Certain solitons on generalized $(\kappa, \mu)$ contact metric manifolds

Abstract

The aim of the present paper is to study some solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. It is proved that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is gradient Einstein soliton then $\mu = \frac{2\kappa}{\kappa - 2}.$ It is shown that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is closed m-quasi Einstein metric then $\kappa = \frac{\lambda}{m + 2}$ and $\mu = 0.$ We also study conformal gradient Ricci solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds.