# Some Curvature Properties of(LCS)n-Manifolds

Authors: Mehmet Atçeken
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The object of the present paper is to study(LCS)n-manifolds with vanishing quasi-conformal curvature tensor. (LCS)n-manifolds satisfying Ricci-symmetric condition are also characterized.


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## Body

## 1. Introduction

Recently, in [1], Shaikh introduced and studied Lorentzian concircular structure manifolds (briefly (LCS)-manifold) which generalizes the notion of LP-Sasakian manifolds, introduced by Matsumoto [2].Generalizing the notion of LP-Sasakian manifold in 2003 [1], Shaikh introduced the notion of (LCS)n-manifolds along with their existence and applications to the general theory of relativity and cosmology. Also, Shaikh and his coauthors studied various types of (LCS)n-manifolds by imposing the curvature restrictions (see [3–6]). In [7, 8], the authors also studied (LCS)2n+1-manifolds.The submanifold of an(LCS)n-manifold is studied by Atceken and Hui [9, 10] and Shukla et al. [11]. In [12], Yano and Sawaki introduced the quasi-conformal curvature tensor, and later it was studied by many authors with curvature restrictions on various structures [13].After then, the same author studied weakly symmetric(LCS)n-manifolds by several examples and obtain various results in such manifolds. In [7], authors shown that a pseudo projectively flat and pseudo projectively recurrent (LCS)n manifolds are η-Einstein manifold.On the other hand, in [5], authors proved the existence of ϕ-recurrent (LCS)3 manifold which is neither locally symmetric nor locally ϕ-symmetric by nontrivial examples. Furthermore, they also give the necessary and sufficient conditions for a (LCS)n-manifold to be locally ϕ-recurrent.In this study, we have investigated the quasi-conformal flat(LCS)n-manifolds satisfying properties such as Ricci-symmetric, locally symmetric, and η-Einstein. Finally, we give an example for η-Einstein manifolds.


