On 3-dimensional generalized (kappa, mu)-contact metric manifolds

Abstract

In the present study, we considered 3-dimensional generalized (kappa, mu)-contact metric manifolds. We proved that a 3-dimensional generalized (kappa, mu)-contact metric manifold is not locally phi-symmetric in the sense of Takahashi. However such a manifold is locally phi-symmetric provided that kappa and mu are constants. Also it is shown that if a 3-dimensional generalized (kappa, mu) -contact metric manifold is Ricci-symmetric, then it is a (kappa, mu)-contact metric manifold. Further we investigated certain conditions under which a generalized (kappa, mu)-contact metric manifold reduces to a (kappa, mu)-contact metric manifold. Then we obtain several necessary and sufficient conditions for the Ricci tensor of a generalized (kappa, mu)-contact metric manifold to be eta-parallel. Finally, we studied Ricci-semisymmetric generalized (kappa, mu)-contact metric manifolds and it is proved that such a manifold is either flat or a Sasakian manifold.