Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 380657 6 pageshttpdxdoiorg1011552013380657

Research ArticleSome Curvature Properties of (LCS)

119899-Manifolds

Mehmet Atccedileken

Department of Mathematics Faculty of Arts and Science Gaziosmanpasa University 60100 Tokat Turkey

Correspondence should be addressed to Mehmet Atceken mehmetatceken382gmailcom

Received 14 January 2013 Revised 4 March 2013 Accepted 6 March 2013

Academic Editor Narcisa C Apreutesei

Copyright copy 2013 Mehmet Atceken This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The object of the present paper is to study (LCS)119899-manifolds with vanishing quasi-conformal curvature tensor (LCS)

119899-manifolds

satisfying Ricci-symmetric condition are also characterized

1 Introduction

Recently in [1] Shaikh introduced and studied Lorentzianconcircular structure manifolds (briefly (LCS)-manifold)which generalizes the notion of LP-Sasakian manifoldsintroduced by Matsumoto [2]

Generalizing the notion of LP-Sasakian manifold in 2003[1] Shaikh introduced the notion of (LCS)

119899-manifolds along

with their existence and applications to the general theoryof relativity and cosmology Also Shaikh and his coauthorsstudied various types of (LCS)

119899-manifolds by imposing the

curvature restrictions (see [3ndash6]) In [7 8] the authors alsostudied (LCS)

2119899+1-manifolds

The submanifold of an (LCS)119899-manifold is studied by

Atceken and Hui [9 10] and Shukla et al [11] In [12] Yanoand Sawaki introduced the quasi-conformal curvature tensorand later it was studied by many authors with curvaturerestrictions on various structures [13]

After then the same author studied weakly symmetric(LCS)

119899-manifolds by several examples and obtain various

results in suchmanifolds In [7] authors shown that a pseudoprojectively flat and pseudo projectively recurrent (LCS)

119899

manifolds are 120578-Einstein manifoldOn the other hand in [5] authors proved the existence

of 120601-recurrent (LCS)3manifold which is neither locally

symmetric nor locally 120601-symmetric by nontrivial examplesFurthermore they also give the necessary and sufficientconditions for a (LCS)

119899-manifold to be locally 120601-recurrent

In this study we have investigated the quasi-conformalflat (LCS)

119899-manifolds satisfying properties such as Ricci-

symmetric locally symmetric and 120578-Einstein Finally wegive an example for 120578-Einstein manifolds

2 Preliminaries

An 119899-dimensional Lorentzian manifold119872 is a smooth con-nected paracompact Hausdorff manifold with a Lorentzianmetric tensor 119892 that is119872 admits a smooth symmetric tensorfield 119892 of the type (2 0) such that for each 119901 isin 119872

119892119901 119879119872(119901) times 119879

119872(119901) 997888rarr R (1)

is a nondegenerate inner product of signature (minus + + +)In such amanifold a nonzero vector119883

119901isin 119879119872(119901) is said to be

timelike (resp nonspacelike null and spacelike) if it satisfiesthe condition 119892

119901(119883119901 119883119901) lt 0 (resp le0 =0 gt0) These cases

are called casual character of the vectors

Definition 1 In a Lorentzianmanifold (119872 119892) a vector field 119875defined by

119892 (119883 119875) = 119860 (119883) (2)for any119883 isin Γ(119879119872) is said to be a concircular vector field if

(nabla119883119860)119884 = 120572 119892 (119883 119884) + 119908 (119883)119860 (119884) (3)

for 119884 isin Γ(119879119872) where 120572 is a nonzero scalar function 119860 is a1-form119908 is also closed 1-form and nabla denotes the Levi-Civitaconnection on119872 [7]

2 Abstract and Applied Analysis

Let119872 be a Lorentzianmanifold admitting a unit timelikeconcircular vector field 120585 called the characteristic vector fieldof the manifold Then we have

119892 (120585 120585) = minus1 (4)

Since 120585 is a unit concircular unit vector field there exists anonzero 1-form 120578 such that

119892 (119883 120585) = 120578 (119883) (5)

The equation of the following form holds

(nabla119883120578) 119884 = 120572 119892 (119883 119884) + 120578 (119883) 120578 (119884) 120572 = 0 (6)

for all 119883119884 isin Γ(119879119872) where 120572 is a nonzero scalar functionsatisfying

nabla119883120572 = 119883 (120572) = 119889120572 (119883) = 120588120578 (119883) (7)

120588 being a certain scalar function given by 120588 = minus120585(120572)Let us put

nabla119883120585 = 120572120601119883 (8)

then from (6) and (8) we can derive

120601119883 = 119883 + 120578 (119883) 120585 (9)

which tell us that 120601 is a symmetric (1 1)-tensor Thus theLorentzian manifold 119872 together with the unit timelikeconcircular vector field 120585 its associated 1-form 120578 and (1 1)-type tensor field 120601 is said to be a Lorentzian concircularstructure manifold

A differentiable manifold 119872 of dimension 119899 is called(LCS)-manifold if it admits a (1 1)-type tensor field 120601 acovariant vector field 120578 and a Lorentzian metric 119892 whichsatisfy

120578 (120585) = 119892 (120585 120585) = minus1 (10)

1206012

119883 = 119883 + 120578 (119883) 120585 (11)

119892 (119883 120585) = 120578 (119883) (12)

120601120585 = 0 120578 ∘ 120601 = 0 (13)

for all119883 isin Γ(119879119872) Particularly if we take 120572 = 1 then we canobtain the 119871119875-Sasakian structure of Matsumoto [2]

Also in an (LCS)119899-manifold 119872 the following relations

are satisfied (see [3ndash6])

120578 (119877 (119883 119884)119885) = (1205722

minus 120588) [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

(14)

119877 (120585 119883) 119884 = (1205722

minus 120588) [119892 (119883 119884) 120585 minus 120578 (119884)119883] (15)

119877 (119883 119884) 120585 = (1205722

minus 120588) [120578 (119884)119883 minus 120578 (119883)119884] (16)

(nabla119883120601)119884 = 120572 [119892 (119883 119884) 120585 + 2120578 (119883) 120578 (119884) 120585 + 120578 (119884)119883] (17)

119878 (119883 120585) = (119899 minus 1) (1205722

minus 120588) 120578 (119883) (18)

119878 (120601119883 120601119884) = 119878 (119883 119884) + (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (119884) (19)

for all vector fields 119883119884 119885 on119872 where 119877 and 119878 denote theRiemannian curvature tensor and Ricci curvature respec-tively 119876 is also the Ricci operator given by 119878(119883 119884) =

119892(119876119883 119884)Now let (119872 119892) be an 119899-dimensional Riemannian man-

ifold then the concircular curvature tensor the Weylconformal curvature tensor 119862 and the pseudo projectivecurvature tensor are respectively defined by

(119883 119884)119885 = 119877 (119883 119884)119885

minus

120591

119899 (119899 minus 1)

[119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(20)

119862 (119883 119884)119885 = 119877 (119883 119884)119885 minus

1

119899 minus 2

times [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884]

+

120591

(119899 minus 1) (119899 minus 2)

[119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(21)

(119883 119884)119885 = 119886119877 (119883 119884)119885

+ 119887 [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884]

minus

120591

119899

[

119886

119899 minus 1

+ 119887] [119892 (119884 119885)119883 minus 119892 (119883119885) 119884]

(22)

where 119886 and 119887 are constants such that 119886 119887 = 0 and 120591 is alsothe scalar curvature of119872 [7]

For an 119899-dimensional (LCS)119899-manifold the quasi-

conformal curvature tensor C is given by

C (119883 119884)119885 = 119886119877 (119883 119884)119885

+ 119887 [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883119885)119876119884]

minus

120591

119899

[

119886

119899 minus 1

+ 2119887] [119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(23)

for all119883119884 119885 isin Γ(119879119872) [14]The notion of quasi-conformal curvature tensor was

defined by Yano and Swaki [12] If 119886 = 1 and 119887 = minus1(119899 minus 1)then quasi-conformal curvature tensor reduces to conformalcurvature tensor

3 Quasi-Conformally Flat (LCS)119899-Manifolds

and Some of Their Properties

For an 119899-dimensional quasi-conformally flat (LCS)119899-mani-

fold we know for 119885 = 120585 from (23)

Abstract and Applied Analysis 3

119886119877 (119883 119884) 120585 + 119887 [119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

minus

120591

119899

[

119886

119899 minus 1

+ 2119887] [119892 (119884 120585)119883 minus 119892 (119883 120585) 119884] = 0

(24)

Here taking into account of (16) we have

[120578 (119884)119883 minus 120578 (119883)119884] [119886 (1205722

minus 120588) + 119887 (119899 minus 1) (1205722

minus 120588)

minus

120591

119899

(

119886

119899 minus 1

+ 2119887)]

+ 119887 [120578 (119884]119876119883 minus 120578 (119883)119876119884] = 0

(25)

Let 119884 = 120585 be in (25) then also by using (18) we obtain

[minus119883 minus 120578 (119883) 120585] [119886 (1205722

minus 120588) minus

120591

119899

(

119886

119899 minus 1

+ 2119887)

+ 119887 (119899 minus 1) (1205722

minus 120588) ]

+ 119887 [minus119876119883 minus 120578 (119883) (119899 minus 1) (1205722

minus 120588) 120585] = 0

(26)

Taking the inner product on both sides of the last equation by119884 we obtain

[119892 (119883 119884) + 120578 (119883) 120578 (119884)] [119886 (1205722

minus 120588) + 119887 (119899 minus 1)

times (1205722

minus 120588) minus

120591

119899

(

119886

119899 minus 1

+ 2119887)]

+ 119887 [119878 (119883 119884) + 120578 (119883) 120578 (119884) (1205722

minus 120588) (119899 minus 1)] = 0

(27)

that is

119878 (119883 119884) = 119892 (119883 119884)

times [

120591

119899119887

(

119886

119899 minus 1

+ 2119887) minus (1205722

minus 120588) (

119886

119887

+ (119899 minus 1))]

+ 120578 (119883) 120578 (119884) [

120591

119899119887

(

119886

119899 minus 1

+ 2119887)

minus (1205722

minus 120588) (

119886

119887

+ 2 (119899 minus 1))]

(28)

Now we are in a proposition to state the following

Theorem 2 If an 119899-dimensional (LCS)119899-manifold119872 is quasi-

conformally flat then119872 is an 120578-Einstein manifold

Now let 1198901 1198902 119890

119899minus1 120585 be an orthonormal basis of the

tangent space at any point of the manifold Then putting119883 =

119884 = 119890119894 120585 in (28) and taking summation for 1 le 119894 le 119899 minus 1 we

have

120591 = 119899 (119899 minus 1) (1205722

minus 120588) if 119886 + (119899 minus 2) 119887 = 0 (29)

In view of (28) and (29) we obtain

119878 (119883 119884) = (119899 minus 1) (1205722

minus 120588) 119892 (119883 119884) (30)

which is equivalent to

119876119883 = (119899 minus 1) (1205722

minus 120588)119883 (31)

for any119883 isin Γ(119879119872)By using (29) and (31) in (23) for a quasi-conformally flat

(LCS)119899-manifold119872 we get

119877 (119883 119884)119885 = (1205722

minus 120588) 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884 (32)

for all 119883119884 119885 isin Γ(119879119872) If we consider Schurrsquos Theorem wecan give the following the theorem

Theorem3 Aquasi-conformally flat (LCS)119899-manifoldM (119899 gt

1) is a manifold of constant curvature (1205722 minus 120588) provided that119886 + 119887(119899 minus 2) = 0

Now let us consider an (LCS)119899-manifold 119872 which is

conformally flat Thus we have from (21) that

119877 (119883 119884)119885 =

1

119899 minus 2

119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884

minus

120591

(119899 minus 1) (119899 minus 2)

119892 (119884 119885)119883 minus 119892 (119883119885) 119884

(33)

for all vector fields119883119884 119885 tangent to119872 Setting119885 = 120585 in (33)and using (16) (18) we have

[

120591

119899 minus 1

minus (1205722

minus 120588)] [120578 (119884)119883 minus 120578 (119883)119884]

= [120578 (119884)119876119883 minus 120578 (119883)119876119884]

(34)

If we put 119884 = 120585 in (34) and also using (18) we obtain

119876119883 = [

120591

119899 minus 1

minus (1205722

minus 120588)]119883 + [

120591

119899 minus 1

minus 119899 (1205722

minus 120588)] 120578 (119883) 120585

(35)

Corollary 4 A conformally flat (LCS)119899-manifold is an 120578-

Einstein manifold

Generalizing the notion of a manifold of constant curva-ture Chen and Yano [15] introduced the notion of a manifoldof quasi-constant curvature which can be defined as follows

