Three dimensional contact metric manifolds with Cotton solitons

Abstract

In this article we study a three dimensional contact metric manifold $M^3$ with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with $Q\xi = \rho\xi$, where $\rho$ is a smooth function on $M$ constant along Reeb vector field $\xi$ and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with $\xi$ or is a gradient vector field. Moreover, if $\rho$ is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: $SU(2)$ or $SO(3)$ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field $\xi$. Secondly, it is proved that a $(\kappa,\mu,\nu)$-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that $M$ is Sasakian, flat, a contact metric $(0,−4)$-space or a contact metric $(\kappa,0)$-space with $\kappa \lt 1$ and $\kappa\neq0$. For the potential vector field being orthogonal to $\xi$, if $\nu$ is constant we prove that $M$ is either Sasakian, or a $(\kappa,\mu)$-contact metric space.