The diffeomorphism type of manifolds with almost maximal volume

摘要

The smallest r so that a metric r-ball covers a metric space M is called the radius of M. The volume of a metric r-ball in the space form of constant curvature k is an upper bound for the volume of any Riemannian manifold with sectional curvature >= k and radius <= r. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.