Certain results on a type of contact metric manifold

Abstract

Let M be a 3-dimensional almost contact metric manifold satisfying (*) condition. We denote such a manifold by M*. At first we study symmetric and skew-synunetric parallel tensor of type (0, 2) in M*. Next we prove that a non-cosymplectic manifold M* is Ricci semisymmetric if and only if it is Einstein. Also we study locally phi-symmetry and eta-parallel Ricci tensor of M*. Finally, we prove that if a non-cosymplectic M* is Einstein, then the manifold is Sasakian.