Definition 5 ARiemannianmanifold is said to be a manifoldof quasi-constant curvature if it is conformally flat and itscurvature tensor of type (0 4) is of the form

(119883 119884 119885119882)

= 119886 119892 (119884 119885) 119892 (119883119882) minus 119892 (119883 119885) 119892 (119884119882)

+ 119887 119892 (119884 119885)119860 (119883)119860 (119882) minus 119892 (119883 119885)119860 (119884)119860 (119882)

+119892 (119883119882)119860 (119884)119860 (119885) minus 119892 (119884119882)119860 (119883)119860 (119885)

(36)

4 Abstract and Applied Analysis

for all 119883119884 119885119882 isin Γ(119879119872) where 119886 119887 are scalars of which119887 = 0 and 119860 is a nonzero 1-form (for more details we refer to[13 16])

Thus we have the following theorem for (LCS)119899-conform-

ally flat manifolds

Theorem 6 A conformally flat (LCS)119899-manifold is a manifold

of quasi-constant curvature

Proof From (33) and (35) we obtain

(119883 119884 119885119882)

= (

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 119892 (119884 119885) minus 119892 (119884119882) 119892 (119883 119885)

+ (

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 120578 (119884) 120578 (119885) minus 119892 (119884119882) 120578 (119883) 120578 (119885)

+119892 (119884 119885) 120578 (119883) 120578 (119882) minus 119892 (119883 119885) 120578 (119884) 120578 (119882)

(37)

This implies (36) for

119886 =

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119887 =

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119860 = 120578

(38)

This proves our assertion

Next differentiating the (19) covariantly with respect to119882 we get

nabla119882119878 (120601119883 120601119884) = nabla

119882119878 (119883 119884) + (119899 minus 1)119882(120572

2

minus 120588)

+ (119899 minus 1) (1205722

minus 120588)119882 [120578 (119883) 120578 (119884)]

(39)

for any119883119884 isin Γ(119879119872) Making use of the definition ofnabla119878 and(8) we have

(nabla119882119878) (120601119883 120601119884) + 119878 (nabla

119882120601119883 120601119884) + 119878 (120601119883 nabla

119882120601119884)

= (nabla119882119878) (119883 119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(40)

Thus we have

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 ((nabla119882120601)119883 + 120601nabla

119882119883 120601119884)

minus 119878 (120601119883 (nabla119882120601)119884 + 120601nabla

119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(41)

Here taking account of (17) we arrive at

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 (120572 119892 (119883119882) 120585 + 2120578 (119883) 120578 (W) 120585 + 120578 (119883)119882 120601119884)

minus 119878 (120601119883 120572 119892 (119884119882) 120585 + 2120578 (119884) 120578 (119882) 120585 + 120578 (119884)119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1) (120572

2

minus 120588) 120578 (119884)

times 120578 (nabla119882119883) + 120572119892 (119883 120601119882) + (119899 minus 1) (120572

2

minus 120588) 120578 (119883)

times 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

= minus120572 119892 (119883119882) 119878 (120585 120601119884) + 2120578 (119883) 120578 (119882) 119878 (120585 120601119884)

+120578 (119883) 119878 (119882 120601119884)

minus 120572 119892 (119884119882) 119878 (120601119883 120585) + 2120578 (119884) 120578 (119882) 119878 (120601119883 120585)

+120578 (119884) 119878 (120601119883119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(42)

Again by using (13) (18) and (19) we reach

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119883)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883)

Abstract and Applied Analysis 5

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588)

times 120578 (nabla119882119883) 120578 (119884) + 120572120578 (119884) 119892 (119883 120601119882)

+120578 (nabla119882119884) 120578 (119883) + 120572120578 (119883) 119892 (119884 120601119882)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

+ 120572 (119899 minus 1) (1205722

minus 120588)

times 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

(43)

Thus we have the following theorem

Theorem7 If an (LCS)119899-manifold119872 is Ricci-symmetric then

1205722

minus 120588 is constant

Proof If 119899 gt 1-dimensional (LCS)119899-manifold 119872 is Ricci-

symmetric then from (43) we conclude that

120572 (119899 minus 1) (1205722

minus 120588) 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

minus 120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882) = 0

(44)

It follows that

120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882) 120585 minus 120578 (119883) 120601119882

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120585

minus 120572120578 (119883) 120601119876119882 minus 120572119878 (120601119883119882) 120585 = 0

(45)

from which

minus 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882)

minus (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) + 119878 (120601119883119882) = 0

(46)

which is equivalent to

minus 120572 (119899 minus 1) (1205722

minus 120588) 120601119882 minus (119899 minus 1)119882(1205722

minus 120588) 120585

+ 120572120601119876119882 = 0

(47)

that is

119882(1205722

minus 120588) = 0 (48)

which proves our assertion

Since nabla119877 = 0 implies that nabla119878 = 0 we can give the follow-ing corollary

Corollary 8 If an 119899-dimensional (119871119862119878)119899-manifold 119872 is

locally symmetric then 1205722 minus 120588 is constant

Now taking the covariant derivation of the both sides of(18) with respect to 119884 we have

119884119878 (119883 120585) = (119899 minus 1)119882 [(1205722

minus 120588) 120578 (119883)] (49)

From the definition of the covariant derivation of Ricci-tensor we have

(nabla119884119878) (119883 120585) = nabla

119884119878 (119883 120585) minus 119878 (nabla

119884119883 120585) minus 119878 (119883 nabla

119884120585)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883) + (1205722

minus 120588)

times [120578 (nabla119884119883) + 120572119892 (119883 120601119884)]

minus (119899 minus 1) (1205722

minus 120588) 120578 (nabla119884119883) minus 120572119878 (119883 120601119884)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883)

+ 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119884) minus 120572119878 (119883 120601119884)

(50)

If an (119871119862119878)119899-manifold 119872 Ricci symmetric then Theorem 7

and (43) imply that

119878 (119883 120601119884) = (119899 minus 1) (1205722

minus 120588) 119892 (120601119884119883) (51)

This leads us to state the following

Theorem9 If an (LCS)119899-manifold119872 is Ricci symmetric then

it is an Einstein manifold

Corollary 10 If an (LCS)119899-manifold119872 is locally symmetric

then it is an Einstein manifold

In this section an example is used to demonstrate that themethod presented in this paper is effective But this exampleis a special case of Example 61 of [6]

Example 11 Now we consider the 3-dimensional manifold

119872 = (119909 119910 119911) isin R3

119911 = 0 (52)

where (119909 119910 119911) denote the standard coordinates in R3 Thevector fields

1198901= 119890119911

(119909

120597

120597119909

+ 119910

120597

120597119910

) 1198902= 119890119911120597

120597119910

1198903=

120597

120597119911

(53)

are linearly independent of each point of 119872 Let 119892 be theLorentzian metric tensor defined by

119892 (1198901 1198901) = 119892 (119890

2 1198902) = minus119892 (119890

3 1198903) = 1

119892 (119890119894 119890119895) = 0 119894 = 119895

(54)

6 Abstract and Applied Analysis

for 119894 119895 = 1 2 3 Let 120578 be the 1-formdefined by 120578(119885) = 119892(119885 1198903)

for any 119885 isin Γ(119879119872) Let 120601 be the (11)-tensor field defined by

1206011198901= 1198901 120601119890

2= 1198902 120601119890

3= 0 (55)

Then using the linearity of 120601 and 119892 we have 120578(1198903) = minus1

1206012

119885 = 119885 + 120578 (119885) 1198903

119892 (120601119885 120601119882) = 119892 (119885119882) + 120578 (119885) 120578 (119882)

(56)

for all 119885119882 isin Γ(119879119872) Thus for 120585 = 1198903 (120601 120585 120578 119892) defines a

Lorentzian paracontact structure on119872Now let nabla be the Levi-Civita connection with respect

to the Lorentzian metric 119892 and let 119877 be the Riemanniancurvature tensor of 119892 Then we have

[1198901 1198902] = minus119890

119911

1198902 [119890

1 1198903] = minus119890

1 [119890

2 1198903] = minus119890

2

(57)

Making use of the Koszul formulae for the Lorentzian metrictensor 119892 we can easily calculate the covariant derivations asfollows

nabla1198901

1198901= minus1198903 nabla

1198902

1198901= 119890119911

1198902 nabla

1198901

1198903= minus1198901

nabla1198902

1198903= minus1198902 nabla

1198902

1198902= minus119890119911

1198901minus 1198903

nabla1198901

1198902= nabla1198903

1198901= nabla1198903

1198902= nabla1198903

1198903= 0

(58)

From the previously mentioned it can be easily seen that(120601 120585 120578 119892) is an (LCS)

3-structure on 119872 that is 119872 is an

(LCS)3-manifold with 120572 = minus1 and 120588 = 0 Using the

previous relations we can easily calculate the components ofthe Riemannian curvature tensor as follows

119877 (1198901 1198902) 1198901= (1198902119911

minus 1) 1198902 119877 (119890

1 1198902) 1198902= (1 minus 119890

2119911

) 1198901

119877 (1198901 1198903) 1198901= minus1198903 119877 (119890

1 1198903) 1198903= minus1198901

119877 (1198902 1198903) 1198902= minus1198903 119877 (119890

2 1198903) 1198903= minus1198902

119877 (1198901 1198902) 1198903= 119877 (119890

1 1198903) 1198902= 119877 (119890

2 1198903) 1198901= 0

(59)

By using the properties of119877 and definition of the Ricci tensorwe obtain

119878 (1198901 1198901) = 119878 (119890

2 1198902) = minus119890

2119911

119878 (1198903 1198903) = minus2

119878 (1198901 1198902) = 119878 (119890

1 1198903) = 119878 (119890

2 1198903) = 0

(60)

Thus the scalar curvature 120591 of119872 is given by

120591 =

3

sum

119894=1

119892 (119890119894 119890119894) 119878 (119890119894 119890119894) = 2 (1 minus 119890

2119911

) (61)

On the other hand for any 119885119882 isin Γ(119879119872) 119885 and119882 can bewritten as 119885 = sum3

119894=1119891119894119890119894and119882 = sum

3

119895=1119892119895119890119895 where 119891

119894and 119892

119894

are smooth functions on119872 By direct calculations we have

119878 (119885119882) = minus 1198902119911

(11989111198921+ 11989121198922) minus 2119891

31198923

= minus1198902119911

(11989111198921+ 11989121198922minus 11989131198923) minus 11989131198923(1198902119911

+ 2)

(62)

Since 120578(119885) = minus1198913and 120578(119882) = minus119892

3and 119892(119885119882) = 119891

11198921+

11989121198922minus 11989131198923 we have

119878 (119885119882) = minus1198902119911

119892 (119885119882) minus (1198902119911

+ 2) 120578 (119885) 120578 (119882) (63)

This tell us that119872 is an 120578-Einstein manifold

Acknowledgment

The authors would like to thank the reviewers for the ex-tremely carefully reading and formany important commentswhich improved the paper considerably

References

[1] A A Shaikh ldquoOn Lorentzian almost paracontact manifoldswith a structure of the concircular typerdquoKyungpookMathemat-ical Journal vol 43 no 2 pp 305ndash314 2003

[2] KMatsumoto ldquoOn Lorentzian paracontactmanifoldsrdquo Bulletinof Yamagata University vol 12 no 2 pp 151ndash156 1989

[3] A A Shaikh ldquoSome results on (119871119862119878)119899-manifoldsrdquo Journal of the

Korean Mathematical Society vol 46 no 3 pp 449ndash461 2009[4] AA Shaikh and S KHui ldquoOn generalized120588-recurrent (119871119862119878)

119899-

manifoldsrdquo in Proceedings of the ICMS International Conferenceon Mathematical Science vol 1309 of American Institute ofPhysics Conference Proceedings pp 419ndash429 2010

[5] A A Shaikh T Basu and S Eyasmin ldquoOn the existence of120601-recurrent (119871119862119878)

119899-manifoldsrdquo ExtractaMathematicae vol 23

no 1 pp 71ndash83 2008[6] A A Shaikh and T Q Binh ldquoOn weakly symmetric (119871119862119878)

119899-

manifoldsrdquo Journal of AdvancedMathematical Studies vol 2 no2 pp 103ndash118 2009

[7] G T Sreenivasa Venkatesha and C S Bagewadi ldquoSome resultson (119871119862119878)

2119899+1-manifoldsrdquo Bulletin of Mathematical Analysis and

Applications vol 1 no 3 pp 64ndash70 2009[8] S K Yadav P K Dwivedi and D Suthar ldquoOn (119871119862119878)

2119899+1-

manifolds satisfying certain conditions on the concircularcurvature tensorrdquoThai Journal of Mathematics vol 9 no 3 pp597ndash603 2011

