1. Introduction
The notion of Lorentzian concircular structure manifolds (briefly (LCS)n-manifold) 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] .
Shaikh and Jana in 2005 [9] introduced and studied a tensor field, called Pseudo-Quasi-Conformal curvature tensor
on a Riemannian manifold of dimension (
). 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.
In this paper, we consider a (LCS)n-manifold satisfying certain conditions on the 3-dimensional pseudo-quasi-conformal curvature tensor. In section 2, we have the preliminaries. In Section 3, we studied a 3-dimensional pseudo-quasi-conformally 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
; 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
). 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.
2. 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
such that for each point
, the tensor
is a non-degenerate inner product of signature
, where
denotes the tangent vector space of M at p and R is the real number space. A non zero vector
is said to be timelike (resp., non-spacelike, null, space like) if it satisfies
[12] .
Definition 2.1. In a Lorentzian manifold
a vector field P defined by
for any
is said to be a concircular vector field if
where
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
is called the characteristic vector field of the manifold. Then we have
(2.1)
Since
is a unit concircular vector field, it follows that there exist a non-zero 1-form
such that for
(2.2)
the equation of the following form holds
(2.3)
for all vector field
, where
denotes the operator of covariant differentiation with respect to the Lorentzian metric g and
is a non-zero scalar function satisfies
(2.4)
being a certain scalar function given by
. Let us put
(2.5)
then from (2.3) and (2.5), we have
(2.6)
which tell us that
is a symmetric
tensor. thus the Lorentzian manifold M together with the unit timelike concircular vector field
, its associated 1-form
and
-type tensor field
is said to be a Lorentzian concircular structure manifold (briefly (LCS)n-manifold) [1] . Especially, we take
, then we can obtain the LP-Sasakian structure of Matsumoto [2] . In a (LCS)n-manifold, the following relation hold [1] .
(2.7)
(2.8)
(2.9)
In a three dimensional (LCS)n-manifolds, the following relation holds [13] .
(2.10)
(2.11)
(2.12)
(2.13)
The pseudo-quasi-conformal curvatur tesor
is defined by [14] .
(2.14)
where
,
and r are the curvature tensor, the Ricci tensor, the symmetric endomorphism of the tangent space at each point corresponding to the Ricci tensor S and the scalar curvature, i.e,
and
are real constants such that
.
In particular, if (1)
; (2)
; (3)
; (4)
; then
reduces to the projective
curvature tensor; quasi-conformal curvature tensor; conformal curvature tensor and concircular curvature tensor, respectively.
3. 3-Dimensional Pseudo-quasi-conformally Flat (LCS)n-Manifold
Definition 3.2. An n-dimensional (
) (LCS)n-manifold M is called a pseudo-quasi-conformally flat, if the condition
, for all
Let us consider the three dimensional (LCS)n-manifold M is a pseudo-quasi-conformally flat, then from (2.6), (2.7) and (2.8) relation to (2.10) that
(3.1)
Putting
in (3.1) and by using (2.10), (2.11), we get
(3.2)
again plugging
in (3.2) by using (2.12) and taking inner product with respect to W, we get
(3.3)
where
and
Hence we can state the following theorem.
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).
4. 3-Dimensional Pseudo-quasi-conformal (LCS)n-Manifold Satisfies
Let us consider a 3-dimensional Riemannian manifold which satisfies the condition
(4.1)
Then we have
(4.2)
Put
in (4.2) by using (2.10), (2.11), (2.12) and (2.14) and also on plugging
, we get
(4.3)
by using (2.13) in (4.3), we get
(4.4)
where
and
Hence we can state the following theorem.
Theorem 4.2. Let a 3-dimensional pseudo-quasi-conformal (LCS)n-manifold satisfying
is an η-Einstein manifold.
5. On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)n-Manifold
Definition 5.3. An (LCS)n-manifold is said to be pseudo-quasi-conformal ϕ-symmetric if the condition
(5.1)
for any vector field
.
Let us consider 3-dimensional (LCS)n-manifold of a pseudo-quasi-conformal curvature tensor has the following from (2.6), we get
(5.2)
which follows that
(5.3)
By virtue of (2.10), (2.11) and (2.14) and contracting we get
(5.4)
On plugging
in (5.4), gives
(5.5)
If the manifold has a constant scalar curvature r, then
.
Hence the Equation (5.5) turns into
(5.6)
Hence we can state the following:
Theorem 5.3. Let M be a 3-dimensional pseudo-quasi-conformal ϕ-symmetric (LCS)n-manifold with constant scalar curvature, then the manifold is reduces to a Einstein manifold.
6. 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
(6.1)
for any vector field
. If
then 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
(6.2)
from which it follows that
(6.3)
By virtue of (2.10), (2.11) and (2.14) and contracting, also plugging
, we get
(6.4)
again putting
in (6.4), we get
(6.5)
If the manifold has a constant scalar curvature r, then
. Hence the Equation (6.5) turns into
(6.6)
Using (6.6) in (6.1), we get
(6.7)
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.