Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds

Abstract

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combining the concentration of measures of Grassmanians with Brendle-Schoen pointwise (weakly)- 1/4-pinching Theorem. A direct corollary of the main result within this paper is the answering of the long standing Hopf Conjecture in sufficiently high dimensions.