On Para-Kenmotsu Manifolds

In this paper we study para-Kenmotsu manifolds. We characterize this manifolds by tensor equations and study their properties. We are devoted to a study of eta-Einstein manifolds. We show that a locally conformally flat para-Kenmotsu manifold is a space of constant negative sectional curvature -1 and we prove that if a para-Kenmotsu manifold is a space of constant phi-para-holomorphic sectional curvature H, then it is a space of constant sectional curvature and H = - 1. Finally the object of the present paper is to study a 3-dimensional para-Kenmotsu manifold, satisfying certain curvature conditions. Among other, it is proved that any 3-dimensional para-Kenmotsu manifold with eta-parallel Ricci tensor is of constant scalar curvature and any 3-dimensional para-Kenmotsu manifold satisfying cyclic Ricci tensor is a manifold of constant negative sectional curvature -1.