Contact Conformal Curvature Tensor of A Metric Sasakian Manifolds

The idea of a conformal Curvature tensor field of Hermitian manifolds [10]. The purpose of the present paper is to introduce K-contact conformal Curvature tensor of a metric Sasakian manifold. The conformal Curvature tensor of N(κ)-contact metric manifolds is studied. We show that a N(κ) - contact metric manifold with vanishing extended conformal curvature tensor is a K-contact Sasakian manifold. It is also show that a n-dimensional N(κ) - contact metric manifold with non-vanishing conformal curvature tensor C₀ satisfies R(ξ,X). C₀ =0if and only if it is locally isometric to Eⁿ⁺¹ × Sⁿ forn>1 and flat forn= 1. Again, we also show that the Ricci tensor S of αN(κ) -contact metric manifold satisfies the condition C₀(ξ,X).S=0 if and only if the manifold is 3-dimensional and flat. In the present paper we have to study contact conformal Curvature tensor of a metric Sasakian manifolds. In the section two, necessary details about contact metric manifolds, K-contact manifolds, Sasakian manifolds and N(κ)-contact metric manifolds are given. In section three, we have to study N(κ)-contact metric manifolds with extended to contact conformal Curvature tensor. As an application, it is obvious that an N(κ)-contact space form with vanishing tensor is a Sasakian space form. In section four, using a result of [1] and proof some theorems of N(κ)-contact metric manifolds satisfying R(ξ,X).C₀ =0. In section five, we show that an N(κ)-contact metric manifolds satisfying C₀(ξ,X).S=0 and proof some important theorem with the help of N(κ)-contact metric manifolds.