypersurface;(iii)if then we say that is a radical anti-invariant lightlike hypersurface.
We note that a radical anti-invariant lightlike hypersurface is a screen invariant lightlike hypersurface.
Remark 3.2. We recall that there are some lightlike hypersurfaces of a semi-Riemannian product manifold which differ from the above definition, that is, this definition does not cover all lightlike hypersurfaces of a semi-Riemannian product manifold . In this paper we will study the hypersurfaces determined above.
Now, let be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold. If we set then we can write where is a dimensional distribution. Hence we have the following decomposition:
Proposition 3.3. The distribution is an invariant distribution with respect to .
Proof. For any and we obtain Thus there are no components of in and . Furthermore, we have Proof is completed.
If we set , we can write From the above proposition we have the following corollary.
Corollary 3.4. The distribution is invariant with respect to .
Example 3.5. Let ) be a 5-dimensional semi-Euclidean space with signature and be the standard coordinate system of . If we set , then and is a product structure on . Consider a hypersurface in by the equation: Then , where It is easy to check that is a lightlike hypersurface and Then take a lightlike transversal vector bundle as follow: It follows that the corresponding screen distribution is spanned by If we set , and , then it can be easily checked that is a screen semi-invariant lightlike hypersurface of .
Example 3.6. Let be the standard coordinate system of and be a semi-Riemannian metric on with index. Let be a product structure on given by . We consider the hypersurface given by [1]. One can easily see that is a lightlike hypersurface and We can easily check that Thus is a screen semi-invariant lightlike hypersurface with , and .
Example 3.7. Let be a 4-dimensional semi-Euclidean space with signature and be the standard coordinate system of . Consider a Monge hypersurface of given by Then the tangent bundle of the hypersurface is spanned by It is easy to check that is a lightlike hypersurface (p.196, Ex.1, [3]) whose radical distribution is spanned by Furthermore, the lightlike transversal vector bundle is given by It follows that the corresponding screen distribution is spanned by If we define a mapping by then and is a product structure on . One can easily check that and . Thus is a radical anti-invariant lightlike hypersurface of . Furthermore, this lightlike hypersurface is a screen invariant lightlike hypersurface.
Theorem 3.8. Let be a semi-Riemannian product manifold and be a screen semi-invariant lightlike hypersurface of . Then the following assertions are equivalent.(i)The distribution is integrable with respect to the induced connection of .(ii), for any .(iii), for any .
Proof. For any , from (2.9), (2.24), and (3.1), we obtain Interchanging role of and we have From (3.19), (3.20) we get and this is . From the first equation of (2.15), we conclude . Thus we have our assertion.
From the decomposition (3.6), we can give the following definition.
Definition 3.9. Let be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold . If , for any ,, then we say that is a mixed geodesic lightlike hypersurface.
Theorem 3.10. Let be a semi-Riemannian product manifold and be a screen semi-invariant lightlike hypersurface of . Then the following assertions are equivalent.(i) is mixed geodesic.(ii)There is no component of .(iii)There is no component of .
Proof. Suppose that is mixed geodesic screen semi-invariant lightlike hypersurface of with respect to the Levi-Civita connection . From (2.24), (2.9), (2.10), and (3.1), we obtain for any . If we take tangential and transversal parts of this last equation we have Furthermore, since , we get . Since , we obtain This is .
From the decomposition (3.6), we have the following theorem.
Theorem 3.11. Let be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold . Then is a locally product manifold according to the decomposition (3.6) if and only if is parallel with respect to induced connection , that is .
Proof. Let be a locally product manifold. Then the leaves of distributions and are both totally geodesic in . Since the distribution is invariant with respect to then, for any , . Thus and belong to , for any . From the Gauss formula, we obtain
Comparing the tangential and normal parts with respect to of (3.25), we have
Since , for any , we get and , that is . Thus we have on .
Conversely, we assume that on . Then we have , for any and , for any . Thus it follows that and . Hence, the leaves of the distributions and are totally geodesic in .
From Theorem 3.11 and (3.27) we have the following corollary.
Corollary 3.12. Let be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold . If has a local product structure, then it is a mixed geodesic lightlike hypersurface.
Let be a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifold . Then we have the following decomposition:
Theorem 3.13. Let be a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifold . Then the screen distribution of is an integrable distribution if and only if .
Proof. If a vector field on belongs to if and only if . Since is a radical anti-invariant lightlike hypersurface, for any , . For any , we can write In this last equation interchanging role of and , we obtain Since , we get Since , if and only if . This is our assertion.
