# Some Results on Lorentzian Para-Sasakian Manifolds

Authors: Venkatesha; C. S. Bagewadi; K. T. Pradeep Kumar
Journal: ISRN Geometry
 (2011)

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## Abstract

The object of the present paper is to study Lorentzian para-Sasakian
(briefly LP-Sasakian) manifolds satisfying certain conditions on theW2-curvature tensor.


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## Body

## 1. Introduction

In 1989, Matsumoto [1] introduced the notion of Lorentzian para-Sasakian manifold. Then,  Mihai and Roşca [2] introduced the same notion independently and they obtained several results on this manifold. LP-Sasakian manifolds have also been studied by Matsumoto and Mihai [3], Mihai et al. [4], and Venkatesha and Bagewadi [5].On the other hand, Pokhariyal and Mishra [6] have introduced new curvature tensor called W2-curvature tensor in a Riemannian manifold and studied their properties. Further, Pokhariyal [7] has studied some properties of this curvature tensor in a Sasakian manifold. Matsumoto et al. [8], and  Yìldìz and De [9] have studied W2-curvature tensor in P-Sasakian and Kenmotsu manifolds, respectively.In the present paper, we study some curvature conditions on LP-Sasakian manifolds. Firstly, we study LP-Sasakian manifolds satisfyingW2=0 and W2-semisymmetric LP-Sasakian manifolds. Further, we study LP-Sasakian manifolds which satisfy P·W2=0, M̃·W2=0 and C·W2=0, where P is the projective curvature tensor, M̃ is the M-projective curvature tensor, and C is the conformal curvature tensor.


## 2. Preliminaries

Ann-dimensional differentiable manifold M is called an LP-Sasakian manifold  [1, 2] if it admits a (1,1) tensor field ϕ, a contravariant vector field ξ, a 1-form η, and a Lorentzian metric g which satisfy(2.1)η(ξ)=-1,ϕ2X=X+η(X)ξ,g(ϕX,ϕY)=g(X,Y)+η(X)η(Y),g(X,ξ)=η(X),∇Xξ=ϕX,(∇Xϕ)Y=g(X,Y)ξ+η(Y)X+2η(X)η(Y)ξ,
where ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric g.It can be easily seen that, in an LP-Sasakian manifold, the following relations hold:(2.2)ϕξ=0,η(ϕX)=0,rank⁡ϕ=n-1.
If we put(2.3)Φ(X,Y)=g(X,ϕY),
for any vector fields X and Y, then the tensor field Φ(X,Y) is a symmetric (0, 2) tensor field [1]. Also, since the 1-form η is closed in an LP-Sasakian manifold, we have  [1, 4](2.4)(∇Xη)(Y)=Φ(X,Y),Φ(X,ξ)=0,
for any vector fields X and Y.Also in an LP-Sasakian manifold, the following relations hold [3, 4]:(2.5)g(R(X,Y)Z,ξ)=η(R(X,Y)Z)=g(Y,Z)η(X)-g(X,Z)η(Y),(2.6)R(ξ,X)Y=g(X,Y)ξ-η(Y)X,(2.7)R(X,Y)ξ=η(Y)X-η(X)Y,(2.8)S(X,ξ)=(n-1)η(X),(2.9)S(ϕX,ϕY)=S(X,Y)+(n-1)η(X)η(Y),
for any vector fields X,Y, and Z, where R is the Riemannian curvature tensor and S is the Ricci tensor of M.An LP-Sasakian manifoldM is said to be Einstein if its Ricci tensor S is of the form(2.10)S(X,Y)=ag(X,Y),
for any vector fields X and Y, where a is a function on M.In [6], Pokhariyal and Mishra have defined the curvature tensor W2, given by(2.11)W2(X,Y,U,V)=R(X,Y,U,V)+1n-1[g(X,U)S(Y,V)-g(Y,U)S(X,V)],
where S is a Ricci tensor of type (0, 2).Now, consider an LP-Sasakian manifold satisfyingW2=0; then, (2.11) becomes(2.12)R(X,Y,U,V)=1n-1[g(Y,U)S(X,V)-g(X,U)S(Y,V)].
Taking X=U=ξ in (2.12) and using (2.7) and (2.8), we have(2.13)S(Y,V)=(n-1)g(Y,V).
Therefore, M is an Einstein manifold.Again using (2.13) in (2.12), we get(2.14)R(X,Y,U,V)=[g(Y,U)g(X,V)-g(X,U)g(Y,V)].Corollary 2.1.
An LP-Sasakian manifold satisfyingW2=0 is a space of constant curvature −1,  that is, it is locally isometric to the hyperbolic space.Definition 2.2.
An LP-Sasakian manifold is calledW2-semisymmetric if it satisfies
(2.15)R(X,Y)⋅W2=0,
where R(X,Y) is to be considered as a derivation of the tensor algebra at each point of the manifold for tangent vectors X and Y.
It can be easily shown that in an LP-Sasakian manifold theW2-curvature tensor satisfies the condition
(2.16)η(W2(X,Y)Z)=0.Theorem 2.3.
AW2-semisymmetric LP-Sasakian manifold M is an Einstein manifold.Proof.
SinceR(X,Y)·W2=0, we have
(2.17)R(X,Y)W2(U,V)Z-W2(R(X,Y)U,V)Z-W2(U,R(X,Y)V)Z-W2(U,V)R(X,Y)Z=0.
Putting X=ξ in (2.17) and then taking the inner product with ξ, we obtain
(2.18)g(R(ξ,Y)W2(U,V)Z,ξ)-g(W2(R(ξ,Y)U,V)Z,ξ)-g(W2(U,R(ξ,Y)V)Z,ξ)-g(W2(U,V)R(ξ,Y)Z,ξ)=0.
Using (2.6) in (2.18), we obtain
(2.19)-g(Y,W2(U,V)Z)-η(W2(U,V)Z)η(Y)-g(Y,U)η(W2(ξ,V)Z)-g(Y,V)η(W2(U,ξ)Z)-g(Y,Z)η(W2(U,V)ξ)+η(U)η(W2(Y,V)Z)+η(V)η(W2(U,Y)Z)+η(Z)η(W2(U,V)Y)=0.
By using (2.16) in (2.19), we get
(2.20)W2(U,V,Z,Y)=0.
In view of (2.11) and (2.20), it follows that
(2.21)R(U,V,Z,Y)=1n-1[g(V,Z)S(U,Y)-g(U,Z)S(V,Y)].
Contracting (2.21), we have
(2.22)S(V,Z)=(n-1)g(V,Z).
Therefore, M is an Einstein manifold.Again using (2.22) in (2.12), we get(2.23)R(U,V,Z,Y)=[g(V,Z)g(U,Y)-g(U,Z)g(V,Y)].Corollary 2.4.
AW2-semisymmetric LP-Sasakian manifold is a space of constant curvature −1, that is, it is locally isometric to the hyperbolic space.


