a Semi-Riemannian Product Manifold and Quarter-Symmetric Nonmetric Connections

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Volume 2012, Article ID 178390,17pages doi:10.1155/2012/178390

Research Article

Lightlike Hypersurfaces of

a Semi-Riemannian Product Manifold and Quarter-Symmetric Nonmetric Connections

Erol Kılıc¸

¹

and O ˘guzhan Bahadır

²

1Department of Mathematics, Faculty of Arts and Sciences, ˙In¨on ¨u University, 44280 Malatya, Turkey

2Department of Mathematics, Faculty of Arts and Sciences, Hitit University, 19030 C¸orum, Turkey

Correspondence should be addressed to Erol Kılıc¸,[email protected]

Received 28 March 2012; Revised 2 July 2012; Accepted 16 July 2012 Academic Editor: Jianya Liu

Copyrightq2012 E. Kılıc¸ and O. 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.

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 in1 see also2,3by Duggal and Bejancu. In4, Atçeken and Kılıç introduced semi-invariant lightlike submanifolds of a semi-Riemannian product manifold. In 5, Kılıç and S¸ahin 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 spacesfor example6,7.

In8, 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, Yas¸ar et al.

have studied lightlike hypersurfaces in semi-Riemannian manifolds with semisymmetric nonmetric connection. The idea of quarter-symmetric linear connections in a diﬀerential

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manifold was introduced by Golab11. A linear connection is said to be a quarter-symmetric connection if its torsion tensorT is of the form:

TX, Y uYϕX−uXϕY, 1.1

for any vector fieldsX, Yon a manifold, whereuis a 1-form andϕis a tensor of type1,1.

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. InSection 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

LetM, gbe anm2-dimensional semi-Riemannian manifold with index g q≥1 and letM, gbe a hypersurface ofM, withgg_|

M. If the induced metricgonMis degenerate, thenMis called a lightlikenull or degeneratehypersurface1 see also2,3. Then there exists a null vector fieldξ /0 onMsuch that

gξ, X 0, ∀X ∈ΓTM. 2.1

The radical or the null space ofTxM, at each pointx∈M, is a subspace RadTxMdefined by

RadT_xM

ξ∈T_xM|gxξ, X 0,∀X∈ΓTM

, 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

RadT_xMT_xM∩T_xM^⊥. 2.3

Since dimT_xM^⊥ 1 and dim RadT_xM 1, we have RadT_xM T_xM^⊥. We call RadTMa radical distribution and it is spanned by the null vector fieldξ. The complementary vector bundleSTMof RadTMinTMis called the screen bundle ofM. We note that any screen bundle is nondegenerate. This means that

TMRadTM⊥STM. 2.4

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Here ⊥ denotes the orthogonal-direct sum. The complementary vector bundle STM^⊥ ofSTMin TMis called screen transversal bundle and it has rank 2. Since Rad TMis a lightlike subbundle ofSTM^⊥there exists a unique local sectionNofSTM^⊥such that

gN, N 0, gξ, N 1. 2.5

Note thatNis transversal toMand{ξ, N}is a local frame field ofSTM^⊥and there exists a line subbundle ltrTM of TM, and it is called the lightlike transversal bundle, locally spanned byN. Hence we have the following decomposition:

TMTM⊕ltrTM STM⊥RadTM⊕ltrTM, 2.6

where⊕is the direct sum but not orthogonal1,3. From the above decomposition of a semi- Riemannian manifoldMalong a lightlike hypersurface M, we can consider the following local quasiorthonormal field of frames ofMalongM:

{X1, . . . , Xm, ξ, N}, 2.7 where{X1, . . . , X_m}is an orthonormal basis ofΓSTM. According to the splitting2.6, we have the following Gauss and Weingarten formulas, respectively:

∇XY ∇XY hX, Y,

∇XN−ANX∇^t_XN,

2.8

for anyX, Y ∈ΓTM, where∇XY, A_NX∈ΓTMandhX, Y,∇^t_XN ∈ΓltrTM. If we set BX, Y ghX, Y, ξandτX g∇^t_XN, ξ, then2.8become

∇XY ∇XYBX, YN, 2.9

∇XN−ANXτXN. 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 ofSTM on M. Then, for any X∈ΓTM, we can write

XP XηXξ, 2.11

whereηis a 1-form given by

ηX gX, N. 2.12

From2.9, we get ∇Xg

Y, Z BX, YηZ BX, ZηY, ∀X, Y, Z∈ΓTM, 2.13

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and the induced connection∇is a nonmetric connection onM. From2.4, we have

∇XW∇^∗_XWh^∗X, W

∇^∗_XWCX, Wξ, X∈ΓTM, W ∈ΓSTM,

∇Xξ −A^∗_ξX−τXξ,

2.14

where∇^∗_XW andA^∗_ξX belong toΓSTM.C,A^∗_ξ and∇^∗are called the local second fundamental form, the local shape operator and the induced connection onSTM, respectively.

