Communications of the
Korean Mathematical Society CKMS

In this paper we study $h$-projectively semisymmetric, $\phi $-projectively semisymmetric, $h$-Weyl semisymmetric and $\phi $-Weyl semisymmetric non-Sasakian $(k,\mu )$-contact metric manifolds. In all the cases the manifold becomes an $\eta $-Einstein manifold. As a consequence of these results we obtain that if a $3$-dimensional non-Sasakian $(k,\mu )$-contact metric manifold satisfies such curvature conditions, then the manifold reduces to an $N(k)$-contact metric manifold.

Keywords: semisymmetric spaces, $(k,\mu )$-contact metric manifolds, non-Sasakian manifolds, $\eta $-Einstein manifolds, $h$-projectively semisymmetric, $\phi $-projectively semisymmetric, $h$-Weyl semisymmetric, $\phi $-Weyl semisymmetric

MSC numbers: 53C25, 53C35, 53D10