On endo curvature tensor of a contact metric manifold

Abstract

We prove that a $( k ,\mu )$-manifold with vanishing Endo curvature tensor is a Sasakian manifold. We find a necessary and sufficient condition for a non-Sasakian $( k ,\mu )$-manifold %$M$ whose Endo curvature tensor $B^{es}$ satisfies $B^{es}(\xi ,X) \cdot S=0$, where $S$ is the Ricci tensor. Using ${\cal D}$-homothetic deformation we obtain an example of an $N\left( k\right)$-contact metric manifold on which $B^{es}(\xi ,X)\cdot S\neq 0$.