Also, we have the following identities:

g

A^∗_ξX, W

BX, W, g

A^∗_ξX, N 0, BX, ξ 0, gANX, N 0.

2.15

Moreover, from the first and third equations of2.15we have

A^∗_ξξ0. 2.16

Now, we will denoteRandRthe curvature tensors of the Levi-Civita connection∇on Mand the induced connection∇onM. Then the Gauss equation ofMis given by

RX, YZRX, YZA_hX,ZY−A_hY,ZX

∇XhY, Z−∇YhX, Z, ∀X, Y, Z∈ΓTM, 2.17 where∇XhY, Z ∇^t_XhY, Z−h∇XY, Z−hY,∇XZ. Then the Gauss-Codazzi equations of a lightlike hypersurface are given by

g

RX, YZ, P W

gRX, YZ, P W

BX, ZCY, P W−BY, ZCX, P W, g

RX, YZ, ξ

∇XBY, Z−∇YBX, Z BY, ZτX−BX, ZτY, g

RX, YZ, N

gRX, YZ, N, g

RX, Yξ, N

gRX, Yξ, N C

Y, A^∗_ξX

−C

X, A^∗_ξY

−2dτX, Y,

2.18

for anyX, Y, Z, W ∈ΓTM, ξ∈ΓRadTM.

For geometries of lightlike submanifolds, hypersurfaces and curves, we refer to1–3.

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2.1. Product Manifolds

LetMbe ann-dimensional diﬀerentiable manifold with a tensor fieldFof type1,1onM such that

F²I. 2.19

ThenMis called an almost product manifold with almost product structureF. If we put π 1

2IF, σ 1

2I−F, 2.20

then we have

πσI, π²π, σ²σ,

σππσ0, Fπ−σ. 2.21

Thusπandσdefine two complementary distributions andFhas the eigenvalue of1 or−1.

If an almost product manifoldMadmits a semi-Riemannian metricgsuch that

gFX, FY gX, Y, 2.22

for any vector fieldsX, Y onM, thenMis called a semi-Riemannian almost product manifold. From2.19and2.22, we have

gFX, Y gX, FY. 2.23

If, for any vector fieldsX, YonM,

∇F 0, that is∇XFY F∇XY, 2.24

thenMis called a semi-Riemannian product manifold, where∇is the Levi-Civita connection onM.

3. Lightlike Hypersurfaces of Semi-Riemannian Product Manifolds

LetMbe a lightlike hypersurface of a semi-Riemannian product manifoldM, g. For any X∈ΓTMwe can write

FXfXwXN, 3.1

wherefis a1,1tensor field andwis a 1-form onMgiven bywX gFX, ξ gX, Fξ.

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Definition 3.1. Let M be a lightlike hypersurface of a semi-Riemannian product manifold M, g:

iif F RadTM ⊂ STMandF ltrTM ⊂ STMthen we say thatMis a screen semi-invariant lightlike hypersurface;

iiifFSTM STMthen we say thatMis a screen invariant lightlike hypersurface;

iiiif F RadTM ltrTMthen we say that Mis 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 manifoldM, g. In this paper we will study the hypersurfaces determined above.

Now, letMbe a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifold. If we setD1F RadTM,D2F ltrTMthen we can write

STM D⊥ {D1⊕D2}, 3.2

whereDis am−2-dimensional distribution. Hence we have the following decomposition:

TMD⊥ {D1⊕D2} ⊥RadTM,

TMD⊥ {D1⊕D2} ⊥ {RadTM⊕ltrTM}. 3.3 Proposition 3.3. The distributionDis an invariant distribution with respect toF.

Proof. For anyX∈ΓDandU∈ΓD1, V ∈ΓD2we obtain gFX, U gX, FU 0,

gFX, V gX, FV 0. 3.4

Thus there are no components ofFXinD1andD2. Furthermore, we have gFX, ξ gX, Fξ 0,

gFX, N gX, FN 0. 3.5

Proof is completed.