[9] M Atceken ldquoOn geometry of submanifolds of (119871119862119878)119899-

manifoldsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2012 Article ID 304647 11 pages 2012

[10] S K Hui and M Atceken ldquoContact warped product semi-slant submanifolds of (119871119862119878)

119899-manifoldsrdquo Acta Universitatis

Sapientiae Mathematica vol 3 no 2 pp 212ndash224 2011[11] S S Shukla M K Shukla and R Prasad ldquoSlant submanifold

of (119871119862119878)119899-manifoldsrdquo to appear in Kyungpook Mathematical

Journal[12] K Yano and S Sawaki ldquoRiemannianmanifolds admitting a con-

formal transformation grouprdquo Journal of Differential Geometryvol 2 pp 161ndash184 1968

[13] A A Shaikh and S K Jana ldquoOn weakly quasi-conformallysymmetric manifoldsrdquo SUT Journal of Mathematics vol 43 no1 pp 61ndash83 2007

[14] R Kumar and B Prasad ldquoOn (119871119862119878)119899-manifoldsrdquo to appear in

Thai Journal of Mathematics[15] B-Y Chen and K Yano ldquoHypersurfaces of a conformally flat

spacerdquo Tensor vol 26 pp 318ndash322 1972[16] A A Shaikh and S K Jana ldquoOnweakly symmetric Riemannian

manifoldsrdquo Publicationes Mathematicae Debrecen vol 71 no 1-2 pp 27ndash41 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Let119872 be a Lorentzianmanifold admitting a unit timelikeconcircular vector field 120585 called the characteristic vector fieldof the manifold Then we have

119892 (120585 120585) = minus1 (4)

Since 120585 is a unit concircular unit vector field there exists anonzero 1-form 120578 such that

119892 (119883 120585) = 120578 (119883) (5)

The equation of the following form holds

(nabla119883120578) 119884 = 120572 119892 (119883 119884) + 120578 (119883) 120578 (119884) 120572 = 0 (6)

for all 119883119884 isin Γ(119879119872) where 120572 is a nonzero scalar functionsatisfying

nabla119883120572 = 119883 (120572) = 119889120572 (119883) = 120588120578 (119883) (7)

120588 being a certain scalar function given by 120588 = minus120585(120572)Let us put

nabla119883120585 = 120572120601119883 (8)

then from (6) and (8) we can derive

120601119883 = 119883 + 120578 (119883) 120585 (9)

which tell us that 120601 is a symmetric (1 1)-tensor Thus theLorentzian manifold 119872 together with the unit timelikeconcircular vector field 120585 its associated 1-form 120578 and (1 1)-type tensor field 120601 is said to be a Lorentzian concircularstructure manifold

A differentiable manifold 119872 of dimension 119899 is called(LCS)-manifold if it admits a (1 1)-type tensor field 120601 acovariant vector field 120578 and a Lorentzian metric 119892 whichsatisfy

120578 (120585) = 119892 (120585 120585) = minus1 (10)

1206012

119883 = 119883 + 120578 (119883) 120585 (11)

119892 (119883 120585) = 120578 (119883) (12)

120601120585 = 0 120578 ∘ 120601 = 0 (13)

for all119883 isin Γ(119879119872) Particularly if we take 120572 = 1 then we canobtain the 119871119875-Sasakian structure of Matsumoto [2]

Also in an (LCS)119899-manifold 119872 the following relations

are satisfied (see [3ndash6])

120578 (119877 (119883 119884)119885) = (1205722

minus 120588) [119892 (119884 119885) 120578 (119883) minus 119892 (119883 119885) 120578 (119884)]

(14)

119877 (120585 119883) 119884 = (1205722

minus 120588) [119892 (119883 119884) 120585 minus 120578 (119884)119883] (15)

119877 (119883 119884) 120585 = (1205722

minus 120588) [120578 (119884)119883 minus 120578 (119883)119884] (16)

(nabla119883120601)119884 = 120572 [119892 (119883 119884) 120585 + 2120578 (119883) 120578 (119884) 120585 + 120578 (119884)119883] (17)

119878 (119883 120585) = (119899 minus 1) (1205722

minus 120588) 120578 (119883) (18)

119878 (120601119883 120601119884) = 119878 (119883 119884) + (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (119884) (19)

for all vector fields 119883119884 119885 on119872 where 119877 and 119878 denote theRiemannian curvature tensor and Ricci curvature respec-tively 119876 is also the Ricci operator given by 119878(119883 119884) =

119892(119876119883 119884)Now let (119872 119892) be an 119899-dimensional Riemannian man-

ifold then the concircular curvature tensor the Weylconformal curvature tensor 119862 and the pseudo projectivecurvature tensor are respectively defined by

(119883 119884)119885 = 119877 (119883 119884)119885

minus

120591

119899 (119899 minus 1)

[119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(20)

119862 (119883 119884)119885 = 119877 (119883 119884)119885 minus

1

119899 minus 2

times [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884]

+

120591

(119899 minus 1) (119899 minus 2)

[119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(21)

(119883 119884)119885 = 119886119877 (119883 119884)119885

+ 119887 [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884]

minus

120591

119899

[

119886

119899 minus 1

+ 119887] [119892 (119884 119885)119883 minus 119892 (119883119885) 119884]

(22)

where 119886 and 119887 are constants such that 119886 119887 = 0 and 120591 is alsothe scalar curvature of119872 [7]

For an 119899-dimensional (LCS)119899-manifold the quasi-

conformal curvature tensor C is given by

C (119883 119884)119885 = 119886119877 (119883 119884)119885

+ 119887 [119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883119885)119876119884]

minus

120591

119899

[

119886

119899 minus 1

+ 2119887] [119892 (119884 119885)119883 minus 119892 (119883 119885) 119884]

(23)

for all119883119884 119885 isin Γ(119879119872) [14]The notion of quasi-conformal curvature tensor was

defined by Yano and Swaki [12] If 119886 = 1 and 119887 = minus1(119899 minus 1)then quasi-conformal curvature tensor reduces to conformalcurvature tensor

3 Quasi-Conformally Flat (LCS)119899-Manifolds

and Some of Their Properties

For an 119899-dimensional quasi-conformally flat (LCS)119899-mani-

fold we know for 119885 = 120585 from (23)

Abstract and Applied Analysis 3

119886119877 (119883 119884) 120585 + 119887 [119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

minus

120591

119899

[

119886

119899 minus 1

+ 2119887] [119892 (119884 120585)119883 minus 119892 (119883 120585) 119884] = 0

(24)

Here taking into account of (16) we have

[120578 (119884)119883 minus 120578 (119883)119884] [119886 (1205722

minus 120588) + 119887 (119899 minus 1) (1205722

minus 120588)

minus

120591

119899

(

119886

119899 minus 1

+ 2119887)]

+ 119887 [120578 (119884]119876119883 minus 120578 (119883)119876119884] = 0

(25)

Let 119884 = 120585 be in (25) then also by using (18) we obtain

[minus119883 minus 120578 (119883) 120585] [119886 (1205722

minus 120588) minus

120591

119899

(

119886

119899 minus 1

+ 2119887)

+ 119887 (119899 minus 1) (1205722

minus 120588) ]

+ 119887 [minus119876119883 minus 120578 (119883) (119899 minus 1) (1205722

minus 120588) 120585] = 0

(26)

Taking the inner product on both sides of the last equation by119884 we obtain

[119892 (119883 119884) + 120578 (119883) 120578 (119884)] [119886 (1205722

minus 120588) + 119887 (119899 minus 1)

times (1205722

minus 120588) minus

120591

119899

(

119886

119899 minus 1

+ 2119887)]

+ 119887 [119878 (119883 119884) + 120578 (119883) 120578 (119884) (1205722

minus 120588) (119899 minus 1)] = 0

(27)

that is

119878 (119883 119884) = 119892 (119883 119884)

times [

120591

119899119887

(

119886

119899 minus 1

+ 2119887) minus (1205722

minus 120588) (

119886

119887

+ (119899 minus 1))]

+ 120578 (119883) 120578 (119884) [

120591

119899119887

(

119886

119899 minus 1

+ 2119887)

minus (1205722

minus 120588) (

119886

119887

+ 2 (119899 minus 1))]

(28)

Now we are in a proposition to state the following

Theorem 2 If an 119899-dimensional (LCS)119899-manifold119872 is quasi-

conformally flat then119872 is an 120578-Einstein manifold

Now let 1198901 1198902 119890

119899minus1 120585 be an orthonormal basis of the

tangent space at any point of the manifold Then putting119883 =

119884 = 119890119894 120585 in (28) and taking summation for 1 le 119894 le 119899 minus 1 we

have

120591 = 119899 (119899 minus 1) (1205722

minus 120588) if 119886 + (119899 minus 2) 119887 = 0 (29)

In view of (28) and (29) we obtain

119878 (119883 119884) = (119899 minus 1) (1205722

minus 120588) 119892 (119883 119884) (30)

which is equivalent to

119876119883 = (119899 minus 1) (1205722

minus 120588)119883 (31)

for any119883 isin Γ(119879119872)By using (29) and (31) in (23) for a quasi-conformally flat

(LCS)119899-manifold119872 we get

119877 (119883 119884)119885 = (1205722

minus 120588) 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884 (32)

for all 119883119884 119885 isin Γ(119879119872) If we consider Schurrsquos Theorem wecan give the following the theorem

Theorem3 Aquasi-conformally flat (LCS)119899-manifoldM (119899 gt

1) is a manifold of constant curvature (1205722 minus 120588) provided that119886 + 119887(119899 minus 2) = 0

Now let us consider an (LCS)119899-manifold 119872 which is

conformally flat Thus we have from (21) that

119877 (119883 119884)119885 =

1

119899 minus 2

119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884

minus

120591

(119899 minus 1) (119899 minus 2)

119892 (119884 119885)119883 minus 119892 (119883119885) 119884

(33)

for all vector fields119883119884 119885 tangent to119872 Setting119885 = 120585 in (33)and using (16) (18) we have

[

120591

119899 minus 1

minus (1205722

minus 120588)] [120578 (119884)119883 minus 120578 (119883)119884]

= [120578 (119884)119876119883 minus 120578 (119883)119876119884]

(34)

If we put 119884 = 120585 in (34) and also using (18) we obtain

119876119883 = [

120591

119899 minus 1

minus (1205722

minus 120588)]119883 + [

120591

119899 minus 1

minus 119899 (1205722

minus 120588)] 120578 (119883) 120585

(35)

Corollary 4 A conformally flat (LCS)119899-manifold is an 120578-

Einstein manifold

Generalizing the notion of a manifold of constant curva-ture Chen and Yano [15] introduced the notion of a manifoldof quasi-constant curvature which can be defined as follows

Definition 5 ARiemannianmanifold is said to be a manifoldof quasi-constant curvature if it is conformally flat and itscurvature tensor of type (0 4) is of the form

(119883 119884 119885119882)

= 119886 119892 (119884 119885) 119892 (119883119882) minus 119892 (119883 119885) 119892 (119884119882)

+ 119887 119892 (119884 119885)119860 (119883)119860 (119882) minus 119892 (119883 119885)119860 (119884)119860 (119882)

+119892 (119883119882)119860 (119884)119860 (119885) minus 119892 (119884119882)119860 (119883)119860 (119885)

(36)

4 Abstract and Applied Analysis

for all 119883119884 119885119882 isin Γ(119879119872) where 119886 119887 are scalars of which119887 = 0 and 119860 is a nonzero 1-form (for more details we refer to[13 16])

Thus we have the following theorem for (LCS)119899-conform-

ally flat manifolds

Theorem 6 A conformally flat (LCS)119899-manifold is a manifold

of quasi-constant curvature

Proof From (33) and (35) we obtain

(119883 119884 119885119882)

= (

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 119892 (119884 119885) minus 119892 (119884119882) 119892 (119883 119885)

+ (

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 120578 (119884) 120578 (119885) minus 119892 (119884119882) 120578 (119883) 120578 (119885)

+119892 (119884 119885) 120578 (119883) 120578 (119882) minus 119892 (119883 119885) 120578 (119884) 120578 (119882)

(37)

This implies (36) for

119886 =

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119887 =

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119860 = 120578

(38)

This proves our assertion

Next differentiating the (19) covariantly with respect to119882 we get

nabla119882119878 (120601119883 120601119884) = nabla

119882119878 (119883 119884) + (119899 minus 1)119882(120572

2

minus 120588)

+ (119899 minus 1) (1205722

minus 120588)119882 [120578 (119883) 120578 (119884)]

(39)

for any119883119884 isin Γ(119879119872) Making use of the definition ofnabla119878 and(8) we have

(nabla119882119878) (120601119883 120601119884) + 119878 (nabla

119882120601119883 120601119884) + 119878 (120601119883 nabla

119882120601119884)

= (nabla119882119878) (119883 119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(40)

Thus we have

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 ((nabla119882120601)119883 + 120601nabla

119882119883 120601119884)

minus 119878 (120601119883 (nabla119882120601)119884 + 120601nabla

119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(41)

Here taking account of (17) we arrive at

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 (120572 119892 (119883119882) 120585 + 2120578 (119883) 120578 (W) 120585 + 120578 (119883)119882 120601119884)

minus 119878 (120601119883 120572 119892 (119884119882) 120585 + 2120578 (119884) 120578 (119882) 120585 + 120578 (119884)119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1) (120572

2

minus 120588) 120578 (119884)

times 120578 (nabla119882119883) + 120572119892 (119883 120601119882) + (119899 minus 1) (120572

2

minus 120588) 120578 (119883)

times 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

= minus120572 119892 (119883119882) 119878 (120585 120601119884) + 2120578 (119883) 120578 (119882) 119878 (120585 120601119884)

+120578 (119883) 119878 (119882 120601119884)

minus 120572 119892 (119884119882) 119878 (120601119883 120585) + 2120578 (119884) 120578 (119882) 119878 (120601119883 120585)

+120578 (119884) 119878 (120601119883119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(42)

Again by using (13) (18) and (19) we reach

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119883)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883)

Abstract and Applied Analysis 5

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588)

times 120578 (nabla119882119883) 120578 (119884) + 120572120578 (119884) 119892 (119883 120601119882)

+120578 (nabla119882119884) 120578 (119883) + 120572120578 (119883) 119892 (119884 120601119882)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

+ 120572 (119899 minus 1) (1205722

minus 120588)

times 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

(43)

Thus we have the following theorem

Theorem7 If an (LCS)119899-manifold119872 is Ricci-symmetric then

1205722

minus 120588 is constant

Proof If 119899 gt 1-dimensional (LCS)119899-manifold 119872 is Ricci-

symmetric then from (43) we conclude that

120572 (119899 minus 1) (1205722

minus 120588) 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

minus 120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882) = 0

(44)

It follows that

120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882) 120585 minus 120578 (119883) 120601119882

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120585

minus 120572120578 (119883) 120601119876119882 minus 120572119878 (120601119883119882) 120585 = 0

(45)

from which

minus 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882)

minus (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) + 119878 (120601119883119882) = 0

(46)

which is equivalent to

minus 120572 (119899 minus 1) (1205722

minus 120588) 120601119882 minus (119899 minus 1)119882(1205722

minus 120588) 120585

+ 120572120601119876119882 = 0

(47)

that is

119882(1205722

minus 120588) = 0 (48)

which proves our assertion

Since nabla119877 = 0 implies that nabla119878 = 0 we can give the follow-ing corollary

Corollary 8 If an 119899-dimensional (119871119862119878)119899-manifold 119872 is

locally symmetric then 1205722 minus 120588 is constant

Now taking the covariant derivation of the both sides of(18) with respect to 119884 we have

119884119878 (119883 120585) = (119899 minus 1)119882 [(1205722

minus 120588) 120578 (119883)] (49)

From the definition of the covariant derivation of Ricci-tensor we have

(nabla119884119878) (119883 120585) = nabla

119884119878 (119883 120585) minus 119878 (nabla

119884119883 120585) minus 119878 (119883 nabla

119884120585)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883) + (1205722

minus 120588)

times [120578 (nabla119884119883) + 120572119892 (119883 120601119884)]

minus (119899 minus 1) (1205722

minus 120588) 120578 (nabla119884119883) minus 120572119878 (119883 120601119884)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883)

+ 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119884) minus 120572119878 (119883 120601119884)

(50)

If an (119871119862119878)119899-manifold 119872 Ricci symmetric then Theorem 7

and (43) imply that

119878 (119883 120601119884) = (119899 minus 1) (1205722

minus 120588) 119892 (120601119884119883) (51)

This leads us to state the following

Theorem9 If an (LCS)119899-manifold119872 is Ricci symmetric then

it is an Einstein manifold

Corollary 10 If an (LCS)119899-manifold119872 is locally symmetric

then it is an Einstein manifold

In this section an example is used to demonstrate that themethod presented in this paper is effective But this exampleis a special case of Example 61 of [6]

Example 11 Now we consider the 3-dimensional manifold

119872 = (119909 119910 119911) isin R3

119911 = 0 (52)

where (119909 119910 119911) denote the standard coordinates in R3 Thevector fields

1198901= 119890119911

(119909

120597

120597119909

+ 119910

120597

120597119910

) 1198902= 119890119911120597

120597119910

1198903=

120597

120597119911

(53)

are linearly independent of each point of 119872 Let 119892 be theLorentzian metric tensor defined by

119892 (1198901 1198901) = 119892 (119890

2 1198902) = minus119892 (119890

3 1198903) = 1

119892 (119890119894 119890119895) = 0 119894 = 119895

(54)

6 Abstract and Applied Analysis

for 119894 119895 = 1 2 3 Let 120578 be the 1-formdefined by 120578(119885) = 119892(119885 1198903)

for any 119885 isin Γ(119879119872) Let 120601 be the (11)-tensor field defined by

1206011198901= 1198901 120601119890

2= 1198902 120601119890

3= 0 (55)

Then using the linearity of 120601 and 119892 we have 120578(1198903) = minus1

1206012

119885 = 119885 + 120578 (119885) 1198903

119892 (120601119885 120601119882) = 119892 (119885119882) + 120578 (119885) 120578 (119882)

(56)

for all 119885119882 isin Γ(119879119872) Thus for 120585 = 1198903 (120601 120585 120578 119892) defines a

Lorentzian paracontact structure on119872Now let nabla be the Levi-Civita connection with respect

to the Lorentzian metric 119892 and let 119877 be the Riemanniancurvature tensor of 119892 Then we have

[1198901 1198902] = minus119890

119911

1198902 [119890

1 1198903] = minus119890

1 [119890

2 1198903] = minus119890

2

(57)

Making use of the Koszul formulae for the Lorentzian metrictensor 119892 we can easily calculate the covariant derivations asfollows

nabla1198901

1198901= minus1198903 nabla

1198902

1198901= 119890119911

1198902 nabla

1198901

1198903= minus1198901

nabla1198902

1198903= minus1198902 nabla

1198902

1198902= minus119890119911

1198901minus 1198903

nabla1198901

1198902= nabla1198903

1198901= nabla1198903

1198902= nabla1198903

1198903= 0

(58)

From the previously mentioned it can be easily seen that(120601 120585 120578 119892) is an (LCS)

3-structure on 119872 that is 119872 is an

(LCS)3-manifold with 120572 = minus1 and 120588 = 0 Using the

previous relations we can easily calculate the components ofthe Riemannian curvature tensor as follows

119877 (1198901 1198902) 1198901= (1198902119911

minus 1) 1198902 119877 (119890

1 1198902) 1198902= (1 minus 119890

2119911

) 1198901

119877 (1198901 1198903) 1198901= minus1198903 119877 (119890

1 1198903) 1198903= minus1198901

119877 (1198902 1198903) 1198902= minus1198903 119877 (119890

2 1198903) 1198903= minus1198902

119877 (1198901 1198902) 1198903= 119877 (119890

1 1198903) 1198902= 119877 (119890

2 1198903) 1198901= 0

(59)

By using the properties of119877 and definition of the Ricci tensorwe obtain

119878 (1198901 1198901) = 119878 (119890

2 1198902) = minus119890

2119911

119878 (1198903 1198903) = minus2

119878 (1198901 1198902) = 119878 (119890

1 1198903) = 119878 (119890

2 1198903) = 0

(60)

Thus the scalar curvature 120591 of119872 is given by

120591 =

3

sum

119894=1

119892 (119890119894 119890119894) 119878 (119890119894 119890119894) = 2 (1 minus 119890

2119911

) (61)

On the other hand for any 119885119882 isin Γ(119879119872) 119885 and119882 can bewritten as 119885 = sum3

119894=1119891119894119890119894and119882 = sum

3

119895=1119892119895119890119895 where 119891

119894and 119892

119894

are smooth functions on119872 By direct calculations we have

119878 (119885119882) = minus 1198902119911

(11989111198921+ 11989121198922) minus 2119891

31198923

= minus1198902119911

(11989111198921+ 11989121198922minus 11989131198923) minus 11989131198923(1198902119911

+ 2)

(62)

Since 120578(119885) = minus1198913and 120578(119882) = minus119892

3and 119892(119885119882) = 119891

11198921+

11989121198922minus 11989131198923 we have

119878 (119885119882) = minus1198902119911

119892 (119885119882) minus (1198902119911

+ 2) 120578 (119885) 120578 (119882) (63)

This tell us that119872 is an 120578-Einstein manifold

Acknowledgment

The authors would like to thank the reviewers for the ex-tremely carefully reading and formany important commentswhich improved the paper considerably

References

[1] A A Shaikh ldquoOn Lorentzian almost paracontact manifoldswith a structure of the concircular typerdquoKyungpookMathemat-ical Journal vol 43 no 2 pp 305ndash314 2003

[2] KMatsumoto ldquoOn Lorentzian paracontactmanifoldsrdquo Bulletinof Yamagata University vol 12 no 2 pp 151ndash156 1989

[3] A A Shaikh ldquoSome results on (119871119862119878)119899-manifoldsrdquo Journal of the

Korean Mathematical Society vol 46 no 3 pp 449ndash461 2009[4] AA Shaikh and S KHui ldquoOn generalized120588-recurrent (119871119862119878)

119899-

manifoldsrdquo in Proceedings of the ICMS International Conferenceon Mathematical Science vol 1309 of American Institute ofPhysics Conference Proceedings pp 419ndash429 2010

[5] A A Shaikh T Basu and S Eyasmin ldquoOn the existence of120601-recurrent (119871119862119878)

119899-manifoldsrdquo ExtractaMathematicae vol 23

no 1 pp 71ndash83 2008[6] A A Shaikh and T Q Binh ldquoOn weakly symmetric (119871119862119878)

119899-

manifoldsrdquo Journal of AdvancedMathematical Studies vol 2 no2 pp 103ndash118 2009

[7] G T Sreenivasa Venkatesha and C S Bagewadi ldquoSome resultson (119871119862119878)

2119899+1-manifoldsrdquo Bulletin of Mathematical Analysis and

Applications vol 1 no 3 pp 64ndash70 2009[8] S K Yadav P K Dwivedi and D Suthar ldquoOn (119871119862119878)

2119899+1-

manifolds satisfying certain conditions on the concircularcurvature tensorrdquoThai Journal of Mathematics vol 9 no 3 pp597ndash603 2011

[9] M Atceken ldquoOn geometry of submanifolds of (119871119862119878)119899-

manifoldsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2012 Article ID 304647 11 pages 2012

[10] S K Hui and M Atceken ldquoContact warped product semi-slant submanifolds of (119871119862119878)

119899-manifoldsrdquo Acta Universitatis

Sapientiae Mathematica vol 3 no 2 pp 212ndash224 2011[11] S S Shukla M K Shukla and R Prasad ldquoSlant submanifold

of (119871119862119878)119899-manifoldsrdquo to appear in Kyungpook Mathematical

Journal[12] K Yano and S Sawaki ldquoRiemannianmanifolds admitting a con-

formal transformation grouprdquo Journal of Differential Geometryvol 2 pp 161ndash184 1968

[13] A A Shaikh and S K Jana ldquoOn weakly quasi-conformallysymmetric manifoldsrdquo SUT Journal of Mathematics vol 43 no1 pp 61ndash83 2007

[14] R Kumar and B Prasad ldquoOn (119871119862119878)119899-manifoldsrdquo to appear in

Thai Journal of Mathematics[15] B-Y Chen and K Yano ldquoHypersurfaces of a conformally flat

spacerdquo Tensor vol 26 pp 318ndash322 1972[16] A A Shaikh and S K Jana ldquoOnweakly symmetric Riemannian

manifoldsrdquo Publicationes Mathematicae Debrecen vol 71 no 1-2 pp 27ndash41 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

119886119877 (119883 119884) 120585 + 119887 [119878 (119884 120585)119883 minus 119878 (119883 120585) 119884

+119892 (119884 120585) 119876119883 minus 119892 (119883 120585) 119876119884]

minus

120591

119899

[

119886

119899 minus 1

+ 2119887] [119892 (119884 120585)119883 minus 119892 (119883 120585) 119884] = 0

(24)

Here taking into account of (16) we have

[120578 (119884)119883 minus 120578 (119883)119884] [119886 (1205722

minus 120588) + 119887 (119899 minus 1) (1205722

minus 120588)

minus

120591

119899

(

119886

119899 minus 1

+ 2119887)]

+ 119887 [120578 (119884]119876119883 minus 120578 (119883)119876119884] = 0

(25)

