Contact Conformal Curvature Tensor of A Metric Sasakian Manifolds

AbstractThe 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(?)-contact metric manifolds is studied. We show that a N(?) - contact metric manifold with vanishing extended conformal curvature tensor is a K-contact Sasakian manifold. It is also show that a n-dimensional N(?) - contact metric manifold with non-vanishing conformal curvature tensor C0 satisfies R(?,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 aN(?) -contact metric manifold satisfies the condition C0(?,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(?)-contact metric manifolds are given. In section three, we have to study N(?)-contact metric manifolds with extended to contact conformal Curvature tensor. As an application, it is obvious that an N(?)-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(?)-contact metric manifolds satisfying R(?,X).C0 =0. In section five, we show that an N(?)-contact metric manifolds satisfying C0(?,X).S=0 and proof some important theorem with the help of N(?)-contact metric manifolds.