Certain Results on Lorentzian Para-Kenmotsu Manifolds

abstract: The object of the present paper is to study Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection. First, we study Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection satisfying the curvature conditions R̄ · S̄ = 0 and S̄ · R̄ = 0. Next, we study φ-conformally flat, φ-conharmonically flat, φ-concircularly flat, φ-projectively flat and conformally flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection and it is shown that in each of these cases the manifold is a generalized η-Einstein manifold.


Introduction
Let (M, g), be an n-dimensional connected semi-Riemannian manifold of class C ∞ and ∇ be its Levi-Civita connection. The Riemannian-Christoffel curvature tensor R, the projective curvature tensor P , the concircular curvature tensor V , the conharmonic curvature tensor K and the conformal curvature tensor C of (M, g) are defined by [16]  [g(Y, Z)X − g(X, Z)Y ], respectively, where r is the scalar curvature, S is the Ricci tensor and Q is the Ricci operator such that S(X, Y ) = g(QX, Y ).
A linear connection∇ defined on (M, g) is said to be a quarter-symmetric connection [8] if its torsion tensor T where η is a 1-form and φ is a (1, 1)-tensor field. If moreover, a quarter-symmetric connection∇ satisfies the condition where X, Y, Z ∈ χ(M ) and χ(M ) is the set of all differentiable vector fields on M , then∇ is said to be a quarter-symmetric metric connection. If we change φX by X, then the connection is known as semi-symmetric metric connection [7]. Thus the notion of quarter-symmetric connection generalizes the notion of semi-symmetric 3 connection. A quarter-symmetric metric connection have been studied by many geometers in several ways to a different extent such as ( [1], [3], [5], [6], [9]- [12], [15]) and many others.
A relation between the quarter-symmetric metric connection∇ and the Levi-Civita connection ∇ in a Lorentzian para-Kenmotsu manifold M is given bȳ (1.8) The paper is organized as follows: In Section 2, we give a brief introduction of Lorentzian para-Kenmotsu manifolds. In Section 3, we establish the relation between the curvature tensors of the Riemannian connection and the quartersymmetric metric connection in a Lorentzian para-Kenmotsu manifold. Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection satisfying the curvature conditionsR·S = 0 andS ·R = 0 have studied in Sections 4 and 5 respectively. Sections 6,7,8,9 and 10 are devoted to study φ-conformally flat, φ-conharmonically flat, φ-concircularly flat, φ-projectively flat and conformally flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection, respectively.

Preliminaries
Let M be an n-dimensional Lorentzian metric manifold. If it is endowed with a structure (φ, ξ, η, g), where φ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form on M and g is a Lorentz metric, satisfying [2] for any X, Y on M , then it is called Lorentzian almost paracontact manifold. In the Lorentzian almost paracontact manifold, the following relations hold: where Φ(X, Y ) = g(X, φY ). If ξ is a killing vector field, the (para) contact structure is called K-(para) contact.
In such a case, we have for any vector fields X, Y on M.
Now, we define a new manifold called Lorentzian para-Kenmostu manifold: for any vector fields X, Y on M.
In the Lorentzian para-Kenmostu manifold, we have where ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric g. Further, on a Lorentzian para-Kenmotsu manifold M , the following relations hold: Example 2.2. We consider the 3-dimensional manifold where (x, y, z) are the standard coordinates in R 3 . Let e 1 , e 2 and e 3 be the vector fields on M 3 given by which are linearly independent at each point of M 3 and hence form a basis of T p M 3 . Define a Lorentzian metric g on M 3 as g(e 1 , e 1 ) = 1, g(e 2 , e 2 ) = 1, g(e 3 , e 3 ) = −1, g(e 1 , e 2 ) = g(e 1 , e 3 ) = g(e 2 , e 3 ) = 0.
By applying linearity of φ and g, we have for all X, Y ∈ χ(M ).
Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g. Then we have The Riemannian connection ∇ of the Lorentzian metric g is given by which is known as Koszul's formula. Using Koszul's formula, we can easily calculate X i e i = X 1 e 1 + X 2 e 2 + X 3 e 3 ∈ χ(M ).
Also, one can easily verify that for all X, Y, Z ∈ χ(M ). It is known that With the help of above expressions of the curvature tensors, it follows that Hence, the manifold (M 3 , φ, ξ, η, g) is a Lorentzian para-Kenmotsu manifold of constant curvature 1 and is locally isometric to the unit sphere S 3 (1).
where a and b are scalar functions on M . A Lorentzian para-Kenmotsu manifold M is said to be a generalized η-Einstein manifold if its Ricci tensor S is of the form where a, b, c are scalar functions on M and Φ(X, Y ) = g(φX, Y ). If c = 0, then the manifold reduces to an η-Einstein manifold.