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If we setDD⊥RadTM⊥F RadTM, we can write

TMD⊕D2. 3.6

From the above proposition we have the following corollary.

Corollary 3.4. The distributionDis invariant with respect toF.

Example 3.5. Let M R⁵₂, g be a 5-dimensional semi-Euclidean space with signature−,,−,,andx, y, z, s, tbe the standard coordinate system ofR⁵₂. If we setFx, y, z, s, t x, y,−z,−s,−t, thenF² I and F is a product structure onR⁵₂. Consider a hyper- surfaceMinMby the equation:

txyz. 3.7

ThenTMSpan{U1, U2, U3, U4}, where

U₁ ∂

∂x ∂

∂t, U₂ ∂

∂y ∂

∂t, U₃ ∂

∂z ∂

∂t, U₄ ∂

∂s. 3.8

It is easy to check thatMis a lightlike hypersurface and

TM^⊥Span{ξU1−U2U3}. 3.9

Then take a lightlike transversal vector bundle as follow:

ltrTM Span

N−1 4

∂

∂x ∂

∂y ∂

∂z− ∂

∂t

. 3.10

It follows that the corresponding screen distributionSTMis spanned by

{W1U4, W2U1−U2−U3, W3U1U2−U3}. 3.11

If we setD Span{W1},D1 Span{W2}andD2 Span{W3}, then it can be easily checked thatMis a screen semi-invariant lightlike hypersurface ofM.

Example 3.6. Letx, y, z, tbe the standard coordinate system ofR⁴andds² −dx²−dy²dz² dt²be a semi-Riemannian metric onR⁴with 2-index. LetFbe a product structure onR⁴given

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byFx, y, z, t z, t, x, y. We consider the hypersurfaceMgiven byt x 1/2yz² 1. One can easily see thatMis a lightlike hypersurface and

RadTMSpan

ξ ∂

∂x

yz ∂

∂y−

yz ∂

∂z ∂

∂t , ltrTM Span

⎧⎨

⎩N− 1 2

1

yz₂ ∂

∂x

yz ∂

∂y

yz ∂

∂z − ∂

∂t

⎫⎬

⎭,

STM Span

W₁−

yz ∂

∂x ∂

∂y, W₂ ∂

∂z

yz∂

∂t .

3.12

We can easily check that

FξW1W2, FN 1 2

1

yz₂{W1−W2}. 3.13 ThusMis a screen semi-invariant lightlike hypersurface withD {0},D1 Span{Fξ}and D2Span{FN}.

Example 3.7. LetR⁴₂, gbe a 4-dimensional semi-Euclidean space with signature−,−,, andx1, x2, x3, x4be the standard coordinate system ofR⁴₂. Consider a Monge hypersurface MofR⁴₂given by

x4Ax1Bx2Cx3, A²B²−C²1, A, B, C∈R. 3.14 Then the tangent bundleTMof the hypersurfaceMis spanned by

U₁ ∂

∂x1 A ∂

∂x4

, U₂ ∂

∂x2 B ∂

∂x4

, U₃ ∂

∂x3 C ∂

∂x4

. 3.15

It is easy to check thatMis a lightlike hypersurfacep.196, Ex.1,3whose radical distribution RadTMis spanned by

ξAU1BU2−CU3A ∂

∂x1 B ∂

∂x2 −C ∂

∂x3 ∂

∂x4. 3.16

Furthermore, the lightlike transversal vector bundle is given by

ltrTM Span

N− 1 2C²1

A ∂

∂x₁ B ∂

∂x₂ C ∂

∂x₃ − ∂

∂x₄ . 3.17 It follows that the corresponding screen distributionSTMis spanned by

W₁ 1 A²B²

B ∂

∂x₁ −A ∂

∂x₂

, W₂ 1 A²B²

∂

∂x₃ C ∂

∂x₄ . 3.18

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If we define a mappingFbyFx1, x2, x3, x4 x1, x2,−x3,−x4thenF²IandFis a product structure on R⁴₂. One can easily check that FSTM STMand F RadTM ltrTM.

Thus Mis a radical anti-invariant lightlike hypersurface ofR⁴₂. Furthermore, this lightlike hypersurface is a screen invariant lightlike hypersurface.

Theorem 3.8. LetM, gbe a semi-Riemannian product manifold andMbe a screen semi-invariant lightlike hypersurface ofM. Then the following assertions are equivalent.

iThe distributionDis integrable with respect to the induced connection∇ofM.

iiBX, fY BY, fX, for anyX, Y∈ΓD.

iiigA^∗_ξX, P fY gA^∗_ξY, P fX, for anyX, Y ∈ΓD.