## 2. Preliminaries

Ann-dimensional Lorentzian manifold M is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric tensor g, that is, M admits a smooth symmetric tensor field g of the type (2,0) such that, for each p∈M,
(1)gp:TM(p)×TM(p)⟶ℝ
is a nondegenerate inner product of signature (-,+,+,…,+). In such a manifold, a nonzero vector Xp∈TM(p) is said to be timelike (resp., nonspacelike, null, and spacelike) if it satisfies the condition gp(Xp,Xp)<0 (resp., ≤0, =0, >0). These cases are called casual character of the vectors.Definition 1.
In a Lorentzian manifold(M,g), a vector field P defined by
(2)g(X,P)=A(X)
for any X∈Γ(TM) is said to be a concircular vector field if
(3)(∇XA)Y=α{g(X,Y)+w(X)A(Y)}
for Y∈Γ(TM), where α is a nonzero scalar function, A is a 1-form, w is also closed 1-form, and ∇ denotes the Levi-Civita connection on M [7].LetM be a Lorentzian manifold admitting a unit timelike concircular vector field ξ, called the characteristic vector field of the manifold. Then we have
(4)g(ξ,ξ)=-1.
Since ξ is a unit concircular unit vector field, there exists a nonzero 1-form η such that
(5)g(X,ξ)=η(X).
The equation of the following form holds:
(6)(∇Xη)Y=α{g(X,Y)+η(X)η(Y)},α≠0
for all X,Y∈Γ(TM), where α is a nonzero scalar function satisfying
(7)∇Xα=X(α)=dα(X)=ρη(X),ρ being a certain scalar function given by ρ=-ξ(α).Let us put(8)∇Xξ=αϕX,
then from (6) and (8), we can derive
(9)ϕX=X+η(X)ξ,
which tell us that ϕ is a symmetric (1,1)-tensor. Thus the Lorentzian manifold M together with the unit timelike concircular vector field ξ, its associated 1-form η, and (1,1)-type tensor field ϕ is said to be a Lorentzian concircular structure manifold.A differentiable manifoldM of dimension n is called (LCS)-manifold if it admits a (1,1)-type tensor field ϕ, a covariant vector field η, and a Lorentzian metric g which satisfy
(10)η(ξ)=g(ξ,ξ)=-1,(11)ϕ2X=X+η(X)ξ,(12)g(X,ξ)=η(X),(13)ϕξ=0,η∘ϕ=0
for all X∈Γ(TM). Particularly, if we take α=1, then we can obtain the LP-Sasakian structure of Matsumoto [2].Also, in an(LCS)n-manifold M, the following relations are satisfied (see [3–6]):
(14)η(R(X,Y)Z)=(α2-ρ)[g(Y,Z)η(X)-g(X,Z)η(Y)],(15)R(ξ,X)Y=(α2-ρ)[g(X,Y)ξ-η(Y)X],(16)R(X,Y)ξ=(α2-ρ)[η(Y)X-η(X)Y],(17)(∇Xϕ)Y=α[g(X,Y)ξ+2η(X)η(Y)ξ+η(Y)X],(18)S(X,ξ)=(n-1)(α2-ρ)η(X),(19)S(ϕX,ϕY)=S(X,Y)+(n-1)(α2-ρ)η(X)η(Y)
for all vector fields X,Y,Z on M, where R and S denote the Riemannian curvature tensor and Ricci curvature, respectively, Q is also the Ricci operator given by S(X,Y)=g(QX,Y).Now let(M,g) be an n-dimensional Riemannian manifold; then the concircular curvature tensor C~, the Weyl conformal curvature tensor C, and the pseudo projective curvature tensor P~ are, respectively, defined by
(20)C~(X,Y)Z=R(X,Y)Z-τn(n-1)[g(Y,Z)X-g(X,Z)Y],(21)C(X,Y)Z=R(X,Y)Z-1n-2×[+g(Y,Z)QX-g(X,Z)QYS(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY]+τ(n-1)(n-2)[g(Y,Z)X-g(X,Z)Y],(22)P~(X,Y)Z=aR(X,Y)Z+b[S(Y,Z)X-S(X,Z)Y]-τn[an-1+b][g(Y,Z)X-g(X,Z)Y],
where a and b are constants such that a,b≠0, and τ  is  also the scalar curvature of M  [7].For ann-dimensional (LCS)n-manifold the quasi-conformal curvature tensor 𝒞~ is given by
(23)𝒞~(X,Y)Z=aR(X,Y)Z+b[+g(Y,Z)QX-g(X,Z)QY]S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY]-τn[an-1+2b][g(Y,Z)X-g(X,Z)Y]
for all X,Y,Z∈Γ(TM) [14].The notion of quasi-conformal curvature tensor was defined by Yano and Swaki [12]. If a=1 and b=-1/(n-1), then quasi-conformal curvature tensor reduces to conformal curvature tensor.


