Almost $*$-Ricci Solitons on Contact Strongly Pseudo-Convex Integrable $\mathcal{CR}$-manifolds

Abstract

We prove that if contact strongly pseudo-convex integrable $\mathcal{CR}$-manifold admits a $*$-Ricci soliton where the soliton vector $Z$ is contact, then the Reeb vector field $\xi$ is an eigenvector of the Ricci operator at each point if and only if $\sigma$ is constant. Then we study contact strongly pseudo-convex integrable $\mathcal{CR}$-manifold such that $g$ is a almost $*$-Ricci soliton with potential vector field $Z$ collinear with $\xi$. To this end, we prove that if a 3-dimensional contact metric manifold $M$ with $Q\varphi=\varphi Q$ which admits a gradient almost $*$-Ricci soliton, then either $M$ is flat or $f$ is constant.