Two classes of conformally flat contact metric 3-manifolds
Let \(M(\eta,\xi,\varphi, g)\) be a 3-dimensional contact manifold. \(M\) is said to be a 3-\(\tau\)-manifold if \(\nabla_\xi\tau= 0\), where \(\nabla\) is the Riemannian connection of \(g\) and \(\tau= L_\xi g\) (Lie derivation). \(M\) is said to be a \((\kappa,\mu)\)-manifold if the curvature tensor of \(M\) satisfies the condition \(R(X,Y)\xi= \kappa(\eta(Y)X- \eta(X)Y)+ \mu(\eta(Y)hX- \eta(X)hY)\) for any vector fields \(X\), \(Y\), where \(2h= L_\xi\varphi\) and \(\kappa\), \(\mu\) are smooth functions on \(M\). The existence of the last kind of manifolds has been recently proved [see \textit{Th. Koufogiorgos} and \textit{Ch. Tsichlias}, `On the existence of a new class of contact metric manifolds', Can. Math. Bul. (to appear)]. In the present paper, the authors prove the following result. Theorem. A 3-dimensional conformally flat, 3-\(\tau\) (or \((\kappa,\mu)\)) manifold is either flat or a Sasakian manifold of constant curvature 1.
scientific article; zbMATH DE number 1292219