The Diffeomorphism Type of Manifolds with Almost Maximal Volume

2012/01/02 by Pro, Curtis, Sill, Michael, Wilhelm, Frederick
#53C20 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1201.0415

Abstract

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.

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