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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 178390, 17 pages
http://dx.doi.org/10.1155/2012/178390
Research Article

Lightlike Hypersurfaces of a Semi-Riemannian Product Manifold and Quarter-Symmetric Nonmetric Connections

1Department of Mathematics, Faculty of Arts and Sciences, İnönü University, 44280 Malatya, Turkey
2Department of Mathematics, Faculty of Arts and Sciences, Hitit University, 19030 Çorum, Turkey

Received 28 March 2012; Revised 2 July 2012; Accepted 16 July 2012

Academic Editor: Jianya Liu

Copyright © 2012 Erol Kılıç and Oğuzhan Bahadır. 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 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 [914]. 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 tensor𝑇is of the form: 𝑇(𝑋,𝑌)=𝑢(𝑌)𝜑𝑋𝑢(𝑋)𝜑𝑌,(1.1) for any vector fields 𝑋,𝑌 on a manifold, where 𝑢 is a 1-form and 𝜑 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 (𝑀,𝑔) be an (𝑚+2)-dimensional semi-Riemannian manifold with index(𝑔)=𝑞1 and let (𝑀,𝑔) be a hypersurface of 𝑀, with 𝑔=𝑔|𝑀. If the induced metric 𝑔 on 𝑀 is degenerate, then 𝑀 is called a lightlike (null or degenerate) hypersurface [1] (see also [2, 3]). Then there exists a null vector field 𝜉0 on 𝑀 such that 𝑔(𝜉,𝑋)=0,𝑋Γ(𝑇𝑀).(2.1) The radical or the null space of 𝑇𝑥𝑀, at each point 𝑥𝑀, is a subspace Rad𝑇𝑥𝑀 defined by Rad𝑇𝑥𝑀=𝜉𝑇𝑥𝑀||𝑔𝑥(𝜉,𝑋)=0,𝑋Γ(𝑇𝑀),(2.2) whose dimension is called the nullity degree of 𝑔. We recall that the nullity degree of 𝑔 for a lightlike hypersurface of 𝑀 is 1. Since 𝑔 is degenerate and any null vector being perpendicular to itself, 𝑇𝑥𝑀 is also null and Rad𝑇𝑥𝑀=𝑇𝑥𝑀𝑇𝑥𝑀.(2.3) Since dim𝑇𝑥𝑀=1 and dimRad𝑇𝑥𝑀=1, we have Rad𝑇𝑥𝑀=𝑇𝑥𝑀. We call Rad𝑇𝑀 a radical distribution and it is spanned by the null vector field 𝜉. The complementary vector bundle 𝑆(𝑇𝑀) of Rad𝑇𝑀 in 𝑇𝑀 is called the screen bundle of 𝑀. We note that any screen bundle is nondegenerate. This means that 𝑇𝑀=Rad𝑇𝑀𝑆(𝑇𝑀).(2.4) Here denotes the orthogonal-direct sum. The complementary vector bundle 𝑆(𝑇𝑀) of𝑆(𝑇𝑀) in 𝑇𝑀 is called screen transversal bundle and it has rank 2. Since Rad𝑇𝑀is a lightlike subbundle of 𝑆(𝑇𝑀) there exists a unique local section 𝑁 of 𝑆(𝑇𝑀) such that 𝑔(𝑁,𝑁)=0,𝑔(𝜉,𝑁)=1.(2.5) Note that 𝑁 is transversal to 𝑀 and {𝜉,𝑁} is a local frame field of 𝑆(𝑇𝑀) and there exists a line subbundle ltr(𝑇𝑀) of 𝑇𝑀, and it is called the lightlike transversal bundle, locally spanned by 𝑁. Hence we have the following decomposition: 𝑇𝑀=𝑇𝑀ltr(𝑇𝑀)=𝑆(𝑇𝑀)Rad𝑇𝑀ltr(𝑇𝑀),(2.6) where is the direct sum but not orthogonal [1, 3]. From the above decomposition of a semi-Riemannian manifold 𝑀 along a lightlike hypersurface 𝑀, we can consider the following local quasiorthonormal field of frames of 𝑀 along 𝑀: 𝑋1,,𝑋𝑚,𝜉,𝑁,(2.7) where {𝑋1,,𝑋𝑚} is an orthonormal basis of Γ(𝑆(𝑇𝑀)). According to the splitting (2.6), we have the following Gauss and Weingarten formulas, respectively: 𝑋𝑌=𝑋𝑌+(𝑋,𝑌),𝑋𝑁=𝐴𝑁𝑋+𝑡𝑋𝑁,(2.8) for any 𝑋,𝑌Γ(𝑇𝑀), where 𝑋𝑌,𝐴𝑁𝑋Γ(𝑇𝑀) and (𝑋,𝑌),𝑡𝑋𝑁Γ(ltr(𝑇𝑀)). If we set 𝐵(𝑋,𝑌)=𝑔((𝑋,𝑌),𝜉) and 𝜏(𝑋)=𝑔(𝑡𝑋𝑁,𝜉), then (2.8) become 𝑋𝑌=𝑋𝑌+𝐵(𝑋,𝑌)𝑁,(2.9)𝑋𝑁=𝐴𝑁𝑋+𝜏(𝑋)𝑁.(2.10)𝐵 and 𝐴 are called the second fundamental form and the shape operator of the lightlike hypersurface 𝑀, respectively [1]. Let 𝑃 be the projection of 𝑆(𝑇𝑀) on 𝑀. Then, for any 𝑋Γ(𝑇𝑀), we can write 𝑋=𝑃𝑋+𝜂(𝑋)𝜉,(2.11) where 𝜂 is a 1-form given by 𝜂(𝑋)=𝑔(𝑋,𝑁).(2.12)