## 3. Quasi-Conformally Flat(LCS)n-Manifolds and Some of Their Properties

For ann-dimensional quasi-conformally flat (LCS)n-manifold, we know for Z=ξ from (23),
(24)aR(X,Y)ξ+b[S(Y,ξ)X-S(X,ξ)Y+g(Y,ξ)QX-g(X,ξ)QY]-τn[an-1+2b][g(Y,ξ)X-g(X,ξ)Y]=0.
Here, taking into account of (16), we have(25)[η(Y)X-η(X)Y][(an-1+2b)a(α2-ρ)+b(n-1)(α2-ρ)-τn(an-1+2b)]+b[η(Y]QX-η(X)QY]=0.
Let Y=ξ be in (25); then also by using (18) we obtain
(26)[-X-η(X)ξ][a(α2-ρ)-τn(an-1+2b)+b(n-1)(α2-ρ)(an-1+2b)]+b[-QX-η(X)(n-1)(α2-ρ)ξ]=0.
Taking the inner product on both sides of the last equation by Y, we obtain
(27)[g(X,Y)+η(X)η(Y)][(an-1+2b)a(α2-ρ)+b(n-1)×(α2-ρ)-τn(an-1+2b)]+b[S(X,Y)+η(X)η(Y)(α2-ρ)(n-1)]=0,
that is,
(28)S(X,Y)=g(X,Y)×[τnb(an-1+2b)-(α2-ρ)(ab+(n-1))]+η(X)η(Y)[τnb(an-1+2b)-(α2-ρ)(ab+2(n-1))].
Now we are in a proposition to state the following.Theorem 2.
If ann-dimensional (LCS)n-manifold M is quasi-conformally flat, then M is an η-Einstein manifold.Now, let{e1,e2,…,en-1,ξ} be an orthonormal basis of the tangent space at any point of the manifold. Then putting X=Y=ei,ξ in (28), and taking summation for 1≤i≤n-1, we have
(29)τ=n(n-1)(α2-ρ)ifa+(n-2)b≠0.
In view of (28) and (29), we obtain
(30)S(X,Y)=(n-1)(α2-ρ)g(X,Y),
which is equivalent to
(31)QX=(n-1)(α2-ρ)X
for any X∈Γ(TM).By using (29) and (31) in (23) for a quasi-conformally flat (LCS)n-manifold M, we get
(32)R(X,Y)Z=(α2-ρ){g(Y,Z)X-g(X,Z)Y},
for all X,Y,Z∈Γ(TM). If we consider Schur's Theorem, we can give the following the theorem.Theorem 3.
A quasi-conformally flat(LCS)n-manifold M(n>1) is a manifold of constant curvature (α2-ρ) provided that a+b(n-2)≠0.Now let us consider an(LCS)n-manifold M which is conformally flat. Thus we have from (21) that
(33)R(X,Y)Z=1n-2{S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY}-τ(n-1)(n-2){g(Y,Z)X-g(X,Z)Y},
for all vector fields X,Y,Z tangent to M. Setting Z=ξ in (33) and using (16), (18) we have
(34)[τn-1-(α2-ρ)][η(Y)X-η(X)Y]=[η(Y)QX-η(X)QY].
If we put Y=ξ in (34) and also using (18), we obtain
(35)QX=[τn-1-(α2-ρ)]X+[τn-1-n(α2-ρ)]η(X)ξ.Corollary 4.
A conformally flat(LCS)n-manifold is an η-Einstein manifold.Generalizing the notion of a manifold of constant curvature, Chen and Yano [15] introduced the notion of a manifold of quasi-constant curvature which can be defined as follows:Definition 5.
A Riemannian manifold is said to be a manifold of quasi-constant curvature if it is conformally flat and its curvature tensorR~ of type (0,4) is of the form
(36)R~(X,Y,Z,W)=a{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)}+b{g(Y,Z)A(X)A(W)-g(X,Z)A(Y)A(W)+g(X,W)A(Y)A(Z)-g(Y,W)A(X)A(Z)},
for all X,Y,Z,W∈Γ(TM), where a,b are scalars of which b≠0 and A is a nonzero 1-form (for more details, we refer to [13, 16]).Thus we have the following theorem for(LCS)n-conformally flat manifolds.Theorem 6.
