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

Authors: Erol Kılıç; Oğuzhan Bahadır
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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.


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## Body

## 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 tensorT¯is of the form:
(1.1)T¯(X,Y)=u(Y)φX-u(X)φY,
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 Section3, 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¯)=q≥1 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 ξ≠0 on M such that
(2.1)g(ξ,X)=0,∀X∈Γ(TM).
The radical or the null space of TxM, at each point x∈M, is a subspace RadTxM defined by
(2.2)RadTxM={ξ∈TxM|gx(ξ,X)=0,∀X∈Γ(TM)},
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
(2.3)RadTxM=TxM∩TxM⊥.
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
(2.4)TM=RadTM⊥S(TM).
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
(2.5)g¯(N,N)=0,g¯(ξ,N)=1.
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:
(2.6)TM¯=TM⊕ltr(TM)=S(TM)⊥RadTM⊕ltr(TM),
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:
(2.7){X1,…,Xm,ξ,N},
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:
(2.8)∇¯XY=∇XY+h(X,Y),∇¯XN=-ANX+∇XtN,
for any X,Y∈Γ(TM), where ∇XY,ANX∈Γ(TM) and h(X,Y),∇XtN∈Γ(ltr(TM)). If we set B(X,Y)=g¯(h(X,Y),ξ) and τ(X)=g¯(∇XtN,ξ), then (2.8) become
(2.9)∇¯XY=∇XY+B(X,Y)N,(2.10)∇¯XN=-ANX+τ(X)N.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 X∈Γ(TM), we can write
(2.11)X=PX+η(X)ξ,
where η is a 1-form given by
(2.12)η(X)=g¯(X,N).From (2.9), we get
(2.13)(∇Xg)(Y,Z)=B(X,Y)η(Z)+B(X,Z)η(Y),∀X,Y,Z∈Γ(TM),
and the induced connection ∇ is a nonmetric connection on M. From (2.4), we have
(2.14)∇XW=∇X*W+h*(X,W)=∇X*W+C(X,W)ξ,X∈Γ(TM),W∈Γ(S(TM)),∇Xξ=-Aξ*X-τ(X)ξ,
where ∇X*W and Aξ*X 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:
(2.15)g(Aξ*X,W)=B(X,W),g(Aξ*X,N)=0,B(X,ξ)=0,g(ANX,N)=0.
Moreover, from the first and third equations of (2.15) we have
(2.16)Aξ*ξ=0.Now, we will denoteR¯ 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
(2.17)R¯(X,Y)Z=R(X,Y)Z+Ah(X,Z)Y-Ah(Y,Z)X+(∇Xh)(Y,Z)-(∇Yh)(X,Z),∀X,Y,Z∈Γ(TM),
where (∇Xh)(Y,Z)=∇Xt(h(Y,Z))-h(∇XY,Z)-h(Y,∇XZ). Then the Gauss-Codazzi equations of a lightlike hypersurface are given by
(2.18)g¯(R¯(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,ξ)=(∇XB)(Y,Z)-(∇YB)(X,Z)+B(Y,Z)τ(X)-B(X,Z)τ(Y),g¯(R¯(X,Y)Z,N)=g(R(X,Y)Z,N),g¯(R¯(X,Y)ξ,N)=g(R(X,Y)ξ,N)=C(Y,Aξ*X)-C(X,Aξ*Y)-2dτ(X,Y),
for any X,Y,Z,W∈Γ(TM),ξ∈Γ(RadTM).For geometries of lightlike submanifolds, hypersurfaces and curves, we refer to [1–3].


### 2.1. Product Manifolds

LetM¯ be an n-dimensional differentiable manifold with a tensor field F of type (1,1) on M¯ such that
(2.19)F2=I.
Then M¯is called an almost product manifold with almost product structure F. If we put
(2.20)π=12(I+F),σ=12(I-F),
then we have
(2.21)π+σ=I,π2=π,σ2=σ,σπ=πσ=0,F=π-σ.
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
(2.22)g¯(FX,FY)=g¯(X,Y),
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
(2.23)g¯(FX,Y)=g¯(X,FY).If, for any vector fieldsX,Y on M¯,
(2.24)∇¯F=0,thatis∇¯XFY=F∇¯XY,
then M¯ is called a semi-Riemannian product manifold, where ∇¯ is the Levi-Civita connection on M¯.


