ON SOME CLASSES OF CONCIRCULAR CURVATURE TENSOR ON LORENTZIAN PARA-SASAKIAN MANIFOLDS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The present paper deals with the study of different classes of concircular curvature tensor on Lorentzian para-Sasakian manifold admitting a quarter-symmetric metric connection.


INTRODUCTION
K. Matsumoto [8] introduced the concept of Lorentzian para-Sasakian manifolds in 1989. Late, the same concept was independently introduced by I. Mihai and R. Rosca [10]. The Lorentzian para-Sasakian manifolds have also been studied by K. Matsumoto and I. Mihai [9], U. C. De and A. A. Shaikh [11] and several others such as ( [12], [14], [15]). K. Matsumoto and I. Mihai obtained some interesting results for conformally recurrent and conformally symmetric P−Sasakian manifold in [1]. In 1924, the notion of semi-symmetric connection on a differentiable manifold was firstly introduced by Friedmann and Schouten [18]. A linear connection∇ on a differentiable manifold M is said to be a semi-symmetric connection if the torsion tensor T of the connection satisfies where η is a 1-form and ξ is a vector field defined by η(U) = g(U, ξ ), for all vector fields U on Γ(T M), Γ(T M) is the set of all differentiable vector fields on M. A. Barman ([2], [3]) studied para-Sasakian manifold admitting semi-symmetric metric and non metric connection.
On the other hand, in 1975, Golab [6] intoduced and studied quarter-symmetric connection in differentiable manifolds along with affine connections.
A liner connection∇ on an n-dimensional Riemannian manifold (M, g) is called a quartersymmetric connection [6] if its torsion tensor T satisfies where φ is a (1,1) tensor field.
The quarter-symmetric connection generalizes the notion of the semi-symmetric connection because if we assume φU = U in the above equation, the quarter-symmetric connection reduces to the semi-symmetric connection [18].
Moreover, if a quarter-symmetric connection∇ satisfies the condition for all U,V,W on Γ(T M), then∇ is said to be a quarter-symmetric metric connection.
Venkatesha and C.S. Bagewadi [19] obtain some interesting results on concircular φ -recurrent Lorentzian para-Sasakian manifolds which generalize the concept of locally concircular φsymmetric Lorentzian para-Sasakian manifolds. If curvature tensor R of Riemannian manifold M satisfies ∇R = 0, then M is called locally symmetric. Later, many geometers have considered semi-symmetric spaces as a generalization of locally symmetric spaces. A Riemannian manifold M is said to be semi-symmetric if its curvature tensor R satisfies R(U,V ).R = 0, where R(U,V ) acts on R as a derivation and also it is called Ricci-semisymmetric manifold if the relation R(U,V ).S = 0 holds, where R(U,V ) the curvature operator.
A transformation which transforms every geodesic circle of a Riemannian manifold M into a geodesic circle, is known as concircular transformation ( [7], [16]), where geodesic circle means a curve in M whose first curvature is constant and second curvature is identically zero.
A concircular transformation is always a conformal transformation [7]. Thus the geometry of concircular transformations is a generalization of inversive geometry in the sense that the change of metric is more general than that induced by a circle preserving diffeomorphism (see also [5]). An invariant of a concircular transformation is the concircular curvature tensor C, which is defined by ( [16], [17]) Using (1.3), we obtain where U,V,W, Z ∈ Γ(T M) and r is the scalar curvature on Lorentzian para-Sasakian manifolds.
Riemannian manifolds with vanishing concircular curvature tensor are of constant curvature.
Thus the concircular curvature tensor is a measure of the failure of a Riemannian manifold to be of constant curvature.
In this paper, we study a type of quarter-symmetric metric connection on Lorentzian para-Sasakian manifolds. The paper is organized as follows: After introduction section two is equipped with some prerequisites of a Lorentzian para-Sasakian manifold. In section three, curvature tensor and Ricci tensor of Lorentzian para-Sasakian manifold with respect to the quarter-symmetric metric connection are given. Section four is devoted to study ξ -concircularly flat Lorentzian para-Sasakian manifold with respect to the quarter-symmetric metric connection. Quasi-concircularly flat and φ -concircularly flat Lorentzian para-Sasakian manifolds with respect to the quarter-symmetric metric connection have been studied in section five and six respectively. In next section, we investigate Ricci-semisymmetric manifolds with respect to the quarter-symmetric metric connection of a Lorentzian para-Sasakian manifold.

PRELIMINARIES
An n-dimensional differentiable manifold M is said to be a Lorentzian almost para-contact manifold, if it admits an almost para-contact structure (φ , ξ , η, g) consisting of a (1, 1) tensor field φ , vector field ξ , 1-form η and a Lorentzian metric g satisfying for any vector fields U, V on M. Such a manifold M is termed as Lorentzian para-contact manifold and the structure (φ , ξ , η, g) a Lorentzian para-contact structure [8].
If moreover (φ , ξ , η, g) satisfies the conditions for U,V tangent to M, then M is called a Lorentzian para-Sasakian manifold or briefly LP-Sasakian manifold, where ∇ denotes the covariant differentiation with respect to Lorentzian metric g.
Moreover, the curvature tensor R, the Ricci tensor S and the Ricci operator Q in a Lorentzian para-Sasakian manifold M with respect to the Levi-Civita connection ∇ satisfies the following relations [13] (2.7) for all vector fields U,V,W ∈ Γ(T M).
Definition 2.1. A Lorentzian para-Sasakian manifold M is said to be an η−Einstein manifold [13] if its Ricci tensor S of the Levi-Civita connection is of the form where a and b are smooth functions on the manifold M.

CURVATURE TENSOR OF LORENTZIAN PARA-SASAKIAN MANIFOLD WITH RESPECT TO THE QUARTER-SYMMETRIC METRIC CONNECTION
A relation between the quarter-symmetric metric connection∇ and the Levi-Civita connection ∇ in an n-dimensional Lorentzian para-Sasakian manifold M is given by [15] (3.1) The curvature tensorR of a Lorentzian para-Sasakian manifold M with respect to the quartersymmetric metric connection∇ is defined by W is the Riemannian curvature tensor with respect to the Levi-Civita connection ∇.
The Ricci tensorS and the Scalar curvaturer in a Lorentzian para-Sasakian manifold M with respect to the quarter-symmetric metric connection∇ are defined by where {e 1 , e 2 , ...., e n−1 , e n = ξ } be a local orthonormal basis of vector fields in M and ε i = g (e i , e i ) .
Again contracting V and W in (3.6) , we get for all U,V,W ∈ Γ(T M) whereR is the curvature tensor andr is the scalar curvature of M with respect to the quarter-symmetric metric connection∇ Putting W = ξ in (4.1) and using (3.8) and (4.2) , we have Putting U = ξ in (4.3) and using (2.1) , we have Taking inner product of (4.4) with W and replacing V by QV , we have Thus we can state the following: From equation (4.1) , we have summing over i = 1 to n − 1, we obtain On LP-Sasakian manifold it can be verify that So by virtue of (5.5) , (5.6) and (5.7) , the equation (5.4) takes the form where a = r(n−2) n(n−1) and b = − r n(n−1) .
From which it follows that the manifold is an η−Einstein manifold with respect to the quarter-symmetric metric connection.
From which it follows that the manifold is an η−Einstein manifold with respect to the quarter-symmetric metric connection.
Hence we can state following: