PROJECTIVE CURVATURE TENSOR WITH RESPECT TO ZAMKOVOY CONNECTION IN LORENTZIAN PARA-SASAKIAN MANIFOLDS

The purpose of the present paper is to study some properties of Projective curvature tensor with respect to Zamkovoy connection in Lorentzian Para Sasakian manifold(briefly, LP-Sasakian manifold). We obtain some results on Lorentzian Para-Sasakian manifold with the help of Zamkovoy connection and Projective curvature tensor. Moreover, we study the LP-Sasakian manifold satisfying P ∗ (ξ, U)◦W ∗ 0 = 0 and P ∗ (ξ, U)◦W ∗ 2 = 0, where P ∗,W ∗ 0 and W ∗ 2 are Projective curvature tensor, W0− curvature tensor and W2−curvature tensor with respect to Zamkovoy connection respectively.


Introduction
In 1989, K. Matsumoto [7] first introduced the notion of Lorentzian Para-Sasakian manifolds. Also, in 1992, I. Mihai and R. Rosca [8] introduced independently the notion of Lorentzian Para Sasakian manifolds(briefly, LP-Sasakian Manifolds) in classical analysis. In an n− dimensional metric manifold the signature of the metric tensor is the number of positive and negative eigenvalues of the metric. If the metric has s positive eigenvalues and t negative eigenvalues then the signature of the metric is (s, t). For a non-degerate metric tensor s + t = n. A Lorentzian manifold is a special case of a semi Riemannian manifold, in which the signature of the metric is (1, n − 1) or (n − 1, 1). And the metric g is called here a Lorentzian metric, which is named after the physicist Hendrik Lorentz. The LP-Sasakian manifold was further studied by several authors. We cite ( [3], [9]) and their references.
The notion of Projective curvature tensor was first introduced by K. Yano and S. Bochner [13] in 1953. This curvature tensor was further studied by U. C. De and J. Sengupta [4], S. Ghosh [5]. If there exists a one -one mapping between each co-ordinate neighbourhood of a manifold M to a domain of R n such that any geodesic of M corresponds to a straight line in R n , then the manifold M is said to be locally projectively flat. Due to [4], the Projective curvature tensor P of rank four for an n-dimensional Riemannian Manifold M is given by for all X, Y, V & Z ∈ χ(M ), set of all vector fields of the manifold M , where P denotes the Projective curvature tensor of type (0, 4) and R denotes the Riemannian curvature tensor of type (0, 4) defined by where R is the Riemannian curvature tensor of type (0, 3), P is the Projective curvature tensor of type (0, 3) and S denotes the Ricci tensor of type (0, 2). In 2008, the notion of Zamkovoy connection on para contact manifold was introduced by S. Zamkovoy [14]. Zamkovoy connection was defined as a canonical paracontact connection whose torsion is the obstruction of paracontact manifold to be a para sasakian manifold. This connection was further studied by many researcher. For instance, we see ( [2], [1], [6]). For an n-dimensional almost contact metric manifold M equipped with an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g, the Zamkovoy connection (∇ * ) in terms of Levi-Civita connection (∇) is given by for all X, Y ∈ χ (M ) . In a LP-Sasakian manifold M of dimension (n > 2), the Projective curvature tensor P, W 0 Curvature tensor [10], W 2 −Curvature tensor [12] with respect to the Levi-Civita connection are given by The Projective curvature tensor, W 0 -Curvature tensor and W 2 -Curvature tensor with respect to the Zamkovoy connection are given by, where R * , S * and Q * are Riemannian curvature tensor, Ricci tensor and Ricci operator with respect to Zamkovoy connection ∇ * respectively.
1. An n-dimensional LP -Sasakian manifold M is said to be generalized η−Einstein manifold if the Ricci tensor of type (0,2) is of the form for all Y, Z ∈ χ (M ) , set of all vector fields of the manifold M and k 1 , k 2 and k 3 are scalars and ω is a 2-form.

Preliminaries
An n-dimensional differentiable manifold is called a LP-Sasakian manifold if it admits a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Lorentzian metric g which satisfies where ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric g. Let us introduced a symmetric (0, 2) tensor field ω such that ω (X, Y ) = g(X, φY ). Also, since the vector field η is closed in LP-Sasakian manifold, we have In LP-Sasakian manifold, the following relations also hold: 3. Some Properties of LP-Sasakian manifolds with respect to Zamkovoy connection Using (15) and (17) in (4), we get with torsion tensor In view of (4) and (17), we have Putting Y = ξ in (25) Using (14), (15) and (16) in (25), we obtain In view of (25), (29), (30) and (31), we have Also we have Let R * be the Riemannian curvature tensor with respect to Zamkovoy connection and it is defined as Using (25), (32), (33) and (34) in (35), we get Consequently one can easily bring out the followings: for all X, Y, Z ∈ χ (M ) , where ψ = trace (φ) Thus we can state the followings: Proposition 3.1. Let M be an n-dimensional LP-Sasakian manifold admitting Zamkovoy connection ∇ * , then (i) The curvature tensor R * of ∇ * is given by (36) (ii) The Ricci tensor S * of ∇ * is given by (37) (iii) The scalar curvature r * of ∇ * is given by (41) (iv) The Ricci tensor S * of ∇ * is symmetric.
( Proof. In view of (8), (36) and (37), the Projective curvature tensor P * with respect to the Zamkovoy connection ∇ * on a LP-Sasakian manifold M of dimension (n > 2) takes the form

Projectively flat LP-Sasakian manifold with respect to the Zamkovoy connection
Let M be projectively flat with respect to Zamkovoy connection, then from (45), we get Taking inner product of (46) with a vector field V, we have Setting X = V = ξ and using (12), (22) in (47), we get where ω (Y, Z) = g (φY, Z) . which shows that M is an η−Einstein manifold. Hence the theorem is proved. Proof. Using (5) in (45), we get Setting Z = ξ in (48), we get Therefore, M is ξ−Projectively flat with respect to Zamkovoy connection iff it is so with respect to Levi-Civita connection.

Locally Projectively φ−symmetric LP-Sasakian manifolds with respect to Zamkovoy connection
In 1977, Takahashi [11] first studied the concept of locally φ-symmetry on Sasakian manifold. In this section we consider a locally projectively φ-symmetric LP-Sasakian manifolds with respect to the connection ∇ * .
Definition 5.1. An n−dimensional LP-Sasakian manifold M is said to be locally projectively φ-symmetric with respect to Zamkovoy connection ∇ * if the projective curvature tensor P * with respect to the connection ∇ * satisfies where X, Y, Z and W are horizontal vector fields on M , i.e X, Y, Z and W are orthonormal to ξ on the manifold M. Proof. In view of (25), we have Taking covariant differentiation of (48) in the direction of W and considering trace (φ) = 0, we obtain In view of (12), (18) and (45), we obtain Using (51) and (52) in (50), we get Applying φ 2 on both sides of (53) and using (12), we obtain where X, Y, Z, W are horizontal vector fields and trace (φ) = 0. Hence the theorem is proved.
Proof. It can be easily seen from (8) and (9), that Let us consider a LP-Sasakian manifold M satisfying the condition Replacing Z by ξ in (58), we get Using (55), (56) and (57) in (59), we have The inner product of the equation (60) with vector field V gives Hence the theorem is proved.