A conformally flat(LCS)n-manifold is a manifold of quasi-constant curvature.Proof.
From (33) and (35), we obtain
(37)R~(X,Y,Z,W)=(τ-2(n-1)(α2-ρ)(n-1)(n-2))×{g(X,W)g(Y,Z)-g(Y,W)g(X,Z)}+(τ-n(n-1)(α2-ρ)(n-1)(n-2))×{g(X,W)η(Y)η(Z)-g(Y,W)η(X)η(Z)+g(Y,Z)η(X)η(W)-g(X,Z)η(Y)η(W)}.
This implies (36) for
(38)a=τ-2(n-1)(α2-ρ)(n-1)(n-2),b=τ-n(n-1)(α2-ρ)(n-1)(n-2),A=η.
This proves our assertion.Next, differentiating the (19) covariantly with respect to W, we get
(39)∇WS(ϕX,ϕY)=∇WS(X,Y)+(n-1)W(α2-ρ)+(n-1)(α2-ρ)W[η(X)η(Y)],
for any X,Y∈Γ(TM). Making use of the definition of ∇S and (8), we have
(40)(∇WS)(ϕX,ϕY)+S(∇WϕX,ϕY)+S(ϕX,∇WϕY)=(∇WS)(X,Y)+S(∇WX,Y)+S(X,∇WY)+(n-1)W(α2-ρ)η(X)η(Y)+(n-1)(α2-ρ)η(Y){η(∇WX)+αg(X,ϕW)}+(n-1)(α2-ρ)η(X){η(∇WY)+αg(Y,ϕW)}.
Thus we have
(41)(∇WS)(ϕX,ϕY)-(∇WS)(X,Y)=-S((∇Wϕ)X+ϕ∇WX,ϕY)-S(ϕX,(∇Wϕ)Y+ϕ∇WY)+S(∇WX,Y)+S(X,∇WY)+(n-1)W(α2-ρ)η(X)η(Y)+(n-1)(α2-ρ)η(Y){η(∇WX)+αg(X,ϕW)}+(n-1)(α2-ρ)η(X){η(∇WY)+αg(Y,ϕW)}.
Here taking account of (17), we arrive at
(42)(∇WS)(ϕX,ϕY)-(∇WS)(X,Y)=-S(α{g(X,W)ξ+2η(X)η(W)ξ+η(X)W},ϕY)-S(ϕX,α{g(Y,W)ξ+2η(Y)η(W)ξ+η(Y)W})-S(ϕX,ϕ∇WY)+S(∇WX,Y)+S(X,∇WY)+(n-1)W(α2-ρ)η(X)η(Y)-S(ϕ∇WX,ϕY)+(n-1)(α2-ρ)η(Y)×{η(∇WX)+αg(X,ϕW)}+(n-1)(α2-ρ)η(X)×{η(∇WY)+αg(Y,ϕW)}=-α{g(X,W)S(ξ,ϕY)+2η(X)η(W)S(ξ,ϕY)+η(X)S(W,ϕY)}-α{g(Y,W)S(ϕX,ξ)+2η(Y)η(W)S(ϕX,ξ)+η(Y)S(ϕX,W)}-S(ϕX,ϕ∇WY)+S(∇WX,Y)+S(X,∇WY)-S(ϕ∇WX,ϕY)+(n-1)W(α2-ρ)η(X)η(Y)+(n-1)(α2-ρ)η(Y){η(∇WX)+αg(X,ϕW)}+(n-1)(α2-ρ)η(X){η(∇WY)+αg(Y,ϕW)}.
Again, by using (13), (18), and (19), we reach
(43)(∇WS)(ϕX,ϕY)-(∇WS)(X,Y)=-αη(X)S(W,ϕY)-αη(Y)S(ϕX,W)-(n-1)(α2-ρ)η(X)η(∇WX)-(n-1)(α2-ρ)η(Y)η(∇WX)+(n-1)W(α2-ρ)η(X)η(Y)+(n-1)(α2-ρ)×{η(∇WX)η(Y)+αη(Y)g(X,ϕW)+η(∇WY)η(X)+αη(X)g(Y,ϕW)}=-αη(X)S(W,ϕY)-αη(Y)S(ϕX,W)+α(n-1)(α2-ρ)×{η(Y)g(X,ϕW)+η(X)g(Y,ϕW)}+(n-1)W(α2-ρ)η(X)η(Y).
Thus we have the following theorem.Theorem 7.
If an(LCS)n-manifold M is Ricci-symmetric; then α2-ρ is constant.Proof.
Ifn>1-dimensional (LCS)n-manifold M is Ricci-symmetric, then from (43) we conclude that
(44)α(n-1)(α2-ρ){η(Y)g(X,ϕW)+η(X)g(Y,ϕW)}+(n-1)W(α2-ρ)η(X)η(Y)-αη(X)S(W,ϕY)-αη(Y)S(ϕX,W)=0.
It follows that
(45)α(n-1)(α2-ρ){g(X,ϕW)ξ-η(X)ϕW}+(n-1)W(α2-ρ)η(X)ξ-αη(X)ϕQW-αS(ϕX,W)ξ=0,
from which
(46)-α(n-1)(α2-ρ)g(X,ϕW)-(n-1)W(α2-ρ)η(X)+S(ϕX,W)=0,
which is equivalent to
(47)-α(n-1)(α2-ρ)ϕW-(n-1)W(α2-ρ)ξ+αϕQW=0,
that is,
(48)W(α2-ρ)=0,
which proves our assertion.Since∇R=0 implies that ∇S=0, we can give the following corollary.Corollary 8.
If ann-dimensional (LCS)n-manifold M is locally symmetric, then α2-ρ is constant.Now, taking the covariant derivation of the both sides of (18) with respect to Y, we have
(49)YS(X,ξ)=(n-1)W[(α2-ρ)η(X)].
From the definition of the covariant derivation of Ricci-tensor, we have