## 3. LP-Sasakian Manifolds SatisfyingP(X,Y)·W2=0

The projective curvature tensorP is defined as [10](3.1)P(X,Y)Z=R(X,Y)Z-1n-1[S(Y,Z)X-S(X,Z)Y].
Using (2.6) and (2.8),   (3.1) reduces to(3.2)P(ξ,Y)Z=g(Y,Z)ξ-1n-1S(Y,Z)ξ.
Let us suppose that in an LP-Sasakian manifold (3.3)P(X,Y)⋅W2=0.
This condition implies that(3.4)P(X,Y)W2(U,V)Z-W2(P(X,Y)U,V)Z-W2(U,P(X,Y)V)Z-W2(U,V)P(X,Y)Z=0.
Putting X=ξ in (3.4) and then taking the inner product with ξ, we obtain(3.5)g(P(ξ,Y)W2(U,V)Z,ξ)-g(W2(P(ξ,Y)U,V)Z,ξ)-g(W2(U,P(ξ,Y)V)Z,ξ)-g(W2(U,V)P(ξ,Y)Z,ξ)=0.
Using (3.2) in (3.5), we obtain(3.6)-g(Y,W2(U,V)Z)-g(Y,U)η(W2(ξ,V)Z)-g(Y,V)η(W2(U,ξ)Z)-g(Y,Z)η(W2(U,V)ξ)+1n-1[S(Y,W2(U,V)Z)+S(Y,U)η(W2(ξ,V)Z)+S(Y,V)η(W2(U,ξ)Z)+S(Y,Z)η(W2(U,V)ξ)]=0.
By using (2.16) in (3.6), we get(3.7)g(Y,W2(U,V)Z)-1n-1S(Y,W2(U,V)Z)=0.
Taking U=Z=ξ in (3.7) and using (2.11) and (2.6), we have(3.8)S(QY,V)=2(n-1)S(Y,V)-(n-1)2g(Y,V).
This implies that(3.9)QY=(n-1)Y.
From this, we get(3.10)S(Y,V)=(n-1)g(Y,V).
Thus, we can state the following.Theorem 3.1.
An LP-Sasakian manifoldM satisfying the condition P(X,Y)·W2=0 is an Einstein manifold.