## 2.1. Product Manifolds

LetM¯ be an n-dimensional differentiable manifold with a tensor field F of type (1,1) on M¯ such that
(2.19)F2=I.
Then M¯is called an almost product manifold with almost product structure F. If we put
(2.20)π=12(I+F),σ=12(I-F),
then we have
(2.21)π+σ=I,π2=π,σ2=σ,σπ=πσ=0,F=π-σ.
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
(2.22)g¯(FX,FY)=g¯(X,Y),
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
(2.23)g¯(FX,Y)=g¯(X,FY).If, for any vector fieldsX,Y on M¯,
(2.24)∇¯F=0,thatis∇¯XFY=F∇¯XY,
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

LetM be a lightlike hypersurface of a semi-Riemannian product manifold (M¯,g¯). For any X∈Γ(TM) we can write
(3.1)FX=fX+w(X)N,
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.
LetM be a lightlike hypersurface of a semi-Riemannian product manifold (M¯,g¯):(i)
ifFRadTM⊂S(TM) and Fltr(TM)⊂S(TM) then we say that M is a screen semi-invariant lightlike hypersurface;(ii)
ifFS(TM)=S(TM) then we say that M is a screen invariant lightlike hypersurface;(iii)
ifFRadTM=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, letM 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
(3.2)S(TM)=𝔻⊥{𝔻1⊕𝔻2},
where 𝔻 is a (m-2)-dimensional distribution. Hence we have the following decomposition:
(3.3)TM=𝔻⊥{𝔻1⊕𝔻2}⊥RadTM,TM¯=𝔻⊥{𝔻1⊕𝔻2}⊥{RadTM⊕ltr(TM)}.Proposition 3.3.
The distribution𝔻 is an invariant distribution with respect to F.Proof.
For anyX∈Γ(𝔻) and U∈Γ(𝔻1),V∈Γ(𝔻2) we obtain
(3.4)g(FX,U)=g(X,FU)=0,g(FX,V)=g(X,FV)=0.
Thus there are no components of FX in 𝔻1 and 𝔻2. Furthermore, we have
(3.5)g(FX,ξ)=g(X,Fξ)=0,g(FX,N)=g(X,FN)=0.
Proof is completed.If we set𝔻¯=𝔻⊥RadTM⊥FRadTM, we can write
(3.6)TM=𝔻¯⊕𝔻2.
From the above proposition we have the following corollary.Corollary 3.4.
The distribution𝔻¯ is invariant with respect to F.Example 3.5.
Let(M¯=R25,g¯) be a 5-dimensional semi-Euclidean space with signature(-,+,-,+,+) and (x,y,z,s,t) be the standard coordinate system of R25. If we set F(x,y,z,s,t)=(x,y,-z,-s,-t), then F2=I and F is a product structure on R25. Consider a hypersurface M in M¯ by the equation:
(3.7)t=x+y+z.
Then TM=Span{U1,U2,U3,U4}, where
(3.8)U1=∂∂x+∂∂t,U2=∂∂y+∂∂t,U3=∂∂z+∂∂t,U4=∂∂s.
It is easy to check that M is a lightlike hypersurface and
(3.9)TM⊥=Span{ξ=U1-U2+U3}.
Then take a lightlike transversal vector bundle as follow:
(3.10)ltr(TM)=Span{N=-14{∂∂x+∂∂y+∂∂z-∂∂t}}.
It follows that the corresponding screen distribution S(TM) is spanned by
(3.11){W1=U4,W2=U1-U2-U3,W3=U1+U2-U3}.
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=-dx2-dy2+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