Let 119884 = 120585 be in (25) then also by using (18) we obtain

[minus119883 minus 120578 (119883) 120585] [119886 (1205722

minus 120588) minus

120591

119899

(

119886

119899 minus 1

+ 2119887)

+ 119887 (119899 minus 1) (1205722

minus 120588) ]

+ 119887 [minus119876119883 minus 120578 (119883) (119899 minus 1) (1205722

minus 120588) 120585] = 0

(26)

Taking the inner product on both sides of the last equation by119884 we obtain

[119892 (119883 119884) + 120578 (119883) 120578 (119884)] [119886 (1205722

minus 120588) + 119887 (119899 minus 1)

times (1205722

minus 120588) minus

120591

119899

(

119886

119899 minus 1

+ 2119887)]

+ 119887 [119878 (119883 119884) + 120578 (119883) 120578 (119884) (1205722

minus 120588) (119899 minus 1)] = 0

(27)

that is

119878 (119883 119884) = 119892 (119883 119884)

times [

120591

119899119887

(

119886

119899 minus 1

+ 2119887) minus (1205722

minus 120588) (

119886

119887

+ (119899 minus 1))]

+ 120578 (119883) 120578 (119884) [

120591

119899119887

(

119886

119899 minus 1

+ 2119887)

minus (1205722

minus 120588) (

119886

119887

+ 2 (119899 minus 1))]

(28)

Now we are in a proposition to state the following

Theorem 2 If an 119899-dimensional (LCS)119899-manifold119872 is quasi-

conformally flat then119872 is an 120578-Einstein manifold

Now let 1198901 1198902 119890

119899minus1 120585 be an orthonormal basis of the

tangent space at any point of the manifold Then putting119883 =

119884 = 119890119894 120585 in (28) and taking summation for 1 le 119894 le 119899 minus 1 we

have

120591 = 119899 (119899 minus 1) (1205722

minus 120588) if 119886 + (119899 minus 2) 119887 = 0 (29)

In view of (28) and (29) we obtain

119878 (119883 119884) = (119899 minus 1) (1205722

minus 120588) 119892 (119883 119884) (30)

which is equivalent to

119876119883 = (119899 minus 1) (1205722

minus 120588)119883 (31)

for any119883 isin Γ(119879119872)By using (29) and (31) in (23) for a quasi-conformally flat

(LCS)119899-manifold119872 we get

119877 (119883 119884)119885 = (1205722

minus 120588) 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884 (32)

for all 119883119884 119885 isin Γ(119879119872) If we consider Schurrsquos Theorem wecan give the following the theorem

Theorem3 Aquasi-conformally flat (LCS)119899-manifoldM (119899 gt

1) is a manifold of constant curvature (1205722 minus 120588) provided that119886 + 119887(119899 minus 2) = 0

Now let us consider an (LCS)119899-manifold 119872 which is

conformally flat Thus we have from (21) that

119877 (119883 119884)119885 =

1

119899 minus 2

119878 (119884 119885)119883 minus 119878 (119883 119885) 119884

+119892 (119884 119885)119876119883 minus 119892 (119883 119885)119876119884

minus

120591

(119899 minus 1) (119899 minus 2)

119892 (119884 119885)119883 minus 119892 (119883119885) 119884

(33)

for all vector fields119883119884 119885 tangent to119872 Setting119885 = 120585 in (33)and using (16) (18) we have

[

120591

119899 minus 1

minus (1205722

minus 120588)] [120578 (119884)119883 minus 120578 (119883)119884]

= [120578 (119884)119876119883 minus 120578 (119883)119876119884]

(34)

If we put 119884 = 120585 in (34) and also using (18) we obtain

119876119883 = [

120591

119899 minus 1

minus (1205722

minus 120588)]119883 + [

120591

119899 minus 1

minus 119899 (1205722

minus 120588)] 120578 (119883) 120585

(35)

Corollary 4 A conformally flat (LCS)119899-manifold is an 120578-

Einstein manifold

Generalizing the notion of a manifold of constant curva-ture Chen and Yano [15] introduced the notion of a manifoldof quasi-constant curvature which can be defined as follows

Definition 5 ARiemannianmanifold is said to be a manifoldof quasi-constant curvature if it is conformally flat and itscurvature tensor of type (0 4) is of the form

(119883 119884 119885119882)

= 119886 119892 (119884 119885) 119892 (119883119882) minus 119892 (119883 119885) 119892 (119884119882)

+ 119887 119892 (119884 119885)119860 (119883)119860 (119882) minus 119892 (119883 119885)119860 (119884)119860 (119882)

+119892 (119883119882)119860 (119884)119860 (119885) minus 119892 (119884119882)119860 (119883)119860 (119885)

(36)

4 Abstract and Applied Analysis

for all 119883119884 119885119882 isin Γ(119879119872) where 119886 119887 are scalars of which119887 = 0 and 119860 is a nonzero 1-form (for more details we refer to[13 16])

Thus we have the following theorem for (LCS)119899-conform-

ally flat manifolds

Theorem 6 A conformally flat (LCS)119899-manifold is a manifold

of quasi-constant curvature

Proof From (33) and (35) we obtain

(119883 119884 119885119882)

= (

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 119892 (119884 119885) minus 119892 (119884119882) 119892 (119883 119885)

+ (

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 120578 (119884) 120578 (119885) minus 119892 (119884119882) 120578 (119883) 120578 (119885)

+119892 (119884 119885) 120578 (119883) 120578 (119882) minus 119892 (119883 119885) 120578 (119884) 120578 (119882)

(37)

This implies (36) for

119886 =

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119887 =

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119860 = 120578

(38)

This proves our assertion

Next differentiating the (19) covariantly with respect to119882 we get

nabla119882119878 (120601119883 120601119884) = nabla

119882119878 (119883 119884) + (119899 minus 1)119882(120572

2

minus 120588)

+ (119899 minus 1) (1205722

minus 120588)119882 [120578 (119883) 120578 (119884)]

(39)

for any119883119884 isin Γ(119879119872) Making use of the definition ofnabla119878 and(8) we have

(nabla119882119878) (120601119883 120601119884) + 119878 (nabla

119882120601119883 120601119884) + 119878 (120601119883 nabla

119882120601119884)

= (nabla119882119878) (119883 119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(40)

Thus we have

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 ((nabla119882120601)119883 + 120601nabla

119882119883 120601119884)

minus 119878 (120601119883 (nabla119882120601)119884 + 120601nabla

119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(41)

Here taking account of (17) we arrive at

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 (120572 119892 (119883119882) 120585 + 2120578 (119883) 120578 (W) 120585 + 120578 (119883)119882 120601119884)

minus 119878 (120601119883 120572 119892 (119884119882) 120585 + 2120578 (119884) 120578 (119882) 120585 + 120578 (119884)119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1) (120572

2

minus 120588) 120578 (119884)

times 120578 (nabla119882119883) + 120572119892 (119883 120601119882) + (119899 minus 1) (120572

2

minus 120588) 120578 (119883)

times 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

= minus120572 119892 (119883119882) 119878 (120585 120601119884) + 2120578 (119883) 120578 (119882) 119878 (120585 120601119884)

+120578 (119883) 119878 (119882 120601119884)

minus 120572 119892 (119884119882) 119878 (120601119883 120585) + 2120578 (119884) 120578 (119882) 119878 (120601119883 120585)

+120578 (119884) 119878 (120601119883119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(42)

Again by using (13) (18) and (19) we reach

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119883)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883)

Abstract and Applied Analysis 5

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588)

times 120578 (nabla119882119883) 120578 (119884) + 120572120578 (119884) 119892 (119883 120601119882)

+120578 (nabla119882119884) 120578 (119883) + 120572120578 (119883) 119892 (119884 120601119882)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

+ 120572 (119899 minus 1) (1205722

minus 120588)

times 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

(43)

Thus we have the following theorem

Theorem7 If an (LCS)119899-manifold119872 is Ricci-symmetric then

1205722

minus 120588 is constant

Proof If 119899 gt 1-dimensional (LCS)119899-manifold 119872 is Ricci-

symmetric then from (43) we conclude that

120572 (119899 minus 1) (1205722

minus 120588) 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

minus 120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882) = 0

(44)

It follows that

120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882) 120585 minus 120578 (119883) 120601119882

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120585

minus 120572120578 (119883) 120601119876119882 minus 120572119878 (120601119883119882) 120585 = 0

(45)

from which

minus 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882)

minus (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) + 119878 (120601119883119882) = 0

(46)

which is equivalent to

minus 120572 (119899 minus 1) (1205722

minus 120588) 120601119882 minus (119899 minus 1)119882(1205722

minus 120588) 120585

+ 120572120601119876119882 = 0

(47)

that is

119882(1205722

minus 120588) = 0 (48)

which proves our assertion

Since nabla119877 = 0 implies that nabla119878 = 0 we can give the follow-ing corollary

Corollary 8 If an 119899-dimensional (119871119862119878)119899-manifold 119872 is

locally symmetric then 1205722 minus 120588 is constant

Now taking the covariant derivation of the both sides of(18) with respect to 119884 we have

119884119878 (119883 120585) = (119899 minus 1)119882 [(1205722

minus 120588) 120578 (119883)] (49)

From the definition of the covariant derivation of Ricci-tensor we have

(nabla119884119878) (119883 120585) = nabla

119884119878 (119883 120585) minus 119878 (nabla

119884119883 120585) minus 119878 (119883 nabla

119884120585)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883) + (1205722

minus 120588)

times [120578 (nabla119884119883) + 120572119892 (119883 120601119884)]

minus (119899 minus 1) (1205722

minus 120588) 120578 (nabla119884119883) minus 120572119878 (119883 120601119884)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883)

+ 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119884) minus 120572119878 (119883 120601119884)

(50)

If an (119871119862119878)119899-manifold 119872 Ricci symmetric then Theorem 7

and (43) imply that

119878 (119883 120601119884) = (119899 minus 1) (1205722

minus 120588) 119892 (120601119884119883) (51)

This leads us to state the following

Theorem9 If an (LCS)119899-manifold119872 is Ricci symmetric then

it is an Einstein manifold

Corollary 10 If an (LCS)119899-manifold119872 is locally symmetric

then it is an Einstein manifold

In this section an example is used to demonstrate that themethod presented in this paper is effective But this exampleis a special case of Example 61 of [6]

Example 11 Now we consider the 3-dimensional manifold

119872 = (119909 119910 119911) isin R3

119911 = 0 (52)

where (119909 119910 119911) denote the standard coordinates in R3 Thevector fields

1198901= 119890119911

(119909

120597

120597119909

+ 119910

120597

120597119910

) 1198902= 119890119911120597

120597119910

1198903=

120597

120597119911

(53)

are linearly independent of each point of 119872 Let 119892 be theLorentzian metric tensor defined by

119892 (1198901 1198901) = 119892 (119890

2 1198902) = minus119892 (119890

3 1198903) = 1

119892 (119890119894 119890119895) = 0 119894 = 119895

(54)

6 Abstract and Applied Analysis

for 119894 119895 = 1 2 3 Let 120578 be the 1-formdefined by 120578(119885) = 119892(119885 1198903)

for any 119885 isin Γ(119879119872) Let 120601 be the (11)-tensor field defined by

1206011198901= 1198901 120601119890

2= 1198902 120601119890

3= 0 (55)

Then using the linearity of 120601 and 119892 we have 120578(1198903) = minus1

1206012

119885 = 119885 + 120578 (119885) 1198903

119892 (120601119885 120601119882) = 119892 (119885119882) + 120578 (119885) 120578 (119882)

(56)

for all 119885119882 isin Γ(119879119872) Thus for 120585 = 1198903 (120601 120585 120578 119892) defines a

Lorentzian paracontact structure on119872Now let nabla be the Levi-Civita connection with respect

to the Lorentzian metric 119892 and let 119877 be the Riemanniancurvature tensor of 119892 Then we have

[1198901 1198902] = minus119890

119911

1198902 [119890

1 1198903] = minus119890

1 [119890

2 1198903] = minus119890

2

(57)

Making use of the Koszul formulae for the Lorentzian metrictensor 119892 we can easily calculate the covariant derivations asfollows

nabla1198901

1198901= minus1198903 nabla

1198902

1198901= 119890119911

1198902 nabla

1198901

1198903= minus1198901

nabla1198902

1198903= minus1198902 nabla

1198902

1198902= minus119890119911

1198901minus 1198903

nabla1198901

1198902= nabla1198903

1198901= nabla1198903

1198902= nabla1198903

1198903= 0

(58)

From the previously mentioned it can be easily seen that(120601 120585 120578 119892) is an (LCS)

3-structure on 119872 that is 119872 is an

(LCS)3-manifold with 120572 = minus1 and 120588 = 0 Using the

previous relations we can easily calculate the components ofthe Riemannian curvature tensor as follows

