Abstract
The purpose of the present paper is to sudy the pseudo-quasi-conformally flat,
Abstract
, pseudo-quasi-conformal ϕ-symmetric and pseudo-quasi-conformal ϕ-recurrent 3-dimensional (LCS) n maifolds.
Introduction
The notion of Lorentzian concircular structure manifolds (briefly (LCS) nmanifold) was introduced by [1] , investigated the application to the general theory of relativity and cosmology with an example, which generalizes the notion of LP-Sasakian manifold introduced by Matsumoto [2] . The notion of Riemannian manifold has been weakened by many authors in a different extent [3] - [8] .
Introduction
Shaikh and Jana in 2005 [9] introduced and studied a tensor field, called Pseudo-Quasi-Conformal curvature tensor C  on a Riemannian manifold of dimension ( 3 n ≥ ). This includes the Projective, Quasi-conformal, Weyl conformal and Concircular curvature tensor as special cases. Recently Kundu and others [10] [11] studied pseudo-quasi-conformal curvature tensor on P-Sasakian manifolds.
Introduction
In this paper, we consider a (LCS) n -manifold satisfying certain conditions on Open Access Library Journal the 3-dimensional pseudo-quasi-conformal curvature tensor. In section 2, we have the preliminaries. In Section 3, we studied a 3-dimensional pseudo-quasiconformally flat (LCS) n -manifold and proved that the manifold is η-Einstein and it is not a conformal curvature tensor. In section 4, we proved a 3-dimensional pseudo-quasi-conformal (LCS) n -manifold satisfies 0 C S ⋅ =  ; this reduces to η-Einstein and it is not a conformal curvature tensor. In section 5, we studied 3-dimensional pseudo-quasi-conformal ϕ-symmetric (LCS) n -manifold with constant scalar curvature and obtained the manifold is Einstein (provided 0 p ≠ ). In section 6, we studied a pseudo-quasi-conformal ϕ-recurrent (LCS) n -manifold with constant scalar curvature, which generalizes the notion of ϕ-symmetric (LCS) n -manifold.
Preliminaries
An n-dimensional Lorentzian manifold M is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type ( ) 0, 2 such that for each point p M ∈ , the tensor : 
Preliminaries
where α is a non-zero scalar and w is a closed 1-form. Let M be a Loretzian manifold admitting a unit timelike concircular vector field ξ is called the characteristic vector field of the manifold. Then we have ( )
Preliminaries
Since ξ is a unit concircular vector field, it follows that there exist a non-zero 1-form η such that for
Preliminaries
the equation of the following form holds
Preliminaries
for all vector field , X Y , where ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric g and α is a non-zero scalar function satisfies
Preliminaries
ρ being a certain scalar function given by
Preliminaries
then from (2.3) and (2.5), we have
Preliminaries
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 (briefly (LCS) n -manifold) [1] . Especially, we take 1 α = , then we can obtain the LP-Sasakian structure of Matsumoto [2] . In a (LCS) nmanifold, the following relation hold [1] .
Preliminaries
In a three dimensional (LCS) n -manifolds, the following relation holds [13] .
Preliminaries
The pseudo-quasi-conformal curvatur tesor C  is defined by [14] . 
Preliminaries
In particular, if (1) 0,
Preliminaries
; then C  reduces to the projective curvature tensor; quasi-conformal curvature tensor; conformal curvature tensor and concircular curvature tensor, respectively. 
3-Dimensional Pseudo-quasi-conformally Flat (LCS)n-Manifold
Putting Z ξ = in (3.1) and by using (2.10), (2.11), we get
3-Dimensional Pseudo-quasi-conformally Flat (LCS)n-Manifold
again plugging Y ξ = in (3.2) by using (2.12) and taking inner product with respect to W, we get
3-Dimensional Pseudo-quasi-conformally Flat (LCS)n-Manifold
Hence we can state the following theorem.
3-Dimensional Pseudo-quasi-conformally Flat (LCS)n-Manifold
Theorem 3.1. Let M be a 3-dimensional pseudo-quasi-conformally flat (LCS) n -manifold is an η-Einstein manifold, provided pseudo-quasi-conformal curvature tensor is not a conformal curvature tensor [15] (p = 1, q = −1 and d = 0). 
3-Dimensional Pseudo-quasi-conformal (LCS)n-Manifold Satisfies
Then we have 
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
Definition 5.
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
3. An (LCS) n -manifold is said to be pseudo-quasi-conformal ϕ-symmetric if the condition
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
for any vector field , , ,
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
Let us consider 3-dimensional (LCS) n -manifold of a pseudo-quasi-conformal curvature tensor has the following from (2.6), we get
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
which follows that
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
By virtue of (2.10), (2.11) and (2.14) and contracting we get
On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
If the manifold has a constant scalar curvature r, then 0 drW = . Hence the Equation ( 
3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)n-Manifold
Definition 6.4. An (LCS) n -manifold is said to be pseudo-quasi-conformal ϕ-recurrent if
3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)n-Manifold
for any vector field , , ,
3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)n-Manifold
pseudo-quasi-conformal ϕ-recurrent reduces to ϕ-symmetric. Let us consider a 3-dimensional pseudo-quasi-conformal ϕ-recurrent (LCS) n -manifold. Then by virtue of (2.6) and (6.1), we have
3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)n-Manifold
from which it follows that If the manifold has a constant scalar curvature r, then 0 drW = . Hence the Equation (6.5) turns into ( ) 0.
3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)n-Manifold
A W = (6.6) Using (6.6) in (6.1), we get
3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)n-Manifold
Hence we can state the following: Theorem 6.4. If M is a 3-dimensional pseudo-quasi-conformal ϕ-recurrent (LCS) n -manifold with constant scalar curvature, then it is ϕ-symmetric.

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