Bulletin of the
Korean Mathematical Society BKMS

We prove that a $(\kappa ,\mu )$-manifold with vanishing $E$-Bochner curvature tensor is a Sasakian manifold. Several interesting corollaries of this result are drawn. Non-Sasakian $(\kappa ,\mu )$-manifolds with $C$-Bochner curvature tensor $B$ satisfying $B\left( \xi ,X\right) \cdot S=0$, where $S$ is the Ricci tensor, are classified. $% N(\kappa )$-contact metric manifolds $M^{2n+1}$, satisfying $B\left( \xi ,X\right) \cdot R=0$ or $B\left( \xi ,X\right) \cdot B=0$ are classified and studied.

Keywords: contact metric manifold, $(\kappa,\mu )$-manifold, $N(\kappa )$-contact metric manifold, Sasakian manifold, $C $-Bochner curvature tensor, $E$-Bochner curvature tensor, $\eta $-Einstein manifold, Einstein manifold

MSC numbers: 53C50, 53B30