119877 (1198901 1198902) 1198901= (1198902119911

minus 1) 1198902 119877 (119890

1 1198902) 1198902= (1 minus 119890

2119911

) 1198901

119877 (1198901 1198903) 1198901= minus1198903 119877 (119890

1 1198903) 1198903= minus1198901

119877 (1198902 1198903) 1198902= minus1198903 119877 (119890

2 1198903) 1198903= minus1198902

119877 (1198901 1198902) 1198903= 119877 (119890

1 1198903) 1198902= 119877 (119890

2 1198903) 1198901= 0

(59)

By using the properties of119877 and definition of the Ricci tensorwe obtain

119878 (1198901 1198901) = 119878 (119890

2 1198902) = minus119890

2119911

119878 (1198903 1198903) = minus2

119878 (1198901 1198902) = 119878 (119890

1 1198903) = 119878 (119890

2 1198903) = 0

(60)

Thus the scalar curvature 120591 of119872 is given by

120591 =

3

sum

119894=1

119892 (119890119894 119890119894) 119878 (119890119894 119890119894) = 2 (1 minus 119890

2119911

) (61)

On the other hand for any 119885119882 isin Γ(119879119872) 119885 and119882 can bewritten as 119885 = sum3

119894=1119891119894119890119894and119882 = sum

3

119895=1119892119895119890119895 where 119891

119894and 119892

119894

are smooth functions on119872 By direct calculations we have

119878 (119885119882) = minus 1198902119911

(11989111198921+ 11989121198922) minus 2119891

31198923

= minus1198902119911

(11989111198921+ 11989121198922minus 11989131198923) minus 11989131198923(1198902119911

+ 2)

(62)

Since 120578(119885) = minus1198913and 120578(119882) = minus119892

3and 119892(119885119882) = 119891

11198921+

11989121198922minus 11989131198923 we have

119878 (119885119882) = minus1198902119911

119892 (119885119882) minus (1198902119911

+ 2) 120578 (119885) 120578 (119882) (63)

This tell us that119872 is an 120578-Einstein manifold

Acknowledgment

The authors would like to thank the reviewers for the ex-tremely carefully reading and formany important commentswhich improved the paper considerably

References

[1] A A Shaikh ldquoOn Lorentzian almost paracontact manifoldswith a structure of the concircular typerdquoKyungpookMathemat-ical Journal vol 43 no 2 pp 305ndash314 2003

[2] KMatsumoto ldquoOn Lorentzian paracontactmanifoldsrdquo Bulletinof Yamagata University vol 12 no 2 pp 151ndash156 1989

[3] A A Shaikh ldquoSome results on (119871119862119878)119899-manifoldsrdquo Journal of the

Korean Mathematical Society vol 46 no 3 pp 449ndash461 2009[4] AA Shaikh and S KHui ldquoOn generalized120588-recurrent (119871119862119878)

119899-

manifoldsrdquo in Proceedings of the ICMS International Conferenceon Mathematical Science vol 1309 of American Institute ofPhysics Conference Proceedings pp 419ndash429 2010

[5] A A Shaikh T Basu and S Eyasmin ldquoOn the existence of120601-recurrent (119871119862119878)

119899-manifoldsrdquo ExtractaMathematicae vol 23

no 1 pp 71ndash83 2008[6] A A Shaikh and T Q Binh ldquoOn weakly symmetric (119871119862119878)

119899-

manifoldsrdquo Journal of AdvancedMathematical Studies vol 2 no2 pp 103ndash118 2009

[7] G T Sreenivasa Venkatesha and C S Bagewadi ldquoSome resultson (119871119862119878)

2119899+1-manifoldsrdquo Bulletin of Mathematical Analysis and

Applications vol 1 no 3 pp 64ndash70 2009[8] S K Yadav P K Dwivedi and D Suthar ldquoOn (119871119862119878)

2119899+1-

manifolds satisfying certain conditions on the concircularcurvature tensorrdquoThai Journal of Mathematics vol 9 no 3 pp597ndash603 2011

[9] M Atceken ldquoOn geometry of submanifolds of (119871119862119878)119899-

manifoldsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2012 Article ID 304647 11 pages 2012

[10] S K Hui and M Atceken ldquoContact warped product semi-slant submanifolds of (119871119862119878)

119899-manifoldsrdquo Acta Universitatis

Sapientiae Mathematica vol 3 no 2 pp 212ndash224 2011[11] S S Shukla M K Shukla and R Prasad ldquoSlant submanifold

of (119871119862119878)119899-manifoldsrdquo to appear in Kyungpook Mathematical

Journal[12] K Yano and S Sawaki ldquoRiemannianmanifolds admitting a con-

formal transformation grouprdquo Journal of Differential Geometryvol 2 pp 161ndash184 1968

[13] A A Shaikh and S K Jana ldquoOn weakly quasi-conformallysymmetric manifoldsrdquo SUT Journal of Mathematics vol 43 no1 pp 61ndash83 2007

[14] R Kumar and B Prasad ldquoOn (119871119862119878)119899-manifoldsrdquo to appear in

Thai Journal of Mathematics[15] B-Y Chen and K Yano ldquoHypersurfaces of a conformally flat

spacerdquo Tensor vol 26 pp 318ndash322 1972[16] A A Shaikh and S K Jana ldquoOnweakly symmetric Riemannian

manifoldsrdquo Publicationes Mathematicae Debrecen vol 71 no 1-2 pp 27ndash41 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

for all 119883119884 119885119882 isin Γ(119879119872) where 119886 119887 are scalars of which119887 = 0 and 119860 is a nonzero 1-form (for more details we refer to[13 16])

Thus we have the following theorem for (LCS)119899-conform-

ally flat manifolds

Theorem 6 A conformally flat (LCS)119899-manifold is a manifold

of quasi-constant curvature

Proof From (33) and (35) we obtain

(119883 119884 119885119882)

= (

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 119892 (119884 119885) minus 119892 (119884119882) 119892 (119883 119885)

+ (

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

)

times 119892 (119883119882) 120578 (119884) 120578 (119885) minus 119892 (119884119882) 120578 (119883) 120578 (119885)

+119892 (119884 119885) 120578 (119883) 120578 (119882) minus 119892 (119883 119885) 120578 (119884) 120578 (119882)

(37)

This implies (36) for

119886 =

120591 minus 2 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119887 =

120591 minus 119899 (119899 minus 1) (1205722

minus 120588)

(119899 minus 1) (119899 minus 2)

119860 = 120578

(38)

This proves our assertion

Next differentiating the (19) covariantly with respect to119882 we get

nabla119882119878 (120601119883 120601119884) = nabla

119882119878 (119883 119884) + (119899 minus 1)119882(120572

2

minus 120588)

+ (119899 minus 1) (1205722

minus 120588)119882 [120578 (119883) 120578 (119884)]

(39)

for any119883119884 isin Γ(119879119872) Making use of the definition ofnabla119878 and(8) we have

(nabla119882119878) (120601119883 120601119884) + 119878 (nabla

119882120601119883 120601119884) + 119878 (120601119883 nabla

119882120601119884)

= (nabla119882119878) (119883 119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(40)

Thus we have

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 ((nabla119882120601)119883 + 120601nabla

119882119883 120601119884)

minus 119878 (120601119883 (nabla119882120601)119884 + 120601nabla

119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(41)

Here taking account of (17) we arrive at

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus119878 (120572 119892 (119883119882) 120585 + 2120578 (119883) 120578 (W) 120585 + 120578 (119883)119882 120601119884)

minus 119878 (120601119883 120572 119892 (119884119882) 120585 + 2120578 (119884) 120578 (119882) 120585 + 120578 (119884)119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884)

+ 119878 (119883 nabla119882119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1) (120572

2

minus 120588) 120578 (119884)

times 120578 (nabla119882119883) + 120572119892 (119883 120601119882) + (119899 minus 1) (120572

2

minus 120588) 120578 (119883)

times 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

= minus120572 119892 (119883119882) 119878 (120585 120601119884) + 2120578 (119883) 120578 (119882) 119878 (120585 120601119884)

+120578 (119883) 119878 (119882 120601119884)

minus 120572 119892 (119884119882) 119878 (120601119883 120585) + 2120578 (119884) 120578 (119882) 119878 (120601119883 120585)

+120578 (119884) 119878 (120601119883119882)

minus 119878 (120601119883 120601nabla119882119884) + 119878 (nabla

119882119883119884) + 119878 (119883 nabla

119882119884)

minus 119878 (120601nabla119882119883 120601119884) + (119899 minus 1)119882(120572

2

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883) + 120572119892 (119883 120601119882)

+ (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119884) + 120572119892 (119884 120601119882)

(42)

Again by using (13) (18) and (19) we reach

(nabla119882119878) (120601119883 120601119884) minus (nabla

119882119878) (119883 119884)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119883) 120578 (nabla119882119883)

minus (119899 minus 1) (1205722

minus 120588) 120578 (119884) 120578 (nabla119882119883)

Abstract and Applied Analysis 5

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588)

times 120578 (nabla119882119883) 120578 (119884) + 120572120578 (119884) 119892 (119883 120601119882)

+120578 (nabla119882119884) 120578 (119883) + 120572120578 (119883) 119892 (119884 120601119882)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

+ 120572 (119899 minus 1) (1205722

minus 120588)

times 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

(43)

Thus we have the following theorem

Theorem7 If an (LCS)119899-manifold119872 is Ricci-symmetric then

1205722

minus 120588 is constant

Proof If 119899 gt 1-dimensional (LCS)119899-manifold 119872 is Ricci-

symmetric then from (43) we conclude that

120572 (119899 minus 1) (1205722

minus 120588) 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

minus 120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882) = 0

(44)

It follows that

120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882) 120585 minus 120578 (119883) 120601119882

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120585

minus 120572120578 (119883) 120601119876119882 minus 120572119878 (120601119883119882) 120585 = 0

(45)

from which

minus 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882)

minus (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) + 119878 (120601119883119882) = 0

(46)

which is equivalent to

minus 120572 (119899 minus 1) (1205722

minus 120588) 120601119882 minus (119899 minus 1)119882(1205722

minus 120588) 120585

+ 120572120601119876119882 = 0

(47)

that is

119882(1205722

minus 120588) = 0 (48)

which proves our assertion

Since nabla119877 = 0 implies that nabla119878 = 0 we can give the follow-ing corollary

Corollary 8 If an 119899-dimensional (119871119862119878)119899-manifold 119872 is

locally symmetric then 1205722 minus 120588 is constant

Now taking the covariant derivation of the both sides of(18) with respect to 119884 we have

119884119878 (119883 120585) = (119899 minus 1)119882 [(1205722

minus 120588) 120578 (119883)] (49)

From the definition of the covariant derivation of Ricci-tensor we have

(nabla119884119878) (119883 120585) = nabla

119884119878 (119883 120585) minus 119878 (nabla

119884119883 120585) minus 119878 (119883 nabla

119884120585)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883) + (1205722

minus 120588)

times [120578 (nabla119884119883) + 120572119892 (119883 120601119884)]

minus (119899 minus 1) (1205722

minus 120588) 120578 (nabla119884119883) minus 120572119878 (119883 120601119884)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883)

+ 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119884) minus 120572119878 (119883 120601119884)

(50)

If an (119871119862119878)119899-manifold 119872 Ricci symmetric then Theorem 7

and (43) imply that

119878 (119883 120601119884) = (119899 minus 1) (1205722

minus 120588) 119892 (120601119884119883) (51)

This leads us to state the following

Theorem9 If an (LCS)119899-manifold119872 is Ricci symmetric then

it is an Einstein manifold

Corollary 10 If an (LCS)119899-manifold119872 is locally symmetric

then it is an Einstein manifold

In this section an example is used to demonstrate that themethod presented in this paper is effective But this exampleis a special case of Example 61 of [6]

Example 11 Now we consider the 3-dimensional manifold

119872 = (119909 119910 119911) isin R3

119911 = 0 (52)

where (119909 119910 119911) denote the standard coordinates in R3 Thevector fields

1198901= 119890119911

(119909

120597

120597119909

+ 119910

120597

120597119910

) 1198902= 119890119911120597

120597119910

1198903=

120597

120597119911

(53)

are linearly independent of each point of 119872 Let 119892 be theLorentzian metric tensor defined by

119892 (1198901 1198901) = 119892 (119890

2 1198902) = minus119892 (119890

3 1198903) = 1

119892 (119890119894 119890119895) = 0 119894 = 119895

(54)

6 Abstract and Applied Analysis

for 119894 119895 = 1 2 3 Let 120578 be the 1-formdefined by 120578(119885) = 119892(119885 1198903)

for any 119885 isin Γ(119879119872) Let 120601 be the (11)-tensor field defined by

