Communications of the
Korean Mathematical Society CKMS

The object of the present paper is to study 3-dimensional normal almost contact metric manifolds satisfying certain curvature conditions. Among others it is proved that a parallel symmetric $(0,2)$ tensor field in a 3-dimensional non-cosympletic normal almost contact metric manifold is a constant multiple of the associated metric tensor and there does not exist a non-zero parallel $2$-form. Also we obtain some equivalent conditions on a 3-dimensional normal almost contact metric manifold and we prove that if a 3-dimensional normal almost contact metric manifold which is not a $\beta $-Sasakian manifold satisfies cyclic parallel Ricci tensor, then the manifold is a manifold of constant curvature. Finally we prove the existence of such a manifold by a concrete example.

Keywords: normal almost contact metric manifolds, non-cosympletic, cyclic parallel Ricci tensor, Einstein manifold

MSC numbers: 53C15, 53C40