S.-S. Chern raised the question of determining those compact 3-manifolds M admitting a contact metric structure whose characteristic vector field generates a one-parameter group of isometries. S. Tachibana showed that the first betti number of these spaces must be even, and H. Sato proved that the second homotopy group of M is zero unless M is homotopy equivalent to S1×S2. A. Weinstein pointed out that M is a Seifert fibre space over an orientable surface. In this paper, it is shown as a consequence of a more general theorem that if, in addition, the scalar curvature is suitably bounded below by a negative constant, then the metric may be deformed to a metric of positive constant sectional curvature. Thus, if the manifold is simply connected it is diffeomorphic with the 3-sphere.
All Science Journal Classification (ASJC) codes
- Contact Riemannian manifolds