## 4. LP-Sasakian Manifold SatisfyingM̃(X,Y)·W2=0

TheM-projective curvature tensor M̃ is defined as [11](4.1)M̃(X,Y)Z=R(X,Y)Z-12(n-1)[S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY].
Using (2.6) and (2.8),   (4.1) reduces to(4.2)M̃(ξ,Y)Z=12[g(Y,Z)ξ-η(Z)Y]-12(n-1)[S(Y,Z)ξ-η(Z)QY].
Suppose that in an LP-Sasakian manifold (4.3)M̃(X,Y)⋅W2=0.
This condition implies that(4.4)M̃(X,Y)W2(U,V)Z-W2(M̃(X,Y)U,V)Z-W2(U,M̃(X,Y)V)Z-W2(U,V)M̃(X,Y)Z=0.
Putting X=ξ in (4.4) and then taking the inner product with ξ, we obtain(4.5)g(M̃(ξ,Y)W2(U,V)Z,ξ)-g(W2(M̃(ξ,Y)U,V)Z,ξ)-g(W2(U,M̃(ξ,Y)V)Z,ξ)-g(W2(U,V)M̃(ξ,Y)Z,ξ)=0.
Using (4.2) in (4.5), we obtain(4.6)12[-g(Y,W2(U,V)Z)-g(Y,U)η(W2(ξ,V)Z)-g(Y,V)η(W2(U,ξ)Z)-g(Y,Z)η(W2(U,V)ξ)+η(U)η(W2(Y,V)Z)+η(V)η(W2(U,Y)Z)+η(Z)η(W2(U,V)Y)]+12(n-1)[S(Y,W2(U,V)Z)+S(Y,U)η(W2(ξ,V)Z)+S(Y,V)η(W2(U,ξ)Z)+S(Y,Z)η(W2(U,V)ξ)-η(U)η(W2(QY,V)Z)-η(V)η(W2(U,QY)Z)-η(Z)η(W2(U,V)QY)]=0.
By using (2.16) in (4.6), we get(4.7)12g(Y,W2(U,V)Z)-12(n-1)S(Y,W2(U,V)Z)=0.
Taking U=Z=ξ in (4.7) and using (2.11) and (2.6), we have(4.8)S(QY,V)=2(n-1)S(Y,V)-(n-1)2g(Y,V).
This implies that(4.9)QY=(n-1)Y,
which gives(4.10)S(Y,V)=(n-1)g(Y,V).
Thus, we can state the following.Theorem 4.1.
An LP-Sasakian manifoldM satisfying the condition M̃(X,Y)·W2=0 is an Einstein manifold.


## 5. LP-Sasakian Manifolds SatisfyingC(X,Y)·W2=0

The conformal curvature tensorC is defined as [12](5.1)C(X,Y)Z=R(X,Y)Z-1n-2[S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY]+r(n-1)(n-2)[g(Y,Z)X-g(X,Z)Y].
Using (2.6) and (2.8), (5.1) reduces to(5.2)C(ξ,Y)Z=1-n+r(n-1)(n-2)[g(Y,Z)ξ-η(Z)Y]-1n-2[S(Y,Z)ξ-η(Z)QY].
Now consider an LP-Sasakian manifold satisfying (5.3)C(X,Y)⋅W2=0.
This condition implies that(5.4)C(X,Y)W2(U,V)Z-W2(C(X,Y)U,V)Z-W2(U,C(X,Y)V)Z-W2(U,V)C(X,Y)Z=0.
Putting X=ξ in (5.4) and then taking the inner product with ξ, we obtain(5.5)g(C(ξ,Y)W2(U,V)Z,ξ)-g(W2(C(ξ,Y)U,V)Z,ξ)-g(W2(U,C(ξ,Y)V)Z,ξ)-g(W2(U,V)C(ξ,Y)Z,ξ)=0.
Using (5.2) in (5.5), we obtain(5.6)1-n+r(n-1)(n-2)[-g(Y,W2(U,V)Z)-g(Y,U)η(W2(ξ,V)Z)-g(Y,V)η(W2(U,ξ)Z)-g(Y,Z)η(W2(U,V)ξ)-η(Y)η(W2(U,V)Z)+η(U)η(W2(Y,V)Z)+η(V)η(W2(U,Y)Z)+η(Z)η(W2(U,V)Y)]+1n-2[S(Y,W2(U,V)Z)+S(Y,U)η(W2(ξ,V)Z)+S(Y,V)η(W2(U,ξ)Z)+S(Y,Z)η(W2(U,V)ξ)+(n-1)η(Y)η(W2(U,V)Z)-η(U)η(W2(QY,V)Z)-η(V)η(W2(U,QY)Z)-η(Z)η(W2(U,V)QY)]=0.
By using (2.16) in (5.6), we get(5.7){1-n+r(n-1)(n-2)}g(Y,W2(U,V)Z)-1n-2S(Y,W2(U,V)Z)=0.
Taking U=Z=ξ in (5.7) and then using (2.11) and (2.6), we have(5.8)S(QY,V)={2+n(n-3)+rn-1}S(Y,V)-(n-1+r)g(Y,V).
This implies that(5.9)QY=(n-1)Y,and it follows that(5.10)S(Y,V)=(n-1)g(Y,V).
Thus, we can state the following.Theorem 5.1.
An LP-Sasakian manifoldM satisfying the condition C(X,Y)·W2=0 is an Einstein manifold.



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*Source: 161523-2011-08-15.xml*