1206011198901= 1198901 120601119890

2= 1198902 120601119890

3= 0 (55)

Then using the linearity of 120601 and 119892 we have 120578(1198903) = minus1

1206012

119885 = 119885 + 120578 (119885) 1198903

119892 (120601119885 120601119882) = 119892 (119885119882) + 120578 (119885) 120578 (119882)

(56)

for all 119885119882 isin Γ(119879119872) Thus for 120585 = 1198903 (120601 120585 120578 119892) defines a

Lorentzian paracontact structure on119872Now let nabla be the Levi-Civita connection with respect

to the Lorentzian metric 119892 and let 119877 be the Riemanniancurvature tensor of 119892 Then we have

[1198901 1198902] = minus119890

119911

1198902 [119890

1 1198903] = minus119890

1 [119890

2 1198903] = minus119890

2

(57)

Making use of the Koszul formulae for the Lorentzian metrictensor 119892 we can easily calculate the covariant derivations asfollows

nabla1198901

1198901= minus1198903 nabla

1198902

1198901= 119890119911

1198902 nabla

1198901

1198903= minus1198901

nabla1198902

1198903= minus1198902 nabla

1198902

1198902= minus119890119911

1198901minus 1198903

nabla1198901

1198902= nabla1198903

1198901= nabla1198903

1198902= nabla1198903

1198903= 0

(58)

From the previously mentioned it can be easily seen that(120601 120585 120578 119892) is an (LCS)

3-structure on 119872 that is 119872 is an

(LCS)3-manifold with 120572 = minus1 and 120588 = 0 Using the

previous relations we can easily calculate the components ofthe Riemannian curvature tensor as follows

119877 (1198901 1198902) 1198901= (1198902119911

minus 1) 1198902 119877 (119890

1 1198902) 1198902= (1 minus 119890

2119911

) 1198901

119877 (1198901 1198903) 1198901= minus1198903 119877 (119890

1 1198903) 1198903= minus1198901

119877 (1198902 1198903) 1198902= minus1198903 119877 (119890

2 1198903) 1198903= minus1198902

119877 (1198901 1198902) 1198903= 119877 (119890

1 1198903) 1198902= 119877 (119890

2 1198903) 1198901= 0

(59)

By using the properties of119877 and definition of the Ricci tensorwe obtain

119878 (1198901 1198901) = 119878 (119890

2 1198902) = minus119890

2119911

119878 (1198903 1198903) = minus2

119878 (1198901 1198902) = 119878 (119890

1 1198903) = 119878 (119890

2 1198903) = 0

(60)

Thus the scalar curvature 120591 of119872 is given by

120591 =

3

sum

119894=1

119892 (119890119894 119890119894) 119878 (119890119894 119890119894) = 2 (1 minus 119890

2119911

) (61)

On the other hand for any 119885119882 isin Γ(119879119872) 119885 and119882 can bewritten as 119885 = sum3

119894=1119891119894119890119894and119882 = sum

3

119895=1119892119895119890119895 where 119891

119894and 119892

119894

are smooth functions on119872 By direct calculations we have

119878 (119885119882) = minus 1198902119911

(11989111198921+ 11989121198922) minus 2119891

31198923

= minus1198902119911

(11989111198921+ 11989121198922minus 11989131198923) minus 11989131198923(1198902119911

+ 2)

(62)

Since 120578(119885) = minus1198913and 120578(119882) = minus119892

3and 119892(119885119882) = 119891

11198921+

11989121198922minus 11989131198923 we have

119878 (119885119882) = minus1198902119911

119892 (119885119882) minus (1198902119911

+ 2) 120578 (119885) 120578 (119882) (63)

This tell us that119872 is an 120578-Einstein manifold

Acknowledgment

The authors would like to thank the reviewers for the ex-tremely carefully reading and formany important commentswhich improved the paper considerably

References

[1] A A Shaikh ldquoOn Lorentzian almost paracontact manifoldswith a structure of the concircular typerdquoKyungpookMathemat-ical Journal vol 43 no 2 pp 305ndash314 2003

[2] KMatsumoto ldquoOn Lorentzian paracontactmanifoldsrdquo Bulletinof Yamagata University vol 12 no 2 pp 151ndash156 1989

[3] A A Shaikh ldquoSome results on (119871119862119878)119899-manifoldsrdquo Journal of the

Korean Mathematical Society vol 46 no 3 pp 449ndash461 2009[4] AA Shaikh and S KHui ldquoOn generalized120588-recurrent (119871119862119878)

119899-

manifoldsrdquo in Proceedings of the ICMS International Conferenceon Mathematical Science vol 1309 of American Institute ofPhysics Conference Proceedings pp 419ndash429 2010

[5] A A Shaikh T Basu and S Eyasmin ldquoOn the existence of120601-recurrent (119871119862119878)

119899-manifoldsrdquo ExtractaMathematicae vol 23

no 1 pp 71ndash83 2008[6] A A Shaikh and T Q Binh ldquoOn weakly symmetric (119871119862119878)

119899-

manifoldsrdquo Journal of AdvancedMathematical Studies vol 2 no2 pp 103ndash118 2009

[7] G T Sreenivasa Venkatesha and C S Bagewadi ldquoSome resultson (119871119862119878)

2119899+1-manifoldsrdquo Bulletin of Mathematical Analysis and

Applications vol 1 no 3 pp 64ndash70 2009[8] S K Yadav P K Dwivedi and D Suthar ldquoOn (119871119862119878)

2119899+1-

manifolds satisfying certain conditions on the concircularcurvature tensorrdquoThai Journal of Mathematics vol 9 no 3 pp597ndash603 2011

[9] M Atceken ldquoOn geometry of submanifolds of (119871119862119878)119899-

manifoldsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2012 Article ID 304647 11 pages 2012

[10] S K Hui and M Atceken ldquoContact warped product semi-slant submanifolds of (119871119862119878)

119899-manifoldsrdquo Acta Universitatis

Sapientiae Mathematica vol 3 no 2 pp 212ndash224 2011[11] S S Shukla M K Shukla and R Prasad ldquoSlant submanifold

of (119871119862119878)119899-manifoldsrdquo to appear in Kyungpook Mathematical

Journal[12] K Yano and S Sawaki ldquoRiemannianmanifolds admitting a con-

formal transformation grouprdquo Journal of Differential Geometryvol 2 pp 161ndash184 1968

[13] A A Shaikh and S K Jana ldquoOn weakly quasi-conformallysymmetric manifoldsrdquo SUT Journal of Mathematics vol 43 no1 pp 61ndash83 2007

[14] R Kumar and B Prasad ldquoOn (119871119862119878)119899-manifoldsrdquo to appear in

Thai Journal of Mathematics[15] B-Y Chen and K Yano ldquoHypersurfaces of a conformally flat

spacerdquo Tensor vol 26 pp 318ndash322 1972[16] A A Shaikh and S K Jana ldquoOnweakly symmetric Riemannian

manifoldsrdquo Publicationes Mathematicae Debrecen vol 71 no 1-2 pp 27ndash41 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

+ (119899 minus 1) (1205722

minus 120588)

times 120578 (nabla119882119883) 120578 (119884) + 120572120578 (119884) 119892 (119883 120601119882)

+120578 (nabla119882119884) 120578 (119883) + 120572120578 (119883) 119892 (119884 120601119882)

= minus120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882)

+ 120572 (119899 minus 1) (1205722

minus 120588)

times 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

(43)

Thus we have the following theorem

Theorem7 If an (LCS)119899-manifold119872 is Ricci-symmetric then

1205722

minus 120588 is constant

Proof If 119899 gt 1-dimensional (LCS)119899-manifold 119872 is Ricci-

symmetric then from (43) we conclude that

120572 (119899 minus 1) (1205722

minus 120588) 120578 (119884) 119892 (119883 120601119882) + 120578 (119883) 119892 (119884 120601119882)

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120578 (119884)

minus 120572120578 (119883) 119878 (119882 120601119884) minus 120572120578 (119884) 119878 (120601119883119882) = 0

(44)

It follows that

120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882) 120585 minus 120578 (119883) 120601119882

+ (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) 120585

minus 120572120578 (119883) 120601119876119882 minus 120572119878 (120601119883119882) 120585 = 0

(45)

from which

minus 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119882)

minus (119899 minus 1)119882(1205722

minus 120588) 120578 (119883) + 119878 (120601119883119882) = 0

(46)

which is equivalent to

minus 120572 (119899 minus 1) (1205722

minus 120588) 120601119882 minus (119899 minus 1)119882(1205722

minus 120588) 120585

+ 120572120601119876119882 = 0

(47)

that is

119882(1205722

minus 120588) = 0 (48)

which proves our assertion

Since nabla119877 = 0 implies that nabla119878 = 0 we can give the follow-ing corollary

Corollary 8 If an 119899-dimensional (119871119862119878)119899-manifold 119872 is

locally symmetric then 1205722 minus 120588 is constant

Now taking the covariant derivation of the both sides of(18) with respect to 119884 we have

119884119878 (119883 120585) = (119899 minus 1)119882 [(1205722

minus 120588) 120578 (119883)] (49)

From the definition of the covariant derivation of Ricci-tensor we have

(nabla119884119878) (119883 120585) = nabla

119884119878 (119883 120585) minus 119878 (nabla

119884119883 120585) minus 119878 (119883 nabla

119884120585)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883) + (1205722

minus 120588)

times [120578 (nabla119884119883) + 120572119892 (119883 120601119884)]

minus (119899 minus 1) (1205722

minus 120588) 120578 (nabla119884119883) minus 120572119878 (119883 120601119884)

= (119899 minus 1) 119884 (1205722

minus 120588) 120578 (119883)

+ 120572 (119899 minus 1) (1205722

minus 120588) 119892 (119883 120601119884) minus 120572119878 (119883 120601119884)

(50)

If an (119871119862119878)119899-manifold 119872 Ricci symmetric then Theorem 7

and (43) imply that

119878 (119883 120601119884) = (119899 minus 1) (1205722

minus 120588) 119892 (120601119884119883) (51)

This leads us to state the following

Theorem9 If an (LCS)119899-manifold119872 is Ricci symmetric then

it is an Einstein manifold

Corollary 10 If an (LCS)119899-manifold119872 is locally symmetric

then it is an Einstein manifold

In this section an example is used to demonstrate that themethod presented in this paper is effective But this exampleis a special case of Example 61 of [6]

Example 11 Now we consider the 3-dimensional manifold

119872 = (119909 119910 119911) isin R3

119911 = 0 (52)

where (119909 119910 119911) denote the standard coordinates in R3 Thevector fields

1198901= 119890119911

(119909

120597

120597119909

+ 119910

120597

120597119910

) 1198902= 119890119911120597

120597119910

1198903=

120597

120597119911

(53)

are linearly independent of each point of 119872 Let 119892 be theLorentzian metric tensor defined by

119892 (1198901 1198901) = 119892 (119890

2 1198902) = minus119892 (119890

3 1198903) = 1

119892 (119890119894 119890119895) = 0 119894 = 119895

(54)

6 Abstract and Applied Analysis

for 119894 119895 = 1 2 3 Let 120578 be the 1-formdefined by 120578(119885) = 119892(119885 1198903)

for any 119885 isin Γ(119879119872) Let 120601 be the (11)-tensor field defined by

1206011198901= 1198901 120601119890

2= 1198902 120601119890

3= 0 (55)

Then using the linearity of 120601 and 119892 we have 120578(1198903) = minus1

1206012

119885 = 119885 + 120578 (119885) 1198903

119892 (120601119885 120601119882) = 119892 (119885119882) + 120578 (119885) 120578 (119882)

(56)

for all 119885119882 isin Γ(119879119872) Thus for 120585 = 1198903 (120601 120585 120578 119892) defines a

Lorentzian paracontact structure on119872Now let nabla be the Levi-Civita connection with respect

to the Lorentzian metric 119892 and let 119877 be the Riemanniancurvature tensor of 119892 Then we have

[1198901 1198902] = minus119890

119911

1198902 [119890

1 1198903] = minus119890

1 [119890

2 1198903] = minus119890

2

(57)

Making use of the Koszul formulae for the Lorentzian metrictensor 119892 we can easily calculate the covariant derivations asfollows

nabla1198901

1198901= minus1198903 nabla

1198902

1198901= 119890119911

1198902 nabla

1198901

1198903= minus1198901

nabla1198902

1198903= minus1198902 nabla

1198902

1198902= minus119890119911

1198901minus 1198903

nabla1198901

1198902= nabla1198903

1198901= nabla1198903

1198902= nabla1198903

1198903= 0

(58)

From the previously mentioned it can be easily seen that(120601 120585 120578 119892) is an (LCS)