From (2.9), we get 𝑋𝑔(𝑌,𝑍)=𝐵(𝑋,𝑌)𝜂(𝑍)+𝐵(𝑋,𝑍)𝜂(𝑌),𝑋,𝑌,𝑍Γ(𝑇𝑀),(2.13) and the induced connection is a nonmetric connection on 𝑀. From (2.4), we have 𝑋𝑊=𝑋𝑊+(𝑋,𝑊)=𝑋𝑊+𝐶(𝑋,𝑊)𝜉,𝑋Γ(𝑇𝑀),𝑊Γ(𝑆(𝑇𝑀)),𝑋𝜉=𝐴𝜉𝑋𝜏(𝑋)𝜉,(2.14) where 𝑋𝑊 and 𝐴𝜉𝑋 belong to Γ(𝑆(𝑇𝑀)). 𝐶, 𝐴𝜉 and are called the local second fundamental form, the local shape operator and the induced connection on 𝑆(𝑇𝑀), respectively. Also, we have the following identities: 𝑔𝐴𝜉𝐴𝑋,𝑊=𝐵(𝑋,𝑊),𝑔𝜉𝐴𝑋,𝑁=0,𝐵(𝑋,𝜉)=0,𝑔𝑁𝑋,𝑁=0.(2.15) Moreover, from the first and third equations of (2.15) we have 𝐴𝜉𝜉=0.(2.16)

Now, we will denote 𝑅 and 𝑅 the curvature tensors of the Levi-Civita connection on 𝑀 and the induced connection on 𝑀. Then the Gauss equation of 𝑀 is given by 𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝐴(𝑋,𝑍)𝑌𝐴(𝑌,𝑍)𝑋+𝑋(𝑌,𝑍)𝑌(𝑋,𝑍),𝑋,𝑌,𝑍Γ(𝑇𝑀),(2.17) where (𝑋)(𝑌,𝑍)=𝑡𝑋((𝑌,𝑍))(𝑋𝑌,𝑍)(𝑌,𝑋𝑍). Then the Gauss-Codazzi equations of a lightlike hypersurface are given by 𝑔𝑅(𝑋,𝑌)𝑍,𝑃𝑊=𝑔(𝑅(𝑋,𝑌)𝑍,𝑃𝑊)+𝐵(𝑋,𝑍)𝐶(𝑌,𝑃𝑊)𝐵(𝑌,𝑍)𝐶(𝑋,𝑃𝑊),𝑔=𝑅(𝑋,𝑌)𝑍,𝜉𝑋𝐵(𝑌,𝑍)𝑌𝐵(𝑋,𝑍)+𝐵(𝑌,𝑍)𝜏(𝑋)𝐵(𝑋,𝑍)𝜏(𝑌),𝑔𝑅(𝑋,𝑌)𝑍,𝑁=𝑔(𝑅(𝑋,𝑌)𝑍,𝑁),𝑔𝑅(𝑋,𝑌)𝜉,𝑁=𝑔(𝑅(𝑋,𝑌)𝜉,𝑁)=𝐶𝑌,𝐴𝜉𝑋𝐶𝑋,𝐴𝜉𝑌2𝑑𝜏(𝑋,𝑌),(2.18) for any 𝑋,𝑌,𝑍,𝑊Γ(𝑇𝑀),𝜉Γ(Rad𝑇𝑀).

For geometries of lightlike submanifolds, hypersurfaces and curves, we refer to [13].

2.1. Product Manifolds

Let 𝑀 be an 𝑛-dimensional differentiable manifold with a tensor field 𝐹 of type (1,1) on 𝑀 such that 𝐹2=𝐼.(2.19) Then 𝑀is called an almost product manifold with almost product structure 𝐹. If we put 1𝜋=21(𝐼+𝐹),𝜎=2(𝐼𝐹),(2.20) then we have 𝜋+𝜎=𝐼,𝜋2=𝜋,𝜎2=𝜎,𝜎𝜋=𝜋𝜎=0,𝐹=𝜋𝜎.(2.21) Thus 𝜋 and 𝜎 define two complementary distributions and 𝐹 has the eigenvalue of+1 or 1. If an almost product manifold 𝑀 admits a semi-Riemannian metric 𝑔 such that 𝑔(𝐹𝑋,𝐹𝑌)=𝑔(𝑋,𝑌),(2.22) for any vector fields 𝑋,𝑌 on 𝑀, then 𝑀 is called a semi-Riemannian almost product manifold. From (2.19) and (2.22), we have 𝑔(𝐹𝑋,𝑌)=𝑔(𝑋,𝐹𝑌).(2.23)

If, for any vector fields 𝑋,𝑌 on 𝑀, 𝐹=0,thatis𝑋𝐹𝑌=𝐹𝑋𝑌,(2.24) then 𝑀 is called a semi-Riemannian product manifold, where is the Levi-Civita connection on 𝑀.

3. Lightlike Hypersurfaces of Semi-Riemannian Product Manifolds

Let 𝑀 be a lightlike hypersurface of a semi-Riemannian product manifold (𝑀,𝑔). For any 𝑋Γ(𝑇𝑀) we can write 𝐹𝑋=𝑓𝑋+𝑤(𝑋)𝑁,(3.1) where 𝑓 is a (1,1) tensor field and 𝑤 is a 1-form on 𝑀 given by 𝑤(𝑋)=𝑔(𝐹𝑋,𝜉)=𝑔(𝑋,𝐹𝜉).

