LP-SASAKIAN MANIFOLDS EQUIPPED WITH ZAMKOVOY CONNECTION AND CONHARMONIC CURVATURE TENSOR

The paper concerns with some results on conharmonically flat, quasiconharmonically flat and φ−conharmonically flat LP-Sasakian manifolds with respect to Zamkovoy connection. Also, it contains study of generalized conharmonic φ−recurrent LP-Sasakian manifolds with respect to Zamkovoy connection. Moreover, the paper deals with LP-Sasakian manifolds satisfying K∗ (ξ, U) .R∗ = 0, where K∗ denotes conharmonic curvature tensor and R∗ denotes Riemannian curvature tensor with respect to Zamkovoy connection, respectively.


Introduction
In 1989, K. Matsumoto [13] first introduced the notion of Lorentzian para-Sasakian manifolds (briefly, LP-Sasakian manifolds). Also, in 1992, I. Mihai and R. Rosca [14] introduced independently the notion of Lorentzian para-Sasakian manifolds in classical analysis. The generalized recurrent manifolds was introduced by Dubey [8] and it was studied by De and Guha et al. [6]. In this context, φ−recurrent LP-Sasakian manifold was first studied by A. A. Shaikh, D. G. Prakasha and Helaluddin Ahmad [15]. On the other hand, φ−conharmonically flat LP-Sasakian manifold was introduced by A. Taleshian [16]. Apart from these, the properties of LP-Sasakian manifolds were studied by several authors, namely U. C. De [7], C. Ozgur [17] and many others.
In 1957, Y. Ishii [9] first studied the notion of a conharmonic curvature tensor. A rank three tensor K, that remains invariant under conharmonic transformation for an n−dimensional Riemannian manifold M is given by for all X, Y, Z ∈ χ(M ), where χ(M ) is the set of all vector fields of the manifold M and R denotes the Riemannian curvature tensor of type (1, 3) , S denotes the Ricci tensor of type (0, 2) , Q is the Ricci operator. The conharmonic curvature tensor(K * ) with respect to Zamkovoy connection is given by for all X, Y, Z ∈ χ (M ) , where R * , S * and Q * are Riemannian curvature tensor, Ricci tensor and Ricci operator with respect to Zamkovoy connection, respectively. Definition 1.1. 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 ) , where k 1 , k 2 and k 3 are scalars and ω is a 2−form.
2. An n-dimensional LP-Sasakian manifold M is said to be conharmonically flat with respect to Zamkovoy connection if K * (X, Y ) Z = 0, for all X, Y, Z ∈ χ (M ) .
3. An n-dimensional LP-Sasakian manifold M is said to be ξ− conharmonically flat with respect to Zamkovoy connection if K (X, Y ) ξ = 0, for all X, Y, Z ∈ χ (M ) .
Definition 1.4. An n−dimensional LP-Sasakian manifold M is said to be generalized conharmonic φ−recurrent with respect to Zamkovoy connection if for all X, Y, Z, W ∈ χ (M ) , where A and B are 1−forms and B is non vanishing such that A (W ) = g (W, ρ 1 ) , B (W ) = g (W, ρ 2 ) and ρ 1 , ρ 2 are vector fields associated with 1−forms A and B, respectively.

Preliminaries
An n-dimensional differentiable manifold is called an 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: g(X, φY ) = g(φX, Y ), η(Y ) = g(Y, ξ), for all X, Y ∈ χ (M ) ,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 Also, since the vector field η is closed in LP-Sasakian manifold M, we have for all X, Y ∈ χ (M ) .
In LP-Sasakian manifold the following relations also hold: Lemma 2.1. The relation between Zamkovoy connection and Levi-Civita connection in an LP-Sasakian manifold is given by where the torsion tensor of Zamkovoy connection is Proof. In view of (1) and (11), we have Suppose that the Zamkovoy connection ∇ * defined on an n− dimensional LP-Sasakian manifold M is connected with the Levi-Civita connection ∇ by the relation where P (X, Y ) is a tensor field of type (1,1). Then by definition of torsion tensor, we have Zamkovoy connection is a non-metric connection and hence from (22), we get In view of (24), (25), (26) and (23), we have Setting in (27), we have which implies that In reference to (20), (28) and (29), we have Using (20), (32) and (33) in (31), we obtain In reference to (22) and (34), we can easily bring out the equation (19).
From the equation (19), it is obvious that Proposition 2.2. The Zamkovoy connection on an n−dimensional LP-Sasakian manifold is a non-metric linear connection with torsion tensor given by equation (20).

Some properties of LP-Sasakian manifold with respect to Zamkovoy connection
Let R * be the Riemannian curvature tensor with respect to Zamkovoy connection and it be defined as Using (5), (8), (9) and (19) in (36), we get the Riemannian curvature R * with respect to Zamkovoy connection as Consequently, one can easily bring out the followings: for all X, Y, Z ∈ χ (M ) , where ψ = trace (φ) .
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 (37), (ii) The Ricci tensor S * of ∇ * is given by (38), (iii) The scalar curvature r * of ∇ * is given by (42), (iv) The Ricci tensor S * of ∇ * is symmetric, Proof. In view of (2) and (3), we have Let us consider an LP-Sasakian manifold M which is conharmonically flat with respect to Zamkovoy connection, then from (3), we have Taking inner product of (47) with a vector field V , we get Taking an orthonormal frame field of M and contracting (48) over X and V, we obtain r = n − 1 − 3ψ 2 . This gives the theorem.  Proof. Setting Z = ξ in (46), we have This gives the theorem. Proof. Setting Z = ξ in (3), we have If M is ξ−conharmonically flat with respect to Zamkovoy connection, then it follows from (50) that Taking inner product of (51) with a vector field V, we obtain Taking an orthonormal frame field of M and contracting (53) over X and V , we get r * = 0.
This gives the theorem.
Theorem 5.1. If an n−dimensional LP-Sasakian manifold M (n > 2) is quasiconharmonically flat with respect to Zamkovoy connection, then its scalar curvature with respect to Zamkovoy connection vanishes.
Proof. Let us consider an LP-Sasakian manifold M which is quasi-conharmonically flat with respect to Zamkovoy connection, i.e., for all X, Y, Z, V ∈ χ (M ). Then, in view of (3), we have Let It can be easily seen that Using (57), (58) and (59) in (56), we get This gives the theorem.
This gives the theorem.
Let {e i } (1 ≤ i ≤ n) be an orthonormal basis of the tangent space at any point of the manifold M . Setting X = V = e i and taking summation over i(1 ≤ i ≤ n) and using (18) in (80), we get 0 = S 2 (Y, U ) + 9ψ 2 g (Y, U ) This gives the theorem.