Nonflatness of certain contact and CR manifolds

Abstract

We prove that, under a suitable geometric condition, a Riemannian manifold of dimension at least 7 endowed with a contact distribution cannot be flat. This result yields nonflatness of some classes of almost contact metric manifolds, contact sub-Riemannian symmetric spaces, locally symmetric C R spaces and C R submanifolds of Kähler manifolds. As an application, we prove that a compact flat Riemannian manifold of odd dimension at least 7 cannot be isometrically immersed as a hypersurface of a simply connected, complete Kähler manifold of nonpositive curvature.