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The purp ose of the present paper is to sudy the pseudo-quasi-conformally flat, C·S=0, pseudo-quasi-conformal ?-symmetric and pseudo-quasi-conformal ?-recurrent 3-dimensional (LCS)n maifolds.

The notion of Lorentzian concircular structure manifolds (briefly (LCS)_{n}-manifold) was introduced by [

Shaikh and Jana in 2005 [

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 C ˜ ⋅ S = 0 ; 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 p ≠ 0 ). 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.

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 g p : T p M × T p M → ℜ is a non-degenerate inner product of signature ( − , + , ⋯ , + ) , where T p M denotes the tangent vector space of M at p and R is the real number space. A non zero vector v ∈ T p M is said to be timelike (resp., non-spacelike, null, space like) if it satisfies g p ( v , v ) < 0 [

Definition 2.1. In a Lorentzian manifold ( M , g ) a vector field P defined by

g ( X , P ) = A ( X ) ,

for any X ∈ χ ( M ) is said to be a concircular vector field if

( ∇ X A ) ( Y ) = α g ( X , Y ) + w ( X ) A ( Y ) ,

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

g ( ξ , ξ ) = − 1. (2.1)

Since ξ is a unit concircular vector field, it follows that there exist a non-zero 1-form η such that for

g ( X , ξ ) = η ( X ) , (2.2)

the equation of the following form holds

( ∇ X η ) ( Y ) = α { g ( X , Y ) + η ( X ) η ( Y ) } ( α ≠ 0 ) , (2.3)

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

∇ X α = ( X α ) = d α ( X ) = ρ η ( X ) , (2.4)

ρ being a certain scalar function given by ρ = − ( ξ α ) . Let us put

ϕ X = 1 α ∇ X ξ , (2.5)

then from (2.3) and (2.5), we have

ϕ X = X = η ( X ) ξ , (2.6)

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) [_{n}-manifold, the following relation hold [

η ( ξ ) = − 1 , ϕ ξ = 0 , η ( ϕ X ) = 0 , (2.7)

g ( ϕ X , ϕ Y ) = g ( X , Y ) + η ( X ) η ( Y ) , (2.8)

η ( R ( X , Y ) Z ) = ( α 2 − ρ ) [ g ( Y , Z ) η ( X ) − g ( X , Z ) η ( Y ) ] . (2.9)

In a three dimensional (LCS)_{n}-manifolds, the following relation holds [

R ( X , Y ) Z = g ( Y , Z ) Q X − g ( X , Z ) Q Y + S ( Y , Z ) X − S ( X , Z ) Y − r 2 [ g ( Y , Z ) X − g ( X , Z ) Y ] , (2.10)

S ( X , ξ ) = 2 ( α 2 − ρ ) η ( X ) , (2.11)

Q ξ = 2 ( α 2 − ρ ) ξ , (2.12)

Q X = [ − ( α 2 − ρ ) + r 2 ] X + [ − 3 ( α 2 − ρ ) + r 2 ] η ( X ) ξ . (2.13)

The pseudo-quasi-conformal curvatur tesor C ˜ is defined by [

C ˜ ( X , Y ) Z = ( p + d ) R ( X , Y ) Z + ( q − d n − 1 ) [ S ( Y , Z ) X − S ( X , Z ) Y ] + q [ g ( Y , Z ) Q X − g ( X , Z ) Q Y ] − r n ( n − 1 ) { p + 2 ( n − 1 ) q } [ g ( Y , Z ) X − g ( X , Z ) Y ] , (2.14)

where X , Y , Z ∈ χ ( M ) , R , S , Q 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, g ( Q X , Y ) = S ( X , Y ) and p , q , d are real constants such that p 2 + q 2 + d 2 > 0 .

In particular, if (1) p = q = 0 , d = 1 ; (2) p ≠ 0 , q ≠ 0 , d = 0 ; (3) p = 1 , q = − 1 n − 2 , d = 0 ; (4) p = 1 , q = d = 0 ; then C ˜ reduces to the projective

curvature tensor; quasi-conformal curvature tensor; conformal curvature tensor and concircular curvature tensor, respectively.

Definition 3.2. An n-dimensional ( n ≥ 3 ) (LCS)_{n}-manifold M is called a pseudo-quasi-conformally flat, if the condition C ˜ ( X , Y ) Z = 0 , for all X , Y , Z ∈ T p M .

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

( p + d ) R ( X , Y ) Z + ( q − d 2 ) [ S ( Y , Z ) X − S ( X , Z ) Y ] + q [ g ( Y , Z ) Q X − g ( X , Z ) Q Y ] − r 6 { p + 4 q } [ g ( Y , Z ) X − g ( X , Z ) Y ] = 0 , (3.1)

Putting Z = ξ in (3.1) and by using (2.10), (2.11), we get

( p + d + q ) [ η ( Y ) Q X − η ( X ) Q Y ] + [ 2 ( α 2 − ρ ) ( p + d 2 + q ) − ( 2 p + 3 d + 4 q ) r 2 ] η ( Y ) X − η ( X ) Y = 0, (3.2)

again plugging Y = ξ in (3.2) by using (2.12) and taking inner product with respect to W, we get

S ( X , W ) = 1 p + d + q { A g ( X , W ) + B η ( X ) η ( W ) } . (3.3)

where

A = { p [ 2 ( α 2 − ρ ) − r 3 ] + d [ ( α 2 − ρ ) − r 2 ] + 2 q [ ( α 2 − ρ ) − r 3 ] } ,

and B = { p [ 4 ( α 2 − ρ ) − r ] + 3 d [ ( α 2 − ρ ) − r 2 ] + 2 q [ 4 ( α 2 − ρ ) − r ] } .