Definition 3.1. Let 𝑀 be a lightlike hypersurface of a semi-Riemannian product manifold (𝑀,𝑔):(i)if 𝐹Rad𝑇𝑀𝑆(𝑇𝑀) and 𝐹ltr(𝑇𝑀)𝑆(𝑇𝑀) then we say that 𝑀 is a screen semi-invariant lightlike hypersurface;(ii)if 𝐹𝑆(𝑇𝑀)=𝑆(𝑇𝑀) then we say that 𝑀 is a screen invariant lightlike hypersurface;(iii)if 𝐹Rad𝑇𝑀=ltr(𝑇𝑀) then we say that 𝑀 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 (𝑀,𝑔). In this paper we will study the hypersurfaces determined above.

Now, let 𝑀 be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold. If we set 𝔻1=𝐹Rad𝑇𝑀,𝔻2=𝐹ltr(𝑇𝑀) then we can write 𝑆𝔻(𝑇𝑀)=𝔻1𝔻2,(3.2) where 𝔻 is a (𝑚2)-dimensional distribution. Hence we have the following decomposition: 𝔻𝑇𝑀=𝔻1𝔻2𝑇Rad𝑇𝑀,𝔻𝑀=𝔻1𝔻2{Rad𝑇𝑀ltr(𝑇𝑀)}.(3.3)

Proposition 3.3. The distribution 𝔻 is an invariant distribution with respect to 𝐹.

Proof. For any 𝑋Γ(𝔻) and 𝑈Γ(𝔻1),𝑉Γ(𝔻2) we obtain 𝑔𝑔(𝐹𝑋,𝑈)=𝑔(𝑋,𝐹𝑈)=0,(𝐹𝑋,𝑉)=𝑔(𝑋,𝐹𝑉)=0.(3.4) Thus there are no components of 𝐹𝑋 in 𝔻1 and 𝔻2. Furthermore, we have 𝑔𝑔(𝐹𝑋,𝜉)=𝑔(𝑋,𝐹𝜉)=0,(𝐹𝑋,𝑁)=𝑔(𝑋,𝐹𝑁)=0.(3.5) Proof is completed.

If we set 𝔻=𝔻Rad𝑇𝑀𝐹Rad𝑇𝑀, we can write 𝑇𝑀=𝔻𝔻2.(3.6) From the above proposition we have the following corollary.

Corollary 3.4. The distribution 𝔻 is invariant with respect to 𝐹.

Example 3.5. Let (𝑀=𝑅52,𝑔) be a 5-dimensional semi-Euclidean space with signature(,+,,+,+) and (𝑥,𝑦,𝑧,𝑠,𝑡) be the standard coordinate system of 𝑅52. If we set 𝐹(𝑥,𝑦,𝑧,𝑠,𝑡)=(𝑥,𝑦,𝑧,𝑠,𝑡), then 𝐹2=𝐼 and 𝐹 is a product structure on 𝑅52. Consider a hypersurface 𝑀 in 𝑀 by the equation: 𝑡=𝑥+𝑦+𝑧.(3.7) Then 𝑇𝑀=Span{𝑈1,𝑈2,𝑈3,𝑈4}, where 𝑈1=𝜕+𝜕𝜕𝑥𝜕𝑡,𝑈2=𝜕+𝜕𝜕𝑦𝜕𝑡,𝑈3=𝜕+𝜕𝜕𝑧𝜕𝑡,𝑈4=𝜕.𝜕𝑠(3.8) It is easy to check that 𝑀 is a lightlike hypersurface and 𝑇𝑀=Span𝜉=𝑈1𝑈2+𝑈3.(3.9) Then take a lightlike transversal vector bundle as follow: 1ltr(𝑇𝑀)=Span𝑁=4𝜕+𝜕𝜕𝑥+𝜕𝜕𝑦𝜕𝜕𝑧𝜕𝑡.(3.10) It follows that the corresponding screen distribution 𝑆(𝑇𝑀) is spanned by 𝑊1=𝑈4,𝑊2=𝑈1𝑈2𝑈3,𝑊3=𝑈1+𝑈2𝑈3.(3.11) If we set 𝔻=Span{𝑊1}, 𝔻1=Span{𝑊2} and 𝔻2=Span{𝑊3}, then it can be easily checked that 𝑀 is a screen semi-invariant lightlike hypersurface of 𝑀.

Example 3.6. Let (𝑥,𝑦,𝑧,𝑡) be the standard coordinate system of 𝑅4 and 𝑑𝑠2=𝑑𝑥2𝑑𝑦2+𝑑𝑧2+𝑑𝑡2 be a semi-Riemannian metric on 𝑅4 with 2-index. Let 𝐹 be a product structure on 𝑅4 given by 𝐹(𝑥,𝑦,𝑧,𝑡)=(𝑧,𝑡,𝑥,𝑦). We consider the hypersurface 𝑀 given by 𝑡=𝑥+(1/2)(𝑦+𝑧)2 [1]. One can easily see that 𝑀 is a lightlike hypersurface and 𝜕Rad𝑇𝑀=Span𝜉=𝜕𝜕𝑥+(𝑦+𝑧)𝜕𝜕𝑦(𝑦+𝑧)+𝜕𝜕𝑧,1𝜕𝑡ltr(𝑇𝑀)=Span𝑁=21+(𝑦+𝑧)2𝜕𝜕𝜕𝑥+(𝑦+𝑧)𝜕𝜕𝑦+(𝑦+𝑧)𝜕𝜕𝑧,𝑊𝜕𝑡𝑆(𝑇𝑀)=Span1𝜕=(𝑦+𝑧)+𝜕𝜕𝑥𝜕𝑦,𝑊2=𝜕𝜕𝜕𝑧+(𝑦+𝑧).𝜕𝑡(3.12) We can easily check that 𝐹𝜉=𝑊1+𝑊21,𝐹𝑁=21+(𝑦+𝑧)2𝑊1𝑊2.(3.13) Thus 𝑀 is a screen semi-invariant lightlike hypersurface with 𝔻={0}, 𝔻1=Span{𝐹𝜉} and 𝔻2=Span{𝐹𝑁}.

