On $C$-Bochner curvature tensor of a contact metric manifold

Bull. Korean Math. Soc. 2005 Vol. 42, No. 4, 713-724 Printed December 1, 2005

Jeong-Sik Kim, Mukut Mani Tripathi, and Jaedong Choi Mathematical Information Yosu National University, Lucknow University, Korea Air Force Academy

Abstract : 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.