(3.12)RadTM=Span{ξ=∂∂x+(y+z)∂∂y-(y+z)∂∂z+∂∂t},ltr(TM)=Span{N=-12(1+(y+z)2)(∂∂x+(y+z)∂∂y+(y+z)∂∂z-∂∂t)},S(TM)=Span{W1=-(y+z)∂∂x+∂∂y,W2=∂∂z+(y+z)∂∂t}.
We can easily check that
(3.13)Fξ=W1+W2,FN=12(1+(y+z)2){W1-W2}.
Thus M is a screen semi-invariant lightlike hypersurface with 𝔻={0}, 𝔻1=Span{Fξ} and 𝔻2=Span{FN}.Example 3.7.
Let(R24,g¯) be a 4-dimensional semi-Euclidean space with signature (-,-,+,+) and (x1,x2,x3,x4) be the standard coordinate system of R24. Consider a Monge hypersurface M of R24 given by
(3.14)x4=Ax1+Bx2+Cx3,A2+B2-C2=1,A,B,C∈R.
Then the tangent bundle TM of the hypersurface M is spanned by
(3.15){U1=∂∂x1+A∂∂x4,U2=∂∂x2+B∂∂x4,U3=∂∂x3+C∂∂x4}.
It is easy to check that M is a lightlike hypersurface (p.196, Ex.1, [3]) whose radical distribution RadTM is spanned by
(3.16)ξ=AU1+BU2-CU3=A∂∂x1+B∂∂x2-C∂∂x3+∂∂x4.
Furthermore, the lightlike transversal vector bundle is given by
(3.17)ltr(TM)=Span{N=-12(C2+1)(A∂∂x1+B∂∂x2+C∂∂x3-∂∂x4)}.
It follows that the corresponding screen distribution S(TM) is spanned by
(3.18){W1=1A2+B2(B∂∂x1-A∂∂x2),W2=1A2+B2(∂∂x3+C∂∂x4)}.
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 R24. One can easily check that FS(TM)=S(TM) and FRadTM=ltr(TM). Thus M is a radical anti-invariant lightlike hypersurface of R24. 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 distribution𝔻¯ is integrable with respect to the induced connection ∇ of M.(ii)
B(X,fY)=B(Y,fX), for any X,Y∈Γ(𝔻¯).(iii)
g(Aξ*X,PfY)=g(Aξ*Y,PfX), for any X,Y∈Γ(𝔻¯).Proof.
For anyX,Y∈Γ(𝔻¯), from (2.9), (2.24), and (3.1), we obtain
(3.19)f∇XY+w(∇XY)N+B(X,Y)FN=∇XfY+B(X,fY)N.
Interchanging role of X and Y we have
(3.20)f∇YX+w(∇YX)N+B(Y,X)FN=∇YfX+B(Y,fX)N.
From (3.19), (3.20) we get
(3.21)w([X,Y])=B(X,fY)-B(Y,fX)
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.
LetM be a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold M¯. If B(X,Y)=0, for any X∈Γ(𝔻¯),Y∈Γ(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 no𝔻2-component of AN.(iii)
There is no𝔻1-component of Aξ*.Proof.
Suppose thatM 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
(3.22)∇XFN+B(X,FN)N=-fANX+τ(X)FN-w(ANX)N,
for any X∈Γ(𝔻¯). If we take tangential and transversal parts of this last equation we have
(3.23)∇XFN=-fANX+τ(X)FN,B(X,FN)=-w(ANX).
Furthermore, since w(ANX)=g(ANX,Fξ), we get (i)⇔(ii). Since g¯(FN,ξ)=g¯(N,Fξ)=0, we obtain
(3.24)g(ANX,Fξ)=-g(Aξ*X,FN).
This is (ii)⇔(iii).From the decomposition (3.6), we have the following theorem.Theorem 3.11.
LetM 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.
LetM 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 Y∈Γ(𝔻¯), FY∈Γ(𝔻¯). Thus ∇XY and ∇XfY belong to Γ(𝔻¯), for any X∈Γ(TM). From the Gauss formula, we obtain
(3.25)∇XfY+B(X,fY)N=f∇XY+w(∇XY)N+B(X,Y)FN.
