$m$-quasi-$*$-Einstein contact metric manifolds

Abstract

The goal of this article is to introduce and study the characterstics of $m$-quasi-$*$-Einstein metric on contact Riemannian manifolds. First, we prove that if a Sasakian manifold admits a gradient $m$-quasi-$*$-Einstein metric, then $M$ is $\eta$-Einstein and $f$ is constant. Next, we show that in a Sasakian manifold if $g$ represents an $m$-quasi-$*$-Einstein metric with a conformal vector field $V$, then $V$ is Killing and $M$ is $\eta$-Einstein. Finally, we prove that if a non-Sasakian $(\kappa,\mu)$-contact manifold admits a gradient $m$-quasi-$*$-Einstein metric, then it is $N(\kappa)$-contact metric manifold or a $*$-Einstein.