(50)(∇YS)(X,ξ)=∇YS(X,ξ)-S(∇YX,ξ)-S(X,∇Yξ)=(n-1){Y(α2-ρ)η(X)+(α2-ρ)×[η(∇YX)+αg(X,ϕY)](α2-ρ)}-(n-1)(α2-ρ)η(∇YX)-αS(X,ϕY)=(n-1)Y(α2-ρ)η(X)+α(n-1)(α2-ρ)g(X,ϕY)-αS(X,ϕY).
If an (LCS)n-manifold M Ricci symmetric, then Theorem 7 and (43) imply that
(51)S(X,ϕY)=(n-1)(α2-ρ)g(ϕY,X).
This leads us to state the following.Theorem 9.
If an(LCS)n-manifold M is Ricci symmetric, then it is an Einstein manifold.Corollary 10.
If an(LCS)n-manifold M is locally symmetric, then it is an Einstein manifold.In this section, an example is used to demonstrate that the method presented in this paper is effective. But this example is a special case of Example 6.1 of [6].Example 11.
Now, we consider the 3-dimensional manifold(52)M={(x,y,z)∈ℝ3,z≠0},
where (x,y,z) denote the standard coordinates in ℝ3. The vector fields
(53)e1=ez(x∂∂x+y∂∂y),e2=ez∂∂y,e3=∂∂z
are linearly independent of each point of M. Let g be the Lorentzian metric tensor defined by
(54)g(e1,e1)=g(e2,e2)=-g(e3,e3)=1,g(ei,ej)=0,i≠j,
for i,j=1,2,3. Let η be the 1-form defined by η(Z)=g(Z,e3) for any Z∈Γ(TM). Let ϕ be the (1,1)-tensor field defined by
(55)ϕe1=e1,ϕe2=e2,ϕe3=0.
Then using the linearity of ϕ and g, we have η(e3)=-1,
(56)ϕ2Z=Z+η(Z)e3,g(ϕZ,ϕW)=g(Z,W)+η(Z)η(W),
for all Z,W∈Γ(TM). Thus for ξ=e3, (ϕ,ξ,η,g) defines a Lorentzian paracontact structure on M.
Now, let∇ be the Levi-Civita connection with respect to the Lorentzian metric g, and let R be the Riemannian curvature tensor of g. Then we have
(57)[e1,e2]=-eze2,[e1,e3]=-e1,[e2,e3]=-e2.
Making use of the Koszul formulae for the Lorentzian metric tensor g, we can easily calculate the covariant derivations as follows:
(58)∇e1e1=-e3,∇e2e1=eze2,∇e1e3=-e1,∇e2e3=-e2,∇e2e2=-eze1-e3,∇e1e2=∇e3e1=∇e3e2=∇e3e3=0.
From the previously mentioned, it can be easily seen that (ϕ,ξ,η,g) is an (LCS)3-structure on M, that is, M is an (LCS)3-manifold with α=-1 and ρ=0. Using the previous relations, we can easily calculate the components of the Riemannian curvature tensor as follows:
(59)R(e1,e2)e1=(e2z-1)e2,R(e1,e2)e2=(1-e2z)e1,R(e1,e3)e1=-e3,R(e1,e3)e3=-e1,R(e2,e3)e2=-e3,R(e2,e3)e3=-e2,R(e1,e2)e3=R(e1,e3)e2=R(e2,e3)e1=0.
By using the properties of R and definition of the Ricci tensor, we obtain
(60)S(e1,e1)=S(e2,e2)=-e2z,S(e3,e3)=-2,S(e1,e2)=S(e1,e3)=S(e2,e3)=0.
Thus the scalar curvature τ of M is given by
(61)τ=∑i=13‍g(ei,ei)S(ei,ei)=2(1-e2z).
On the other hand, for any Z,W∈Γ(TM), Z and W can be written as Z=∑i=13‍fiei and W=∑j=13‍gjej, where fi and gi are smooth functions on M. By direct calculations, we have
(62)S(Z,W)=-e2z(f1g1+f2g2)-2f3g3=-e2z(f1g1+f2g2-f3g3)-f3g3(e2z+2).
Since η(Z)=-f3 and η(W)=-g3 and g(Z,W)=f1g1+f2g2-f3g3, we have
(63)S(Z,W)=-e2zg(Z,W)-(e2z+2)η(Z)η(W).
This tell us that M is an η-Einstein manifold.



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