Comparing the tangential and normal parts with respect to 𝔻¯ of (3.25), we have
(3.26)∇XfY=f∇XY,thatis(∇Xf)Y=0,(3.27)B(X,Y)=0.
Since fZ=0, for any Z∈Γ(𝔻2), we get ∇XfZ=0 and f∇XZ=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=f∇XY, for any X,Y∈Γ(𝔻¯) and ∇UfW=f∇UW=0, for any U,W∈Γ(𝔻2). Thus it follows that ∇XfY∈Γ(𝔻¯) and ∇UW∈Γ(𝔻2). Hence, the leaves of the distributions 𝔻¯ and 𝔻2 are totally geodesic in M.From Theorem3.11 and (3.27) we have the following corollary.Corollary 3.12.
LetM 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.LetM be a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifold M¯. Then we have the following decomposition:
(3.28)TM¯=S(TM)⊥{RadTM⊕FRadTM}.Theorem 3.13.
LetM 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 fieldX on M belongs to S(TM) if and only if η(X)=0. Since M is a radical anti-invariant lightlike hypersurface, for any X∈Γ(S(TM)), FX∈Γ(S(TM)). For any X,Y∈Γ(S(TM)), we can write
(3.29)∇¯XFY=∇XFY+B(X,FY)N.
In this last equation interchanging role of X and Y, we obtain
(3.30)F[X,Y]=∇XFY-∇YFX+(B(X,FY)-B(Y,FX))N.
Since η([X,Y])=g¯([X,Y],N)=g¯(F[X,Y],FN), we get
(3.31)η([X,Y])=(B(X,FY)-B(Y,FX))g¯(N,FN).
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
(4.1)D¯XY=∇¯XY+u(Y)FX,
for any X,Y∈Γ(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
(4.2)u(X)=g¯(X,U).
The torsion tensor of D¯ on M¯ denoted by T¯. Then we obtain
(4.3)T¯(X,Y)=u(Y)FX-u(X)FY,(4.4)(D¯Xg¯)(Y,Z)=-u(Y)g¯(FX,Z)-u(Z)g¯(FX,Y),
for any X,Y∈Γ(TM¯). Thus D¯ is a quarter-symmetric nonmetric connection on M¯. From (2.24) and (4.1) we have
(4.5)(D¯XF)Y=u(FY)FX-u(Y)X.
Replacing X by FX and Y by FY in (4.5) and using (2.19) we obtain
(4.6)(D¯FXF)FY=u(Y)X-u(FY)FX.
Thus we have
(4.7)(D¯XF)Y+(D¯FXF)FY=0.
If we set
(4.8)′F(X,Y)=g¯(FX,Y),
for any X,Y∈Γ(TM¯), from (4.1) we get
(4.9)(D¯X′F)(Y,Z)=(∇¯X′F)(Y,Z)-u(Y)g¯(X,Z)-u(Z)g¯(X,Y).
From (4.1) the curvature tensor R¯D of the quarter-symmetric nonmetric connection D¯ is given by
(4.10)R¯D(X,Y)Z=R¯(X,Y)Z+λ¯(X,Z)FY-λ¯(Y,Z)FX,
for any X,Y,Z∈Γ(TM¯), where λ¯ is a (0,2)-tensor given by λ¯(X,Z)=(∇¯Xu)(Z)-u(Z)u(FX). If we set R¯D(X,Y,Z,W)=g¯(R¯D(X,Y)Z,W), then, from (4.10), we obtain
(4.11)R¯D(X,Y,Z,W)=-R¯D(Y,X,Z,W).
We note that the Riemannian curvature tensor R¯D of D¯ does not satisfy the other curvature-like properties. But, from (4.10), we have
(4.12)R¯D(X,Y)Z+R¯D(Y,Z)X+R¯D(Z,X)Y=(λ¯(Z,Y)-λ¯(Y,Z))FX+(λ¯(X,Z)-λ¯(Z,X))FY+(λ¯(Y,X)-λ¯(X,Y))FZ.
Thus we have the following proposition.Proposition 4.1.
LetM 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.LetM 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,
(4.13)D¯XY=DXY+B¯(X,Y)N(4.14)D¯XN=-A¯NX+τ¯(X)N
