$N(k)$-contact metric as $\ast$-conformal Ricci soliton

Abstract

The aim of this paper is characterize a class of contact metric manifolds admitting * -conformal Ricci soliton. It is shown that if a (2n + 1)-dimensional N (k)-contact metric manifold M admits * -conformal Ricci soliton or * -conformal gradient Ricci soliton, then the manifold M is * -Ricci flat and locally isometric to the Riemannian of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of constant curvature 4 for n > 1 and flat for n = 1. Further, for the first case, the soliton vector field is conformal and for the * -gradient case, the potential function f is either harmonic or satisfy a Poisson equation. Finally, an example is presented to support the results.