Proof. For anyX, Y ∈ΓD, from2.9,2.24, and3.1, we obtain f∇XYw∇XYNBX, YFN∇XfYB

X, fY

N. 3.19

Interchanging role ofXandY we have

f∇YXw∇YXNBY, XFN∇YfXB Y, fX

N. 3.20

From3.19,3.20we get

wX, Y B X, fY

−B Y, fX

3.21

and this isi⇔ii. From the first equation of2.15, we concludeii⇔iii. Thus we have our assertion.

From the decomposition3.6, we can give the following definition.

Definition 3.9. LetMbe a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifoldM. IfBX, Y 0, for anyX ∈ ΓD,Y ∈ ΓD2, then we say thatMis a mixed geodesic lightlike hypersurface.

Theorem 3.10. LetM, gbe a semi-Riemannian product manifold andMbe a screen semi-invariant lightlike hypersurface ofM. Then the following assertions are equivalent.

iMis mixed geodesic.

iiThere is noD2-component ofAN. iiiThere is noD1-component ofA^∗_ξ.

Proof. Suppose thatMis mixed geodesic screen semi-invariant lightlike hypersurface ofM with respect to the Levi-Civita connection∇. From2.24,2.9,2.10, and3.1, we obtain

∇XFNBX, FNN−fANXτXFN−wANXN, 3.22

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for anyX∈ΓD. If we take tangential and transversal parts of this last equation we have

∇XFN−fANXτXFN,

BX, FN −wANX. 3.23

Furthermore, sincewANX gANX, Fξ, we geti⇔ii. SincegFN, ξ gN, Fξ 0, we obtain

gANX, Fξ −g

A^∗_ξX, FN

. 3.24

This isii⇔iii.

From the decomposition3.6, we have the following theorem.

Theorem 3.11. LetMbe a screen semi-invariant lightlike hypersurface of a semi-Riemannian product manifoldM. ThenMis a locally product manifold according to the decomposition3.6if and only if fis parallel with respect to induced connection∇, that is∇f0.

Proof. LetM be a locally product manifold. Then the leaves of distributionsD andD2 are both totally geodesic inM. Since the distributionDis invariant with respect toF then, for anyY ∈ ΓD,FY ∈ΓD. Thus∇XY and∇XfY belong toΓD, for anyX ∈ΓTM. From the Gauss formula, we obtain

∇XfY B X, fY

Nf∇XYw∇XYNBX, YFN. 3.25

Comparing the tangential and normal parts with respect toDof3.25, we have

∇XfY f∇XY, that is

∇Xf

Y 0, 3.26

BX, Y 0. 3.27

SincefZ0, for anyZ∈ΓD2, we get∇XfZ0 andf∇XZ0, that is∇XfZ0. Thus we have∇f 0 onM.

Conversely, we assume that∇f 0 onM. Then we have ∇XfY f∇XY, for any X, Y ∈ ΓDand∇UfW f∇UW 0, for anyU, W ∈ ΓD2. Thus it follows that∇XfY ∈ ΓDand∇UW ∈ΓD2. Hence, the leaves of the distributionsDandD2 are totally geodesic inM.

FromTheorem 3.11and3.27we 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.

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LetMbe a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifoldM. Then we have the following decomposition:

TMSTM⊥ {RadTM⊕F RadTM}. 3.28 Theorem 3.13. Let M be a radical anti-invariant lightlike hypersurface of a semi-Riemannian product manifoldM. Then the screen distributionSTMofMis an integrable distribution if and only ifBX, FY BY, FX.

Proof. If a vector fieldXonMbelongs toSTMif and only ifηX 0. SinceMis a radical anti-invariant lightlike hypersurface, for anyX∈ΓSTM,FX ∈ΓSTM. For anyX, Y ∈ ΓSTM, we can write

∇XFY ∇XFYBX, FYN. 3.29

In this last equation interchanging role ofXandY, we obtain

FX, Y ∇XFY− ∇YFX BX, FY−BY, FXN. 3.30

SinceηX, Y gX, Y, N gFX, Y, FN, we get

ηX, Y BX, FY−BY, FXgN, FN. 3.31

SincegN, FN/0,ηX, Y 0 if and only ifBX, FY BY, FX. This is our assertion.