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 [

Let us consider a 3-dimensional Riemannian manifold which satisfies the condition

C ˜ ( X , Y ) ⋅ S = 0. (4.1)

Then we have

S ( C ˜ ( X , Y ) U , V ) + S ( U , C ˜ ( X , Y ) V ) = 0. (4.2)

Put X = ξ in (4.2) by using (2.10), (2.11), (2.12) and (2.14) and also on plugging U = ξ , we get

4 ( p + d + q ) ( α 2 − ρ ) 2 η ( V ) η ( Y ) + ( p + q + d ) S ( Q Y , V ) − r 6 ( 4 p + 3 d + 4 q ) S ( Y , V ) − 2 ( α 2 − ρ ) ( p + d + q ) η ( V ) η ( Q Y ) + r 3 ( α 2 − ρ ) ( 4 p + 3 d + 4 q ) g ( Y , V ) = 0, (4.3)

by using (2.13) in (4.3), we get

S ( V , Y ) = A g ( V , Y ) + B η ( V ) η ( Y ) . (4.4)

where

A = 2 r ( α 2 − ρ ) ( 4 p + 3 d + 4 q ) 6 ( α 2 − ρ ) ( p + d + q ) + r ( p + q ) ,

and B = 6 ( α 2 − ρ ) [ r − 6 ( α 2 − ρ ) ] ( p + d + q ) 6 ( α 2 − ρ ) ( p + d + q ) + r ( p + q ) .

Hence we can state the following theorem.

Theorem 4.2. Let a 3-dimensional pseudo-quasi-conformal (LCS)_{n}-manifold satisfying C ˜ ⋅ S = 0 is an η-Einstein manifold.

Definition 5.3. An (LCS)_{n}-manifold is said to be pseudo-quasi-conformal ϕ-symmetric if the condition

ϕ 2 ( ( ∇ W C ˜ ) ( X , Y ) Z ) = 0 , (5.1)

for any vector field X , Y , Z , W ∈ T p M .

Let us consider 3-dimensional (LCS)_{n}-manifold of a pseudo-quasi-conformal curvature tensor has the following from (2.6), we get

( ∇ W C ˜ ) ( X , Y ) Z + η ( ( ∇ W C ˜ ) ( X , Y ) Z ) ξ = 0. (5.2)

which follows that

g ( ( ∇ W C ˜ ) ( X , Y ) Z , U ) + η ( ( ∇ W C ˜ ) ( X , Y ) Z ) η ( U ) = 0. (5.3)

By virtue of (2.10), (2.11) and (2.14) and contracting we get

( q − d ) ( ∇ W S ) ( Y , Z ) + d r W 6 [ ( 2 p + 2 q + 3 d ) g ( Y , Z ) ( 4 p + 3 d + 4 q ) η ( Y ) η ( Z ) ] + ( − ( p + d ) + q ) η ( Z ) ( ∇ W S ) ( Y , ξ ) + ( − p − 3 d 2 + q ) η ( Y ) ( ∇ W S ) ( Z , ξ ) = 0.

(5.4)

On plugging Z = ξ in (5.4), gives

S ( Y , W ) = 2 ( α 2 − ρ ) g ( Y , W ) − 1 p α ( p + q ) η ( Y ) d r W 3 . (5.5)

If the manifold has a constant scalar curvature r, then d r W = 0 .

Hence the Equation (5.5) turns into

S ( Y , W ) = 2 ( α 2 − ρ ) g ( Y , W ) . (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.

Definition 6.4. An (LCS)_{n}-manifold is said to be pseudo-quasi-conformal ϕ-recurrent if

ϕ 2 ( ( ∇ W C ˜ ) ( X , Y ) Z ) = A ( W ) C ˜ ( X , Y ) Z , (6.1)

for any vector field X , Y , Z , W ∈ T p M . If A ( W ) = 0 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

( ∇ W C ˜ ) ( X , Y ) Z + η ( ( ∇ W C ˜ ) ( X , Y ) Z ) ξ = A ( W ) C ˜ ( X , Y ) Z , (6.2)

from which it follows that

g ( ( ∇ W C ˜ ) ( X , Y ) Z , U ) + η ( ( ∇ W C ˜ ) ( X , Y ) Z ) η ( U ) = A ( W ) g ( C ˜ ( X , Y ) Z , U ) . (6.3)

By virtue of (2.10), (2.11) and (2.14) and contracting, also plugging Z = ξ , we get

S ( Y , W ) = 2 ( α 2 − ρ ) g ( Y , W ) − 2 3 [ r ( 2 p + 3 d − q ) + 6 ( p + q ) ( α 2 − ρ ) ( 2 p + d + 2 q ) α ] η ( Y ) A ( W ) + d r W η ( Y ) 2 ( p + d ) α ( 2 p + d + 2 q ) , (6.4)

again putting Y = ξ in (6.4), we get

A ( W ) = 3 ( p + d ) r ( q − 2 p − 3 d ) − 6 ( p + q ) ( α 2 − ρ ) d r W . (6.5)

If the manifold has a constant scalar curvature r, then d r W = 0 . Hence the Equation (6.5) turns into

A ( W ) = 0. (6.6)

Using (6.6) in (6.1), we get

ϕ 2 ( ( ∇ W C ˜ ) ( X , Y ) Z ) = 0. (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.

The authors declare no conflicts of interest regarding the publication of this paper.

Murthy, B. and Venkatesha (2019) On 3-Dimensional Pseudo-Quasi-Conformal Curvature Tensor on (LCS)_{n}-Manifolds. Open Access Library Journal, 6: e5474. https://doi.org/10.4236/oalib.1105474