CONFORMALLY FLAT CONTACT THREE-MANIFOLDS

In this paper, we consider contact metric three-manifolds (M; eta, g, phi,xi) which satisfy the condition del xi h = mu h phi+nu h for some smooth functions mu and nu, where 2h = pound xi phi. We prove that if M is conformally flat and if mu is constant, then M is either a flat manifold or a Sasakian manifold of constant curvature + 1. We cannot extend this result for a smooth function mu. Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.