4. Quarter-Symmetric Nonmetric Connections

LetM, g, Fbe a semi-Riemannian product manifold and∇be the Levi-Civita connection onM. If we set

DXY ∇XYuYFX, 4.1

for anyX, Y ∈ΓTM, thenDis a linear connection onM, whereuis a 1-form onMwithU as associated vector field, that is

uX gX, U. 4.2

The torsion tensor ofDonMdenoted byT. Then we obtain

TX, Y uYFX−uXFY, 4.3 D_Xg

Y, Z −uYgFX, Z−uZgFX, Y, 4.4

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for anyX, Y ∈ ΓTM. ThusD is a quarter-symmetric nonmetric connection onM. From 2.24and4.1we have

D_XF

Y uFYFX−uYX. 4.5

ReplacingXbyFXandYbyFYin4.5and using2.19we obtain

D_FXF

FY uYX−uFYFX. 4.6

Thus we have

DXF

Y DFXF

FY 0. 4.7

If we set

FX, Y gFX, Y, 4.8

for anyX, Y ∈ΓTM, from4.1we get

DX F

Y, Z

∇X F

Y, Z−uYgX, Z−uZgX, Y. 4.9

From4.1the curvature tensorR^Dof the quarter-symmetric nonmetric connectionDis given by

R^DX, YZRX, YZλX, ZFY−λY, ZFX, 4.10 for anyX, Y, Z∈ΓTM, whereλis a0,2-tensor given byλX, Z ∇XuZ−uZuFX.

If we setR^DX, Y, Z, W gR^DX, YZ, W, then, from4.10, we obtain

R^DX, Y, Z, W −R^DY, X, Z, W. 4.11

We note that the Riemannian curvature tensorR^DofDdoes not satisfy the other curvature- like properties. But, from4.10, we have

R^DX, YZR^DY, ZXR^DZ, XY

λZ, Y−λY, Z FX

λX, Z−λZ, X FY

λY, X−λX, Y FZ.

4.12

Thus we have the following proposition.

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Proposition 4.1. LetMbe a lightlike hypersurface of a semi-Riemannian product manifoldM. Then the first Bianchi identity of the quarter-symmetric nonmetric connectionDonMis provided if and only ifλis symmetric.

LetMbe a lightlike hypersurface of a semi-Riemannian product manifoldM, gwith quarter-symmetric nonmetric connectionD. Then the Gauss and Weingarten formulas with respect toDare given by, respectively,

D_XY D_XYBX, YN 4.13

DXN−ANXτXN 4.14

for anyX, Y ∈ΓTM, whereDXY,ANX ∈ΓTM,BX, Y gDXY, ξ,τX gDXN, ξ.

Here,D,BandA_Nare called the induced connection onM, the second fundamental form, and the Weingarten mapping with respect toD. From2.9,2.10,3.1,4.1,4.13, and 4.14we obtain

D_XY ∇XYuYfX, 4.15

BX, Y BX, Y uYwX, 4.16

A_NXA_NX−uNfX,

τX τX uNwX, 4.17

for anyX,Y ∈ΓTM. From4.1,4.4,4.13, and4.16we get D_Xg

Y, Z BX, YηZ BX, ZηY

−uYg fX, Z

−uZg fX, Y

. 4.18

On the other hand, the torsion tensor of the induced connectionDis

T^DX, Y uYfX−uXfY. 4.19 From last two equations we have the following proposition.

Proposition 4.2. LetMbe a lightlike hypersurface of a semi-Riemannian product manifoldM, g with quarter-symmetric nonmetric connection D. Then the induced connection D is a quarter- symmetric nonmetric connection on the lightlike hypersurfaceM.

For anyX, Y ∈ΓTM, we can write

DX P YD^∗_XP YCX, P Yξ,

D_X ξ−A^∗_ξXεXξ, 4.20

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where D_X^∗P Y A^∗_ξX ∈ ΓSTM, CX, P Y gDXP Y, N, and εX gDXξ, N. From 2.14,16, and4.15, we obtain

CX, P Y CX, P Y uP Yη fX

, 4.21

A^∗_ξXA^∗_ξX−uξP fX, εX −τX uξη fX

. 4.22

Using2.15,4.16and4.22we obtain

BX, P Y g

A^∗_ξX, P Y

uP YwX uξgFX, P Y,

4.23

for anyX, Y ∈ΓTM.

Now, we consider a screen semi-invariant lightlike hypersurface M of a semi- Rieamannian product manifoldMwith respect to the quarter symmetric connectionDgiven by4.1. SincewX gFX, ξ, for anyX ∈ ΓD, wX 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 formBof quarter-symmetric nonmetric connectionDis degenerate.