4. Quarter-Symmetric Nonmetric Connections
Let be a semi-Riemannian product manifold and be the Levi-Civita connection on . If we set for any , then is a linear connection on , where is a 1-form on with as associated vector field, that is The torsion tensor of on denoted by . Then we obtain for any . Thus is a quarter-symmetric nonmetric connection on . From (2.24) and (4.1) we have Replacing by and by in (4.5) and using (2.19) we obtain Thus we have If we set for any , from (4.1) we get From (4.1) the curvature tensor of the quarter-symmetric nonmetric connection is given by for any , where is a -tensor given by . If we set , then, from (4.10), we obtain We note that the Riemannian curvature tensor of does not satisfy the other curvature-like properties. But, from (4.10), we have Thus we have the following proposition.
Proposition 4.1. Let be a lightlike hypersurface of a semi-Riemannian product manifold . Then the first Bianchi identity of the quarter-symmetric nonmetric connection on is provided if and only if is symmetric.
Let be a lightlike hypersurface of a semi-Riemannian product manifold with quarter-symmetric nonmetric connection . Then the Gauss and Weingarten formulas with respect to are given by, respectively, for any , where , , , . Here, , and are called the induced connection on , the second fundamental form, and the Weingarten mapping with respect to . From (2.9), (2.10), (3.1), (4.1), (4.13), and (4.14) we obtain for any , . From (4.1), (4.4), (4.13), and (4.16) we get On the other hand, the torsion tensor of the induced connection is From last two equations we have the following proposition.
Proposition 4.2. Let be a lightlike hypersurface of a semi-Riemannian product manifold with quarter-symmetric nonmetric connection . Then the induced connection is a quarter-symmetric nonmetric connection on the lightlike hypersurface .
For any , we can write where , , and . From (2.14), (16), and (4.15), we obtain Using (2.15), (4.16) and (4.22) we obtain for any .
Now, we consider a screen semi-invariant lightlike hypersurface of a semi-Rieamannian product manifold with respect to the quarter symmetric connection given by (4.1). Since , for any . Thus we have the following propositions.
Proposition 4.3. Let be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold with quarter-symmetric nonmetric connection. The second fundamental form of quarter-symmetric nonmetric connection is degenerate.
Proposition 4.4. Let be a semi-Riemannian product manifold and be a screen semi-invariant lightlike hypersurfaces of . If is totally geodesic with respect to , then is totally geodesic with respect to quarter-symmetric nonmetric connection.
Theorem 4.5. Let be a semi-Riemannian product manifold and be a screen semi-invariant lightlike hypersurfaces of . Then the following assertions are equivalent.(i)The distribution is integrable with respect to the quarter symmetric nonmetric connection .(ii), for any , .(iii), for any , .
The proof of this theorem is similar to the proof of the Theorem 3.8.
From (4.23), for any and , we have . If we set , then, from Theorem 3.10, we have the following corollary.
Corollary 4.6. Let be a semi-Riemannian product manifold and be a screen semi-invariant lightlike hypersurface of . Then the distribution is a mixed geodesic foliation defined with respect to quarter symmetric nonmetric connection if and only if there is no component of .
From (4.15), we obtain where is a tensor on given by .
From (4.24), we have the following proposition which is similar to the Proposition 4.1.
Proposition 4.7. Let be a lightlike hypersurface of a semi-Riemannian product manifold . One assumes that is parallel on . Then the first Bianchi identity of the quarter-symmetric nonmetric connection on is provided if and only if is symmetric.
Now we will compute Gauss-Codazzi equations of lightlike hypersurfaces with respect to the quarter-symmetric nonmetric connection: for any , , , .
Now, let be a screen semi-invariant lightlike hypersurface of a dimensional semi-Riemannian product manifold with the quarter-symmetric nonmetric connection such that the tensor field is parallel on . We consider the local quasiorthonormal basis , , of along , where is an orthonormal basis of . Then, the Ricci tensor of with respect to is given by From (4.24) we have where is the Ricci tensor of . Thus we have the following corollary.
Corollary 4.8. Let a screen semi-invariant lightlike hypersurface of a dimensional semi-Riemannian product manifold with the quarter-symmetric nonmetric connection such that the tensor field is parallel on and is symmetric. Then is symmetric on the distribution if and only if is symmetric and .
Acknowledgment
The authors have greatly benefited from the referee's report. So we wish to express our gratitude to the reviewer for his/her valuable suggestions which improved the content and presentation of the paper. This paper is dedicated to Professor Sadık Keleş on his sixtieth birthday.