Example 3.7. Let (𝑅42,𝑔) be a 4-dimensional semi-Euclidean space with signature (,,+,+) and (𝑥1,𝑥2,𝑥3,𝑥4) be the standard coordinate system of 𝑅42. Consider a Monge hypersurface 𝑀 of 𝑅42 given by 𝑥4=𝐴𝑥1+𝐵𝑥2+𝐶𝑥3,𝐴2+𝐵2𝐶2=1,𝐴,𝐵,𝐶𝑅.(3.14) Then the tangent bundle 𝑇𝑀 of the hypersurface 𝑀 is spanned by 𝑈1=𝜕𝜕𝑥1𝜕+𝐴𝜕𝑥4,𝑈2=𝜕𝜕𝑥2𝜕+𝐵𝜕𝑥4,𝑈3=𝜕𝜕𝑥3𝜕+𝐶𝜕𝑥4.(3.15) It is easy to check that 𝑀 is a lightlike hypersurface (p.196, Ex.1, [3]) whose radical distribution Rad𝑇𝑀 is spanned by 𝜉=𝐴𝑈1+𝐵𝑈2𝐶𝑈3𝜕=𝐴𝜕𝑥1𝜕+𝐵𝜕𝑥2𝜕𝐶𝜕𝑥3+𝜕𝜕𝑥4.(3.16) Furthermore, the lightlike transversal vector bundle is given by 1ltr(𝑇𝑀)=Span𝑁=2𝐶2𝐴𝜕+1𝜕𝑥1𝜕+𝐵𝜕𝑥2𝜕+𝐶𝜕𝑥3𝜕𝜕𝑥4.(3.17) It follows that the corresponding screen distribution 𝑆(𝑇𝑀) is spanned by 𝑊1=1𝐴2+𝐵2𝐵𝜕𝜕𝑥1𝜕𝐴𝜕𝑥2,𝑊2=1𝐴2+𝐵2𝜕𝜕𝑥3𝜕+𝐶𝜕𝑥4.(3.18) If we define a mapping 𝐹 by 𝐹(𝑥1,𝑥2,𝑥3,𝑥4)=(𝑥1,𝑥2,𝑥3,𝑥4) then 𝐹2=𝐼 and 𝐹 is a product structure on 𝑅42. One can easily check that 𝐹𝑆(𝑇𝑀)=𝑆(𝑇𝑀) and 𝐹Rad𝑇𝑀=ltr(𝑇𝑀). Thus 𝑀 is a radical anti-invariant lightlike hypersurface of 𝑅42. Furthermore, this lightlike hypersurface is a screen invariant lightlike hypersurface.

Theorem 3.8. Let (𝑀,𝑔) be a semi-Riemannian product manifold and 𝑀 be a screen semi-invariant lightlike hypersurface of 𝑀. Then the following assertions are equivalent.(i)The distribution 𝔻 is integrable with respect to the induced connection of 𝑀.(ii)𝐵(𝑋,𝑓𝑌)=𝐵(𝑌,𝑓𝑋), for any 𝑋,𝑌Γ(𝔻).(iii)𝑔(𝐴𝜉𝑋,𝑃𝑓𝑌)=𝑔(𝐴𝜉𝑌,𝑃𝑓𝑋), for any 𝑋,𝑌Γ(𝔻).

Proof. For any 𝑋,𝑌Γ(𝔻), from (2.9), (2.24), and (3.1), we obtain 𝑓𝑋𝑌+𝑤𝑋𝑌𝑁+𝐵(𝑋,𝑌)𝐹𝑁=𝑋𝑓𝑌+𝐵(𝑋,𝑓𝑌)𝑁.(3.19) Interchanging role of 𝑋 and 𝑌 we have 𝑓𝑌𝑋+𝑤𝑌𝑋𝑁+𝐵(𝑌,𝑋)𝐹𝑁=𝑌𝑓𝑋+𝐵(𝑌,𝑓𝑋)𝑁.(3.20) From (3.19), (3.20) we get []𝑤(𝑋,𝑌)=𝐵(𝑋,𝑓𝑌)𝐵(𝑌,𝑓𝑋)(3.21) and this is (i)(ii). From the first equation of (2.15), we conclude (ii)(iii). Thus we have our assertion.

From the decomposition (3.6), we can give the following definition.

Definition 3.9. Let 𝑀 be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold 𝑀. If 𝐵(𝑋,𝑌)=0, for any 𝑋Γ(𝔻),𝑌Γ(𝔻2), then we say that 𝑀 is a mixed geodesic lightlike hypersurface.

Theorem 3.10. Let (𝑀,𝑔) be a semi-Riemannian product manifold and 𝑀 be a screen semi-invariant lightlike hypersurface of 𝑀. Then the following assertions are equivalent.(i)𝑀 is mixed geodesic.(ii)There is no 𝔻2-component of 𝐴𝑁.(iii)There is no 𝔻1-component of 𝐴𝜉.

Proof. Suppose that 𝑀 is mixed geodesic screen semi-invariant lightlike hypersurface of 𝑀 with respect to the Levi-Civita connection . From (2.24), (2.9), (2.10), and (3.1), we obtain 𝑋𝐹𝑁+𝐵(𝑋,𝐹𝑁)𝑁=𝑓𝐴𝑁𝐴𝑋+𝜏(𝑋)𝐹𝑁𝑤𝑁𝑋𝑁,(3.22) for any 𝑋Γ(𝔻). If we take tangential and transversal parts of this last equation we have 𝑋𝐹𝑁=𝑓𝐴𝑁𝐴𝑋+𝜏(𝑋)𝐹𝑁,𝐵(𝑋,𝐹𝑁)=𝑤𝑁𝑋.(3.23) Furthermore, since 𝑤(𝐴𝑁𝑋)=𝑔(𝐴𝑁𝑋,𝐹𝜉), we get (i)(ii). Since 𝑔(𝐹𝑁,𝜉)=𝑔(𝑁,𝐹𝜉)=0, we obtain 𝑔𝐴𝑁𝐴𝑋,𝐹𝜉=𝑔𝜉𝑋,𝐹𝑁.(3.24) This is (ii)(iii).