Proposition 4.4. Let M, g be a semi-Riemannian product manifold and M be a screen semi- invariant lightlike hypersurfaces ofM. IfMis Dtotally geodesic with respect to∇, thenM isD totally geodesic with respect to quarter-symmetric nonmetric connection.

Theorem 4.5. LetM, gbe a semi-Riemannian product manifold andMbe a screen semi-invariant lightlike hypersurfaces ofM. Then the following assertions are equivalent.

iThe distributionDis integrable with respect to the quarter symmetric nonmetric connection D.

iiBX, fY BY, fX, for anyX,Y ∈ΓD.

iiigA^∗_ξX, P fY gA^∗_ξY, P fX, for anyX,Y ∈ΓD.

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

From4.23, for anyX ∈ΓDandY ∈ΓD2, we haveBX, P Y gA^∗_ξX, P Y. If we setDD⊥D2, then, fromTheorem 3.10, we have the following corollary.

Corollary 4.6. LetM, gbe a semi-Riemannian product manifold andMbe a screen semi-invariant lightlike hypersurface ofM. Then the distributionDis a mixed geodesic foliation defined with respect to quarter symmetric nonmetric connection if and only if there is noD1component ofA^∗_ξ.

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From4.15, we obtain

R^DX, YZRX, YZuZ

∇Xf Y−

∇Yf X

λX, ZfY−λY, ZfX, 4.24

whereλis a0,2tensor onMgiven byλX, Z ∇XuZ−uZufX.

From4.24, we have the following proposition which is similar to theProposition 4.1.

Proposition 4.7. LetMbe a lightlike hypersurface of a semi-Riemannian product manifoldM. One assumes thatfis parallel onM. Then the first Bianchi identity of the quarter-symmetric nonmetric connectionDonMis provided if and only ifλis symmetric.

Now we will compute Gauss-Codazzi equations of lightlike hypersurfaces with respect to the quarter-symmetric nonmetric connection:

g

R^DX, YZ, P W

gRX, YZ, P W

BX, ZCY, P W−BY, ZCX, P W λX, Zg

fY, P W

−λY, Zg

fX, P W , g

R^DX, YZ, ξ

∇XBY, Z−∇YBX, Z λX, ZwY−λY, ZwX, g

R^DX, YZ, N

gRX, YZ, N λX, Zη

fY

−λY, Zη fX

,

4.25

for anyX,Y,Z,W∈ΓTM.

Now, letMbe a screen semi-invariant lightlike hypersurface of am2-dimensional semi-Riemannian product manifold with the quarter-symmetric nonmetric connection D such that the tensor fieldf is parallel onM. We consider the local quasiorthonormal basis {Ei, Fξ, FN, ξ, N},i 1, . . . m−2, ofM alongM, where {E1, . . . , E_m−2} is an orthonormal basis ofΓD. Then, the Ricci tensor ofMwith respect toDis given by

R^D0,2X, Y ^m−2

i1

ε_ig

R^DX, EiY, Ei

g

R^DX, FξY, FN g

R^DX, FNY, Fξ g

R^DX, ξY, N .

4.26

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From4.24we have

R^D0,2X, Y R^0,2X, Y ^m−2

i1

ε_i

λX, Yg fE_i, E_i

−λEi, Yg

fX, E_i

−λFξ, YηX−λξ, Yη fX

,

4.27

whereR^0,2X, Yis the Ricci tensor ofM. Thus we have the following corollary.

Corollary 4.8. LetMa screen semi-invariant lightlike hypersurface of am2-dimensional semi- Riemannian product manifold with the quarter-symmetric nonmetric connection D such that the tensor field f is parallel on M and R^0,2X, Y is symmetric. Then R^D0,2 is symmetric on the distributionDif and only ifλis symmetric andλfX, Y λfY, 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 Keles¸ on his sixtieth birthday.

References

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a Semi-Riemannian Product Manifold and Quarter-Symmetric Nonmetric Connections

Research Article

Lightlike Hypersurfaces of

a Semi-Riemannian Product Manifold and Quarter-Symmetric Nonmetric Connections

Erol Kılıc¸

and O ˘guzhan Bahadır

1. Introduction

2. Lightlike Hypersurfaces

3. Lightlike Hypersurfaces of Semi-Riemannian Product Manifolds

4. Quarter-Symmetric Nonmetric Connections

Acknowledgment

References

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