A NOTE ON CONCIRCULAR CURVATURE TENSOR IN LORENTZIAN ALMOST PARA-CONTACT GEOMETRY

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The paper deals with the notion of different classes of concircular curvature tensor on Lorentzian almost para-contact manifolds admitting a quarter-symmetric metric connection. In this paper we study Lorentzian almost para-contact manifolds with respect to the quarter-symmetric metric connection satisfying the curvature condition Z.S = 0. We also investigate the properties of ξ−concircularly flat, φ−concircularly flat and quasi-concircularly flat Lorentzian almost para-contact manifolds admitting a quarter-symmetric metric connection and it is found that in each of above cases the manifold is generalized η−Einstein manifold.


INTRODUCTION
Differential Geometry is the most important and very interesting branch of mathematics and physics from ancient days. In differential geometry, there are various topics which have very important applications in mathematics and physics both.
A transformation of an n−dimensional Riemannian manifold M, which transforms every geodesic circle of M into a geodesic circle is called a concircular transformation ( [10], [23]).
A concircular transformation is always a conformal transformation [10]. Here geodesic circle means a curve in M whose first curvature is constant and whose second curvature is identically zero. Thus the geometry of concircular transformations, that is, the concircular geometry, 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 [6]). An interesting invariant of a concircular transformation is the concircular curvature tensor Z. It is defined by ( [23], [24]) where R is the curvature tensor, r is the scalar curvature and In 1977, D. E. Blair stated that the tangent sphere bundle of a flat Riemannian manifold admits a contact metric structure satisfying R(U,V )ξ = 0 [5]. On the other hand, we know that on a manifold M equipped with a Sasakian stucture (η, ξ , ψ, g), In 1989, K. Matsumoto introduced the concept of Lorentzian para-Sasakian manifolds [11].
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 [8]. 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 studied para-Sasakian manifold admitting semi-symmetric metric and non metric connection.
On the other hand, in 1975, Golab [9] 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 [9] 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 [8].
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.
In 2008,Venkatesha and C.S. Bagewadi [22] obtain some interesting results on concircular φ -recurrent Lorentzian para-Sasakian manifolds which generalize the concept of locally con- In 2005, D. E. Blair and the authors [7] started a syudy of concircular curvature tensor of contact metric manifolds. B. J. Papantoniou [16] and D. Perrone [17] incuded the studies of contact metric manifolds satisfying R(X, ξ ).S = 0, where S is the Ricci tensor. Motivated by these studies, we study a concircular curvature tensor in Lorentzian almost para contact geometry.
In this paper, we study a type of quarter-symmetric metric connection on Lorentzian almost para-contact manifolds. The paper is organized as follows: After introduction the section two is equipped with some prerequisites of a Lorentzian para contact manifolds. In section three the curvature tensor of Lorentzian almost para contact manifold with respect to the quartersymmetric metric connection is defined. The section four is the study of Lorentzian almost para-contact manifolds with respect to the quarter-symmetric metric connection satisfying the curvature condition Z.S = 0. The section five is devoted to study of ξ −concircularly flat in a Lorentzian almost para-Sasakian manifolds with respect to the quarter-symmetric metric connection. In the section six, we define φ −concircularly flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection and in the last last section, we investigate the properties of quasi-concircularly flat Lorentzian almost para-Sasakian manifolds admitting a quarter-symmetric metric connection.

PRELIMINARIES
An n-dimensional differentiable manifold M is said to be an 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 field 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 [11].
Definition 2.1. A Lorentzian almost para contact mnifold M is called Lorentzian para-Sasakian manifold or briefly LP−Sasakian manifold if (φ , ξ , η, g) satisfies the conditions for U,V tangent to M, 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 for all vector fields U,V,W ∈ Γ(T M).

Definition 2.2.
A Lorentzian para-Sasakian manifold M is said to be an η−Einstein manifold [18] 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.
for any vector field U,V on M.
In the Lorentzian para-Kenmotsu manifold, we have where ∇ the operator of covariant differentiation with respect to the Lorentzian metric g.
R and i = 1, 2, . . . , m)} denote an n(= 2m + 1)-dimensional smooth manifold. Let us define the structure tensor φ as: If g represents the Lorentzian metric of M defined by then by linearity properties, we can easily show that the relations hold for all vector fields X on R 2m+1 . Thus, (M, φ , ξ , η, g) forms a Lorentzian para-Kenmotsu

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 [20] (3.1) The curvature tensorR of a Lorentzian para-Sasakian manifold M with respect to the quartersymmetric metric connection∇ is defined by From the equations (2.1) − (2.6) , (3.1) and (3.2) , we obtain 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     This implies thatr = 0.
Hence we can state following:  The Riemannian connection ∇ of the metric g is given by which is known as Koszul's formula, we can easily calculate Putting W = ξ in (5.1) and using (3.8) and (5.2) , we have Putting U = ξ in (5.3) and using (2.1) , we have Taking inner product of (5.4) with W and replacing V by QV , we have where R and r are the Riemannian curvature tensor and the scalar curvature with respect to the connection ∇, respectively on M. From the equations (6.1) and (6.3), we have As we know that the curvature tensor R of Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection ∇ is defined by In view of (6.5), (6.4) takes the form Let {e  and summing over i = 1 to n − 1, we obtain (6.7) For φ −conformally flat Lorentzian para-Kenmotsu manifold with respect to the quarter-symmetric metric connection, it can be easily verify that g(R(φ e i , φV )φW, φ e i ) = S(φV, φW ) − g(φV, φW ), g(e i , φW )g(V, φ e i ) = g(V,W ) + η(V )η(W ), So by virtue of (6.8) − (6.13) ,the equation (6.7) takes the form (6.14) In view of (2.2), (2.3) and (2.12), (6.14) takes the form ] and c = (n + ψ − 3).
Contracting (6.15) over V and W gives By using the value of r in (6.15), we get 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 theorem: So by virtue of (7.5) , (7.6) and (7.7) , the equation ( From which it follows that the manifold is an η−Einstein manifold with respect to the quarter-symmetric metric connection.
Hence we can state the following theorem: Theorem 7.2. If a Lorentzian para-Sasakian manifold admitting a quarter-symmetric metric connection is quasi-concircularly flat, then the manifold with respect to the quarter-symmetric metric connection is an η−Einstein manifold.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.