Contact 3-manifolds with the Reeb-flow symmetry

Abstract

We prove that the Ricci operator on a contact Riemannian 3-manifold $M$ is invariant along the Reeb flow if and only if $M$ is Sasakian or locally isometric to $\mathrm{SU}(2)$ (or $\mathrm{SO}(3)$), $\mathrm{SL}(2,\boldsymbol{R})$ (or $O(1,2)$), the group $E(2)$ of rigid motions of Euclidean 2-plane with a contact left invariant Riemannian metric.