## Tohoku Mathematical Journal 2014 December SECOND SERIES VOL. 66, NO. 4

 Tohoku Math. J. 66 (2014), 491-500

Title CONTACT 3-MANIFOLDS WITH THE REEB-FLOW SYMMETRY

Author Jong Taek Cho

(Received February 13, 2013, revised October 16, 2013)
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.

Mathematics Subject Classification. Primary 53C25; Secondary 53B20, 53D10.

Key words and phrases. Contact 3-manifold, Reeb flow, Lie group.