Curvature tensor of Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection
The curvature tensorR of a Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection∇ is defined bȳ is the Riemannian curvature tensor of the connection ∇. Taking inner product of (3.2) with W , we havē Certain Results on Lorentzian Para-Kenmotsu Manifolds 7 where S andS are the Ricci tensors with respect to the connections ∇ and∇, respectively on M and ψ=trace φ.
where Q andQ are the Ricci operators with respect to the connections ∇ and∇, respectively on M . Contracting (3.4) over Y and Z, we get where r andr are the scalar curvatures with respect to the connections ∇ and∇, respectively on M . Writing two more equations by the cyclic permutations of X, Y and Z, we havē By adding (3.2), (3.7) and (3.8) and using the fact that Thus we can state that, if the manifold is a Lorentzian para-Kenmotsu, then the curvature tensor with respect to the quarter-symmetric metric connection satisfies the first Bianchi identity.
Combining equations (3.10)-(3.12), we havē Thus, in view of the equations (3.10)-(3.12), we can state the following theorem: Theorem 3.1. The curvature tensor of type (0, 4) of a Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection is (i) Skew-symmetric in first two slots, (ii) Skew-symmetric in last two slots, (iii) Symmetric in pair of slots.
for all X, Y on M .
4. Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection satisfyingR.S = 0 In this section we consider a Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection∇ satisfying the condition R(X, Y ).S = 0.
6. φ-conformally flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection Analogous to the equation (1.5), the conformal curvature tensorC with respect to the quarter-symmetric metric connection is defined bȳ whereR,S andr are the Riemannian curvature tensor, the Ricci tensor and the scalar curvature with respect to the connection∇, respectively on M .

φ-conharmonically flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection
Analogous to the equation (1.4), the conharmonic curvature tensorK with respect to the quarter-symmetric metric connection is defined bȳ 1) whereR,S andQ are the Riemannian curvature tensor, the Ricci tensor and the Ricci operator with respect to the connection∇, respectively on M . for all X, Y, Z on M .
2. An n-dimensional φ-conhamonically flat Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection is a generalized η-Einstein manifold with the scalar curvature r given by (7.10).
8. φ-concircularly flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection Analogous to the equation (1.3), the concircular curvature tensorV with respect to the quarter-symmetric metric connection is defined bȳ whereR andr are the Riemannian curvature tensor and the scalar curvature with respect to the connection∇, respectively on M . for all X, Y, Z on M .

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A. Haseeb and R. Prasad By virtue of (6.7)-(6.10), (6.13) and (6.14), the equation (8.6) becomes In view of (2.1) and (2.16), (8.7) takes the form Contracting (8.8) over Y and Z gives By using this value of r in (8.8), we get Thus we can state the following theorem: Theorem 8.2. An n-dimensional φ-concircularly flat Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection is a generalized η-Einstein manifold with the scalar curvature r given by (8.9).
9. φ-projectively flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection Analogous to the equation (1.2), the projective curvature tensorP with respect to the quarter-symmetric metric connection is defined bȳ whereR andQ are the Riemannian curvature tensor and the Ricci operator with respect to the connection∇, respectively on M . for all X, Y, Z on M .