3-structure on 119872 that is 119872 is an

(LCS)3-manifold with 120572 = minus1 and 120588 = 0 Using the

previous relations we can easily calculate the components ofthe Riemannian curvature tensor as follows

119877 (1198901 1198902) 1198901= (1198902119911

minus 1) 1198902 119877 (119890

1 1198902) 1198902= (1 minus 119890

2119911

) 1198901

119877 (1198901 1198903) 1198901= minus1198903 119877 (119890

1 1198903) 1198903= minus1198901

119877 (1198902 1198903) 1198902= minus1198903 119877 (119890

2 1198903) 1198903= minus1198902

119877 (1198901 1198902) 1198903= 119877 (119890

1 1198903) 1198902= 119877 (119890

2 1198903) 1198901= 0

(59)

By using the properties of119877 and definition of the Ricci tensorwe obtain

119878 (1198901 1198901) = 119878 (119890

2 1198902) = minus119890

2119911

119878 (1198903 1198903) = minus2

119878 (1198901 1198902) = 119878 (119890

1 1198903) = 119878 (119890

2 1198903) = 0

(60)

Thus the scalar curvature 120591 of119872 is given by

120591 =

3

sum

119894=1

119892 (119890119894 119890119894) 119878 (119890119894 119890119894) = 2 (1 minus 119890

2119911

) (61)

On the other hand for any 119885119882 isin Γ(119879119872) 119885 and119882 can bewritten as 119885 = sum3

119894=1119891119894119890119894and119882 = sum

3

119895=1119892119895119890119895 where 119891

119894and 119892

119894

are smooth functions on119872 By direct calculations we have

119878 (119885119882) = minus 1198902119911

(11989111198921+ 11989121198922) minus 2119891

31198923

= minus1198902119911

(11989111198921+ 11989121198922minus 11989131198923) minus 11989131198923(1198902119911

+ 2)

(62)

Since 120578(119885) = minus1198913and 120578(119882) = minus119892

3and 119892(119885119882) = 119891

11198921+

11989121198922minus 11989131198923 we have

119878 (119885119882) = minus1198902119911

119892 (119885119882) minus (1198902119911

+ 2) 120578 (119885) 120578 (119882) (63)

This tell us that119872 is an 120578-Einstein manifold

Acknowledgment

The authors would like to thank the reviewers for the ex-tremely carefully reading and formany important commentswhich improved the paper considerably

References

[1] A A Shaikh ldquoOn Lorentzian almost paracontact manifoldswith a structure of the concircular typerdquoKyungpookMathemat-ical Journal vol 43 no 2 pp 305ndash314 2003

[2] KMatsumoto ldquoOn Lorentzian paracontactmanifoldsrdquo Bulletinof Yamagata University vol 12 no 2 pp 151ndash156 1989

[3] A A Shaikh ldquoSome results on (119871119862119878)119899-manifoldsrdquo Journal of the

Korean Mathematical Society vol 46 no 3 pp 449ndash461 2009[4] AA Shaikh and S KHui ldquoOn generalized120588-recurrent (119871119862119878)

119899-

manifoldsrdquo in Proceedings of the ICMS International Conferenceon Mathematical Science vol 1309 of American Institute ofPhysics Conference Proceedings pp 419ndash429 2010

[5] A A Shaikh T Basu and S Eyasmin ldquoOn the existence of120601-recurrent (119871119862119878)

119899-manifoldsrdquo ExtractaMathematicae vol 23

no 1 pp 71ndash83 2008[6] A A Shaikh and T Q Binh ldquoOn weakly symmetric (119871119862119878)

119899-

manifoldsrdquo Journal of AdvancedMathematical Studies vol 2 no2 pp 103ndash118 2009

[7] G T Sreenivasa Venkatesha and C S Bagewadi ldquoSome resultson (119871119862119878)

2119899+1-manifoldsrdquo Bulletin of Mathematical Analysis and

Applications vol 1 no 3 pp 64ndash70 2009[8] S K Yadav P K Dwivedi and D Suthar ldquoOn (119871119862119878)

2119899+1-

manifolds satisfying certain conditions on the concircularcurvature tensorrdquoThai Journal of Mathematics vol 9 no 3 pp597ndash603 2011

[9] M Atceken ldquoOn geometry of submanifolds of (119871119862119878)119899-

manifoldsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2012 Article ID 304647 11 pages 2012

[10] S K Hui and M Atceken ldquoContact warped product semi-slant submanifolds of (119871119862119878)

119899-manifoldsrdquo Acta Universitatis

Sapientiae Mathematica vol 3 no 2 pp 212ndash224 2011[11] S S Shukla M K Shukla and R Prasad ldquoSlant submanifold

of (119871119862119878)119899-manifoldsrdquo to appear in Kyungpook Mathematical

Journal[12] K Yano and S Sawaki ldquoRiemannianmanifolds admitting a con-

formal transformation grouprdquo Journal of Differential Geometryvol 2 pp 161ndash184 1968

[13] A A Shaikh and S K Jana ldquoOn weakly quasi-conformallysymmetric manifoldsrdquo SUT Journal of Mathematics vol 43 no1 pp 61ndash83 2007

[14] R Kumar and B Prasad ldquoOn (119871119862119878)119899-manifoldsrdquo to appear in

Thai Journal of Mathematics[15] B-Y Chen and K Yano ldquoHypersurfaces of a conformally flat

spacerdquo Tensor vol 26 pp 318ndash322 1972[16] A A Shaikh and S K Jana ldquoOnweakly symmetric Riemannian

manifoldsrdquo Publicationes Mathematicae Debrecen vol 71 no 1-2 pp 27ndash41 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

for 119894 119895 = 1 2 3 Let 120578 be the 1-formdefined by 120578(119885) = 119892(119885 1198903)

for any 119885 isin Γ(119879119872) Let 120601 be the (11)-tensor field defined by

1206011198901= 1198901 120601119890

2= 1198902 120601119890

3= 0 (55)

Then using the linearity of 120601 and 119892 we have 120578(1198903) = minus1

1206012

119885 = 119885 + 120578 (119885) 1198903

119892 (120601119885 120601119882) = 119892 (119885119882) + 120578 (119885) 120578 (119882)

(56)

for all 119885119882 isin Γ(119879119872) Thus for 120585 = 1198903 (120601 120585 120578 119892) defines a

Lorentzian paracontact structure on119872Now let nabla be the Levi-Civita connection with respect

to the Lorentzian metric 119892 and let 119877 be the Riemanniancurvature tensor of 119892 Then we have

[1198901 1198902] = minus119890

119911

1198902 [119890

1 1198903] = minus119890

1 [119890

2 1198903] = minus119890

2

(57)

Making use of the Koszul formulae for the Lorentzian metrictensor 119892 we can easily calculate the covariant derivations asfollows

nabla1198901

1198901= minus1198903 nabla

1198902

1198901= 119890119911

1198902 nabla

1198901

1198903= minus1198901

nabla1198902

1198903= minus1198902 nabla

1198902

1198902= minus119890119911

1198901minus 1198903

nabla1198901

1198902= nabla1198903

1198901= nabla1198903

1198902= nabla1198903

1198903= 0

(58)

From the previously mentioned it can be easily seen that(120601 120585 120578 119892) is an (LCS)

3-structure on 119872 that is 119872 is an

(LCS)3-manifold with 120572 = minus1 and 120588 = 0 Using the

previous relations we can easily calculate the components ofthe Riemannian curvature tensor as follows

119877 (1198901 1198902) 1198901= (1198902119911

minus 1) 1198902 119877 (119890

1 1198902) 1198902= (1 minus 119890

2119911

) 1198901

119877 (1198901 1198903) 1198901= minus1198903 119877 (119890

1 1198903) 1198903= minus1198901

119877 (1198902 1198903) 1198902= minus1198903 119877 (119890

2 1198903) 1198903= minus1198902

119877 (1198901 1198902) 1198903= 119877 (119890

1 1198903) 1198902= 119877 (119890

2 1198903) 1198901= 0

(59)

By using the properties of119877 and definition of the Ricci tensorwe obtain

119878 (1198901 1198901) = 119878 (119890

2 1198902) = minus119890

2119911

119878 (1198903 1198903) = minus2

119878 (1198901 1198902) = 119878 (119890

1 1198903) = 119878 (119890

2 1198903) = 0

(60)

Thus the scalar curvature 120591 of119872 is given by

120591 =

3

sum

119894=1

119892 (119890119894 119890119894) 119878 (119890119894 119890119894) = 2 (1 minus 119890

2119911

) (61)

On the other hand for any 119885119882 isin Γ(119879119872) 119885 and119882 can bewritten as 119885 = sum3

119894=1119891119894119890119894and119882 = sum

3

119895=1119892119895119890119895 where 119891

119894and 119892

119894

are smooth functions on119872 By direct calculations we have

119878 (119885119882) = minus 1198902119911

(11989111198921+ 11989121198922) minus 2119891

31198923

= minus1198902119911

(11989111198921+ 11989121198922minus 11989131198923) minus 11989131198923(1198902119911

+ 2)

(62)

Since 120578(119885) = minus1198913and 120578(119882) = minus119892

3and 119892(119885119882) = 119891

11198921+

11989121198922minus 11989131198923 we have

119878 (119885119882) = minus1198902119911

119892 (119885119882) minus (1198902119911

+ 2) 120578 (119885) 120578 (119882) (63)

This tell us that119872 is an 120578-Einstein manifold

Acknowledgment

The authors would like to thank the reviewers for the ex-tremely carefully reading and formany important commentswhich improved the paper considerably

References

[1] A A Shaikh ldquoOn Lorentzian almost paracontact manifoldswith a structure of the concircular typerdquoKyungpookMathemat-ical Journal vol 43 no 2 pp 305ndash314 2003

[2] KMatsumoto ldquoOn Lorentzian paracontactmanifoldsrdquo Bulletinof Yamagata University vol 12 no 2 pp 151ndash156 1989

[3] A A Shaikh ldquoSome results on (119871119862119878)119899-manifoldsrdquo Journal of the

Korean Mathematical Society vol 46 no 3 pp 449ndash461 2009[4] AA Shaikh and S KHui ldquoOn generalized120588-recurrent (119871119862119878)

119899-

manifoldsrdquo in Proceedings of the ICMS International Conferenceon Mathematical Science vol 1309 of American Institute ofPhysics Conference Proceedings pp 419ndash429 2010

[5] A A Shaikh T Basu and S Eyasmin ldquoOn the existence of120601-recurrent (119871119862119878)

119899-manifoldsrdquo ExtractaMathematicae vol 23

no 1 pp 71ndash83 2008[6] A A Shaikh and T Q Binh ldquoOn weakly symmetric (119871119862119878)

119899-

manifoldsrdquo Journal of AdvancedMathematical Studies vol 2 no2 pp 103ndash118 2009

[7] G T Sreenivasa Venkatesha and C S Bagewadi ldquoSome resultson (119871119862119878)

2119899+1-manifoldsrdquo Bulletin of Mathematical Analysis and

Applications vol 1 no 3 pp 64ndash70 2009[8] S K Yadav P K Dwivedi and D Suthar ldquoOn (119871119862119878)

2119899+1-

manifolds satisfying certain conditions on the concircularcurvature tensorrdquoThai Journal of Mathematics vol 9 no 3 pp597ndash603 2011

[9] M Atceken ldquoOn geometry of submanifolds of (119871119862119878)119899-

manifoldsrdquo International Journal of Mathematics and Mathe-matical Sciences vol 2012 Article ID 304647 11 pages 2012

[10] S K Hui and M Atceken ldquoContact warped product semi-slant submanifolds of (119871119862119878)

119899-manifoldsrdquo Acta Universitatis

Sapientiae Mathematica vol 3 no 2 pp 212ndash224 2011[11] S S Shukla M K Shukla and R Prasad ldquoSlant submanifold

of (119871119862119878)119899-manifoldsrdquo to appear in Kyungpook Mathematical

Journal[12] K Yano and S Sawaki ldquoRiemannianmanifolds admitting a con-

formal transformation grouprdquo Journal of Differential Geometryvol 2 pp 161ndash184 1968

[13] A A Shaikh and S K Jana ldquoOn weakly quasi-conformallysymmetric manifoldsrdquo SUT Journal of Mathematics vol 43 no1 pp 61ndash83 2007

[14] R Kumar and B Prasad ldquoOn (119871119862119878)119899-manifoldsrdquo to appear in

Thai Journal of Mathematics[15] B-Y Chen and K Yano ldquoHypersurfaces of a conformally flat

spacerdquo Tensor vol 26 pp 318ndash322 1972[16] A A Shaikh and S K Jana ldquoOnweakly symmetric Riemannian

manifoldsrdquo Publicationes Mathematicae Debrecen vol 71 no 1-2 pp 27ndash41 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of