From the decomposition (3.6), we have the following theorem.

Theorem 3.11. Let 𝑀 be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold 𝑀. Then 𝑀 is a locally product manifold according to the decomposition (3.6) if and only if 𝑓 is parallel with respect to induced connection , that is 𝑓=0.

Proof. Let 𝑀 be a locally product manifold. Then the leaves of distributions 𝔻 and 𝔻2 are both totally geodesic in 𝑀. Since the distribution 𝔻 is invariant with respect to 𝐹 then, for any 𝑌Γ(𝔻), 𝐹𝑌Γ(𝔻). Thus 𝑋𝑌 and 𝑋𝑓𝑌 belong to Γ(𝔻), for any 𝑋Γ(𝑇𝑀). From the Gauss formula, we obtain 𝑋𝑓𝑌+𝐵(𝑋,𝑓𝑌)𝑁=𝑓𝑋𝑌+𝑤𝑋𝑌𝑁+𝐵(𝑋,𝑌)𝐹𝑁.(3.25) Comparing the tangential and normal parts with respect to 𝔻 of (3.25), we have 𝑋𝑓𝑌=𝑓𝑋𝑌,thatis𝑋𝑓𝑌=0,(3.26)𝐵(𝑋,𝑌)=0.(3.27) Since 𝑓𝑍=0, for any 𝑍Γ(𝔻2), we get 𝑋𝑓𝑍=0 and 𝑓𝑋𝑍=0, that is (𝑋𝑓)𝑍=0. Thus we have 𝑓=0 on 𝑀.
Conversely, we assume that 𝑓=0 on 𝑀. Then we have 𝑋𝑓𝑌=𝑓𝑋𝑌, for any 𝑋,𝑌Γ(𝔻) and 𝑈𝑓𝑊=𝑓𝑈𝑊=0, for any 𝑈,𝑊Γ(𝔻2). Thus it follows that 𝑋𝑓𝑌Γ(𝔻) and 𝑈𝑊Γ(𝔻2). Hence, the leaves of the distributions 𝔻 and 𝔻2 are totally geodesic in 𝑀.

From Theorem 3.11 and (3.27) we have the following corollary.

Corollary 3.12. Let 𝑀 be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold 𝑀. If 𝑀 has a local product structure, then it is a mixed geodesic lightlike hypersurface.

Let 𝑀 be a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifold 𝑀. Then we have the following decomposition: 𝑇𝑀=𝑆(𝑇𝑀){Rad𝑇𝑀𝐹Rad𝑇𝑀}.(3.28)

Theorem 3.13. Let 𝑀 be a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifold 𝑀. Then the screen distribution 𝑆(𝑇𝑀) of 𝑀 is an integrable distribution if and only if 𝐵(𝑋,𝐹𝑌)=𝐵(𝑌,𝐹𝑋).

Proof. If a vector field 𝑋 on 𝑀 belongs to 𝑆(𝑇𝑀) if and only if 𝜂(𝑋)=0. Since 𝑀 is a radical anti-invariant lightlike hypersurface, for any 𝑋Γ(𝑆(𝑇𝑀)), 𝐹𝑋Γ(𝑆(𝑇𝑀)). For any 𝑋,𝑌Γ(𝑆(𝑇𝑀)), we can write 𝑋𝐹𝑌=𝑋𝐹𝑌+𝐵(𝑋,𝐹𝑌)𝑁.(3.29) In this last equation interchanging role of 𝑋 and 𝑌, we obtain 𝐹[]𝑋,𝑌=𝑋𝐹𝑌𝑌𝐹𝑋+(𝐵(𝑋,𝐹𝑌)𝐵(𝑌,𝐹𝑋))𝑁.(3.30) Since 𝜂([𝑋,𝑌])=𝑔([𝑋,𝑌],𝑁)=𝑔(𝐹[𝑋,𝑌],𝐹𝑁), we get []𝜂(𝑋,𝑌)=(𝐵(𝑋,𝐹𝑌)𝐵(𝑌,𝐹𝑋))𝑔(𝑁,𝐹𝑁).(3.31) Since 𝑔(𝑁,𝐹𝑁)0, 𝜂([𝑋,𝑌])=0 if and only if 𝐵(𝑋,𝐹𝑌)=𝐵(𝑌,𝐹𝑋). This is our assertion.