for any X,Y∈Γ(TM), where DXY, A¯NX∈Γ(TM), B¯(X,Y)=g¯(D¯XY,ξ), τ¯(X)=g¯(D¯XN,ξ). Here, D, B¯ and A¯N 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
(4.15)DXY=∇XY+u(Y)fX,(4.16)B¯(X,Y)=B(X,Y)+u(Y)w(X),(4.17)A¯NX=ANX-u(N)fX,τ¯(X)=τ(X)+u(N)w(X),
for any X, Y∈Γ(TM). From (4.1), (4.4), (4.13), and (4.16) we get
(4.18)(DXg)(Y,Z)=B(X,Y)η(Z)+B(X,Z)η(Y)-u(Y)g(fX,Z)-u(Z)g(fX,Y).
On the other hand, the torsion tensor of the induced connection D is
(4.19)TD(X,Y)=u(Y)fX-u(X)fY.
From last two equations we have the following proposition.Proposition 4.2.
LetM 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 anyX,Y∈Γ(TM), we can write
(4.20)DXPY=DX*PY+C¯(X,PY)ξ,DXξ=-A¯ξ*X+ɛ(X)ξ,
where DX*PYA¯ξ*X∈Γ(S(TM)), C¯(X,PY)=g¯(DXPY,N), and ɛ(X)=g¯(DXξ,N). From (2.14), (16), and (4.15), we obtain
(4.21)C¯(X,PY)=C(X,PY)+u(PY)η(fX),(4.22)A¯ξ*X=Aξ*X-u(ξ)PfX,ε(X)=-τ(X)+u(ξ)η(fX).
Using (2.15), (4.16) and (4.22) we obtain
(4.23)B¯(X,PY)=g(A¯ξ*X,PY)+u(PY)w(X)+u(ξ)g¯(FX,PY),
for any X,Y∈Γ(TM).Now, we consider a screen semi-invariant lightlike hypersurfaceM 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 X∈Γ(𝔻),w(X)=0. Thus we have the following propositions.Proposition 4.3.
LetM 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 distribution𝔻¯ is integrable with respect to the quarter symmetric nonmetric connection D.(ii)
B¯(X,fY)=B¯(Y,fX), for any X, Y∈Γ(𝔻¯).(iii)
g(A¯ξ*X,PfY)=g(A¯ξ*Y,PfX), for any X, Y∈Γ(𝔻¯).The proof of this theorem is similar to the proof of the Theorem3.8.From (4.23), for any X∈Γ(𝔻) and Y∈Γ(𝔻2), we have B¯(X,PY)=g(A¯ξ*X,PY). If we set 𝔻′=𝔻⊥𝔻2, 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 A¯ξ*.From (4.15), we obtain
(4.24)RD(X,Y)Z=R(X,Y)Z+u(Z){(∇Xf)Y-(∇Yf)X}+λ(X,Z)fY-λ(Y,Z)fX,
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.
LetM 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:(4.25)g¯(R¯D(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),g¯(R¯D(X,Y)Z,ξ)=(∇XB)(Y,Z)-(∇YB)(X,Z)+λ¯(X,Z)w(Y)-λ¯(Y,Z)w(X),g¯(R¯D(X,Y)Z,N)=g(R(X,Y)Z,N)+λ¯(X,Z)η(fY)-λ¯(Y,Z)η(fX),
for any X, Y, Z, W∈Γ(TM).Now, letM 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,…m-2, of M¯ along M, where {E1,…,Em-2} is an orthonormal basis of Γ(𝔻). Then, the Ricci tensor of M with respect to D is given by
(4.26)RD(0,2)(X,Y)=∑i=1m-2‍ɛig(RD(X,Ei)Y,Ei)+g(RD(X,Fξ)Y,FN)+g(RD(X,FN)Y,Fξ)+g(RD(X,ξ)Y,N).
From (4.24) we have
(4.27)RD(0,2)(X,Y)=R(0,2)(X,Y)+∑i=1m-2‍ɛi{λ(X,Y)g(fEi,Ei)-λ(Ei,Y)g(fX,Ei)}-λ(Fξ,Y)η(X)-λ(ξ,Y)η(fX),
where R(0,2)(X,Y) is the Ricci tensor of M. Thus we have the following corollary.Corollary 4.8.
LetM 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).



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*Source: 178390-2012-08-30.xml*