Ricci solitons and gradient Ricci solitons on $N(k)$-paracontact manifolds

An $\eta$-Einstein paracontact manifold $M$ admits a Ricci soliton $(g,\xi)$ if and only if $M$ is a $K$-paracontact Einstein manifold provided one of the associated scalars $\alpha$ or $\beta$ is constant. Also we prove the non-existence of Ricci soliton in an $N(k)$-paracontact metric manifold $M$ whose potential vector field is the Reeb vector field $\xi$. Moreover, if the metric $g$ of an $N(k)$-paracontact metric manifold $M^{2n+1}$ is a gradient Ricci soliton, then either the manifold is locally isometric to a product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of negative constant curvature equal to $-4$, or $M^{2n+1}$ is an Einstein manifold. Finally, an illustrative example is given.