4. Quarter-Symmetric Nonmetric Connections

Let (𝑀,𝑔,𝐹) be a semi-Riemannian product manifold and be the Levi-Civita connection on 𝑀. If we set 𝐷𝑋𝑌=𝑋𝑌+𝑢(𝑌)𝐹𝑋,(4.1) for any 𝑋,𝑌Γ(𝑇𝑀), then 𝐷 is a linear connection on 𝑀, where 𝑢 is a 1-form on 𝑀 with 𝑈 as associated vector field, that is 𝑢(𝑋)=𝑔(𝑋,𝑈).(4.2) The torsion tensor of 𝐷 on 𝑀 denoted by 𝑇. Then we obtain 𝑇(𝑋,𝑌)=𝑢(𝑌)𝐹𝑋𝑢(𝑋)𝐹𝑌,(4.3)𝐷𝑋𝑔(𝑌,𝑍)=𝑢(𝑌)𝑔(𝐹𝑋,𝑍)𝑢(𝑍)𝑔(𝐹𝑋,𝑌),(4.4) for any 𝑋,𝑌Γ(𝑇𝑀). Thus 𝐷 is a quarter-symmetric nonmetric connection on 𝑀. From (2.24) and (4.1) we have 𝐷𝑋𝐹𝑌=𝑢(𝐹𝑌)𝐹𝑋𝑢(𝑌)𝑋.(4.5) Replacing 𝑋 by 𝐹𝑋 and 𝑌 by 𝐹𝑌 in (4.5) and using (2.19) we obtain 𝐷𝐹𝑋𝐹𝐹𝑌=𝑢(𝑌)𝑋𝑢(𝐹𝑌)𝐹𝑋.(4.6) Thus we have 𝐷𝑋𝐹𝑌+𝐷𝐹𝑋𝐹𝐹𝑌=0.(4.7) If we set 𝐹(𝑋,𝑌)=𝑔(𝐹𝑋,𝑌),(4.8) for any 𝑋,𝑌Γ(𝑇𝑀), from (4.1) we get 𝐷𝑋𝐹(𝑌,𝑍)=𝑋𝐹(𝑌,𝑍)𝑢(𝑌)𝑔(𝑋,𝑍)𝑢(𝑍)𝑔(𝑋,𝑌).(4.9) From (4.1) the curvature tensor 𝑅𝐷 of the quarter-symmetric nonmetric connection 𝐷 is given by 𝑅𝐷(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝜆(𝑋,𝑍)𝐹𝑌𝜆(𝑌,𝑍)𝐹𝑋,(4.10) for any 𝑋,𝑌,𝑍Γ(𝑇𝑀), where 𝜆 is a (0,2)-tensor given by 𝜆(𝑋,𝑍)=(𝑋𝑢)(𝑍)𝑢(𝑍)𝑢(𝐹𝑋). If we set 𝑅𝐷(𝑋,𝑌,𝑍,𝑊)=𝑔(𝑅𝐷(𝑋,𝑌)𝑍,𝑊), then, from (4.10), we obtain 𝑅𝐷(𝑋,𝑌,𝑍,𝑊)=𝑅𝐷(𝑌,𝑋,𝑍,𝑊).(4.11) We note that the Riemannian curvature tensor 𝑅𝐷 of 𝐷 does not satisfy the other curvature-like properties. But, from (4.10), we have 𝑅𝐷(𝑋,𝑌)𝑍+𝑅𝐷(𝑌,𝑍)𝑋+𝑅𝐷(𝑍,𝑋)𝑌=𝜆(𝑍,𝑌)+𝜆(𝑌,𝑍)𝐹𝑋𝜆(𝑋,𝑍)+𝜆(𝑍,𝑋)𝐹𝑌𝜆(𝑌,𝑋)𝜆(𝑋,𝑌)𝐹𝑍.(4.12) Thus we have the following proposition.

Proposition 4.1. Let 𝑀 be a lightlike hypersurface of a semi-Riemannian product manifold 𝑀. Then the first Bianchi identity of the quarter-symmetric nonmetric connection 𝐷 on 𝑀 is provided if and only if 𝜆 is symmetric.

Let 𝑀 be a lightlike hypersurface of a semi-Riemannian product manifold (𝑀,𝑔) with quarter-symmetric nonmetric connection 𝐷. Then the Gauss and Weingarten formulas with respect to 𝐷 are given by, respectively, 𝐷𝑋𝑌=𝐷𝑋𝑌+𝐵(𝑋,𝑌)𝑁(4.13)𝐷𝑋𝑁=𝐴𝑁𝑋+𝜏(𝑋)𝑁(4.14) for any 𝑋,𝑌Γ(𝑇𝑀), where 𝐷𝑋𝑌, 𝐴𝑁𝑋Γ(𝑇𝑀), 𝐵(𝑋,𝑌)=𝑔(𝐷𝑋𝑌,𝜉), 𝜏(𝑋)=𝑔(𝐷𝑋𝑁,𝜉). Here, 𝐷, 𝐵 and 𝐴𝑁 are called the induced connection on 𝑀, the second fundamental form, and the Weingarten mapping with respect to 𝐷. From (2.9), (2.10), (3.1), (4.1), (4.13), and (4.14) we obtain 𝐷𝑋𝑌=𝑋𝑌+𝑢(𝑌)𝑓𝑋,(4.15)𝐵(𝑋,𝑌)=𝐵(𝑋,𝑌)+𝑢(𝑌)𝑤(𝑋),(4.16)𝐴𝑁𝑋=𝐴𝑁𝑋𝑢(𝑁)𝑓𝑋,𝜏(𝑋)=𝜏(𝑋)+𝑢(𝑁)𝑤(𝑋),(4.17) for any 𝑋, 𝑌Γ(𝑇𝑀). From (4.1), (4.4), (4.13), and (4.16) we get 𝐷𝑋𝑔(𝑌,𝑍)=𝐵(𝑋,𝑌)𝜂(𝑍)+𝐵(𝑋,𝑍)𝜂(𝑌)𝑢(𝑌)𝑔(𝑓𝑋,𝑍)𝑢(𝑍)𝑔(𝑓𝑋,𝑌).(4.18) On the other hand, the torsion tensor of the induced connection 𝐷 is 𝑇𝐷(𝑋,𝑌)=𝑢(𝑌)𝑓𝑋𝑢(𝑋)𝑓𝑌.(4.19) From last two equations we have the following proposition.

Proposition 4.2. Let 𝑀 be a lightlike hypersurface of a semi-Riemannian product manifold (𝑀,𝑔) with quarter-symmetric nonmetric connection 𝐷. Then the induced connection 𝐷 is a quarter-symmetric nonmetric connection on the lightlike hypersurface 𝑀.

