On three-dimensional N(k)-paracontact metric manifolds and Ricci solitons

The aim of this paper is to characterize 3-dimensional N(k)-paracontact metric manifolds satisfying certain curvature conditions. We prove that a 3-dimensional N(k)-paracontact metric manifold M admits a Ricci soliton whose potential vector field is the Reeb vector field xi if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discussed. Finally, an illustrative example is constructed.