For any 𝑋,𝑌Γ(𝑇𝑀), we can write 𝐷𝑋𝑃𝑌=𝐷𝑋𝑃𝑌+𝐶𝐷(𝑋,𝑃𝑌)𝜉,𝑋𝜉=𝐴𝜉𝑋+𝜀(𝑋)𝜉,(4.20) where 𝐷𝑋𝑃𝑌𝐴𝜉𝑋Γ(𝑆(𝑇𝑀)), 𝐶(𝑋,𝑃𝑌)=𝑔(𝐷𝑋𝑃𝑌,𝑁), and 𝜀(𝑋)=𝑔(𝐷𝑋𝜉,𝑁). From (2.14), (16), and (4.15), we obtain 𝐶(𝑋,𝑃𝑌)=𝐶(𝑋,𝑃𝑌)+𝑢(𝑃𝑌)𝜂(𝑓𝑋),(4.21)𝐴𝜉𝑋=𝐴𝜉𝑋𝑢(𝜉)𝑃𝑓𝑋,𝜀(𝑋)=𝜏(𝑋)+𝑢(𝜉)𝜂(𝑓𝑋).(4.22) Using (2.15), (4.16) and (4.22) we obtain 𝐵(𝑋,𝑃𝑌)=𝑔𝐴𝜉𝑋,𝑃𝑌+𝑢(𝑃𝑌)𝑤(𝑋)+𝑢(𝜉)𝑔(𝐹𝑋,𝑃𝑌),(4.23) for any 𝑋,𝑌Γ(𝑇𝑀).

Now, we consider a screen semi-invariant lightlike hypersurface 𝑀 of a semi-Rieamannian product manifold 𝑀 with respect to the quarter symmetric connection 𝐷 given by (4.1). Since 𝑤(𝑋)=𝑔(𝐹𝑋,𝜉), for any 𝑋Γ(𝔻),𝑤(𝑋)=0. Thus we have the following propositions.

Proposition 4.3. Let 𝑀 be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold (𝑀,𝑔) with quarter-symmetric nonmetric connection. The second fundamental form 𝐵 of quarter-symmetric nonmetric connection 𝐷 is degenerate.

Proposition 4.4. Let (𝑀,𝑔) be a semi-Riemannian product manifold and 𝑀 be a screen semi-invariant lightlike hypersurfaces of 𝑀. If 𝑀 is 𝔻 totally geodesic with respect to , then 𝑀 is 𝔻 totally geodesic with respect to quarter-symmetric nonmetric connection.

Theorem 4.5. Let (𝑀,𝑔) be a semi-Riemannian product manifold and 𝑀 be a screen semi-invariant lightlike hypersurfaces of 𝑀. Then the following assertions are equivalent.(i)The distribution 𝔻 is integrable with respect to the quarter symmetric nonmetric connection 𝐷.(ii)𝐵(𝑋,𝑓𝑌)=𝐵(𝑌,𝑓𝑋), for any 𝑋, 𝑌Γ(𝔻).(iii)𝑔(𝐴𝜉𝑋,𝑃𝑓𝑌)=𝑔(𝐴𝜉𝑌,𝑃𝑓𝑋), for any 𝑋, 𝑌Γ(𝔻).

The proof of this theorem is similar to the proof of the Theorem 3.8.

From (4.23), for any 𝑋Γ(𝔻) and 𝑌Γ(𝔻2), we have 𝐵(𝑋,𝑃𝑌)=𝑔(𝐴𝜉𝑋,𝑃𝑌). If we set 𝔻=𝔻𝔻2, then, from Theorem 3.10, we have the following corollary.

Corollary 4.6. Let (𝑀,𝑔) be a semi-Riemannian product manifold and 𝑀 be a screen semi-invariant lightlike hypersurface of 𝑀. Then the distribution 𝔻 is a mixed geodesic foliation defined with respect to quarter symmetric nonmetric connection if and only if there is no 𝔻1 component of 𝐴𝜉.

From (4.15), we obtain 𝑅𝐷(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝑢(𝑍)𝑋𝑓𝑌𝑌𝑓𝑋+𝜆(𝑋,𝑍)𝑓𝑌𝜆(𝑌,𝑍)𝑓𝑋,(4.24) where 𝜆 is a (0,2) tensor on 𝑀 given by 𝜆(𝑋,𝑍)=(𝑋𝑢)(𝑍)𝑢(𝑍)𝑢(𝑓𝑋).

From (4.24), we have the following proposition which is similar to the Proposition 4.1.

Proposition 4.7. Let 𝑀 be a lightlike hypersurface of a semi-Riemannian product manifold 𝑀. One assumes that 𝑓 is parallel on 𝑀. Then the first Bianchi identity of the quarter-symmetric nonmetric connection 𝐷 on 𝑀 is provided if and only if 𝜆 is symmetric.

Now we will compute Gauss-Codazzi equations of lightlike hypersurfaces with respect to the quarter-symmetric nonmetric connection: 𝑔𝑅𝐷+(𝑋,𝑌)𝑍,𝑃𝑊=𝑔(𝑅(𝑋,𝑌)𝑍,𝑃𝑊)+𝐵(𝑋,𝑍)𝐶(𝑌,𝑃𝑊)𝐵(𝑌,𝑍)𝐶(𝑋,𝑃𝑊)𝜆(𝑋,𝑍)𝑔(𝑓𝑌,𝑃𝑊)𝜆(𝑌,𝑍)𝑔(𝑓𝑋,𝑃𝑊),𝑔𝑅𝐷=(𝑋,𝑌)𝑍,𝜉𝑋𝐵(𝑌,𝑍)𝑌𝐵+(𝑋,𝑍)𝜆(𝑋,𝑍)𝑤(𝑌)𝜆(𝑌,𝑍)𝑤(𝑋),𝑔𝑅𝐷(+𝑋,𝑌)𝑍,𝑁=𝑔(𝑅(𝑋,𝑌)𝑍,𝑁)𝜆(𝑋,𝑍)𝜂(𝑓𝑌)𝜆(𝑌,𝑍)𝜂(𝑓𝑋),(4.25) for any 𝑋, 𝑌, 𝑍, 𝑊Γ(𝑇𝑀).

Now, let 𝑀 be a screen semi-invariant lightlike hypersurface of a (𝑚+2)-dimensional semi-Riemannian product manifold with the quarter-symmetric nonmetric connection 𝐷 such that the tensor field 𝑓 is parallel on 𝑀. We consider the local quasiorthonormal basis {𝐸𝑖,𝐹𝜉,𝐹𝑁,𝜉,𝑁}, 𝑖=1,𝑚2, of 𝑀 along 𝑀, where {𝐸1,,𝐸𝑚2} is an orthonormal basis of Γ(𝔻). Then, the Ricci tensor of 𝑀 with respect to 𝐷 is given by 𝑅𝐷(0,2)(𝑋,𝑌)=𝑚2𝑖=1𝜀𝑖𝑔𝑅𝐷𝑋,𝐸𝑖𝑌,𝐸𝑖𝑅+𝑔𝐷𝑅(𝑋,𝐹𝜉)𝑌,𝐹𝑁+𝑔𝐷𝑅(𝑋,𝐹𝑁)𝑌,𝐹𝜉+𝑔𝐷.(𝑋,𝜉)𝑌,𝑁(4.26) From (4.24) we have 𝑅𝐷(0,2)(𝑋,𝑌)=𝑅(0,2)+(𝑋,𝑌)𝑚2𝑖=1𝜀𝑖𝜆(𝑋,𝑌)𝑔𝑓𝐸𝑖,𝐸𝑖𝐸𝜆𝑖𝑔,𝑌𝑓𝑋,𝐸𝑖𝜆(𝐹𝜉,𝑌)𝜂(𝑋)𝜆(𝜉,𝑌)𝜂(𝑓𝑋),(4.27) where 𝑅(0,2)(𝑋,𝑌) is the Ricci tensor of 𝑀. Thus we have the following corollary.

Corollary 4.8. Let 𝑀 a screen semi-invariant lightlike hypersurface of a (𝑚+2)-dimensional semi-Riemannian product manifold with the quarter-symmetric nonmetric connection 𝐷 such that the tensor field 𝑓 is parallel on 𝑀 and 𝑅(0,2)(𝑋,𝑌) is symmetric. Then 𝑅𝐷(0,2) is symmetric on the distribution 𝔻 if and only if 𝜆 is symmetric and 𝜆(𝑓𝑋,𝑌)=𝜆(𝑓𝑌,𝑋).

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.

References

  1. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996. View at Zentralblatt MATH
  2. K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhäuser, Boston, Mass, USA, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, 2007. View at Zentralblatt MATH
  4. M. Atçeken and E. Kılıç, “Semi-invariant lightlike submanifolds of a semi-Riemannian product manifold,” Kodai Mathematical Journal, vol. 30, no. 3, pp. 361–378, 2007. View at Publisher · View at Google Scholar
  5. E. Kılıç and B. Şahin, “Radical anti-invariant lightlike submanifolds of semi-Riemannian product manifolds,” Turkish Journal of Mathematics, vol. 32, no. 4, pp. 429–449, 2008. View at Zentralblatt MATH
  6. F. Massamba, “Killing and geodesic lightlike hypersurfaces of indefinite Sasakian manifolds,” Turkish Journal of Mathematics, vol. 32, no. 3, pp. 325–347, 2008. View at Zentralblatt MATH
  7. F. Massamba, “Lightlike hypersurfaces of indefinite Sasakian manifolds with parallel symmetric bilinear forms,” Differential Geometry—Dynamical Systems, vol. 10, pp. 226–234, 2008. View at Zentralblatt MATH
  8. H. A. Hayden, “Sub-spaces of a space with torsion,” Proceedings of the London Mathematical Society, vol. 34, pp. 27–50, 1932. View at Publisher · View at Google Scholar
  9. U. C. De and D. Kamilya, “Hypersurfaces of a Riemannian manifold with semi-symmetric non-metric connection,” Journal of the Indian Institute of Science, vol. 75, no. 6, pp. 707–710, 1995. View at Zentralblatt MATH
  10. B. G. Schmidt, “Conditions on a connection to be a metric connection,” Communications in Mathematical Physics, vol. 29, pp. 55–59, 1973. View at Publisher · View at Google Scholar
  11. S. Golab, “On semi-symmetric and quarter-symmetric linear connections,” The Tensor Society, vol. 29, no. 3, pp. 249–254, 1975.
  12. N. S. Agashe and M. R. Chafle, “A semi-symmetric nonmetric connection on a Riemannian manifold,” Indian Journal of Pure and Applied Mathematics, vol. 23, no. 6, pp. 399–409, 1992.
  13. Y. X. Liang, “On semi-symmetric recurrent-metric connection,” The Tensor Society, vol. 55, no. 2, pp. 107–112, 1994. View at Zentralblatt MATH
  14. M. M. Tripathi, “A new connection in a Riemannian manifold,” International Electronic Journal of Geometry, vol. 1, no. 1, pp. 15–24, 2008. View at Zentralblatt MATH
  15. E. Yaşar, A. C. Çöken, and A. Yücesan, “Lightlike hypersurfaces in semi-Riemannian manifold with semi-symmetric non-metric connection,” Mathematica Scandinavica, vol. 102, no. 2, pp. 253–264, 2008.