Curvature, sphere theorems, and the Ricci flow

2010/01/13 by S. Brendle, Simon Brendle, Brendle, S. +2 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG #math.GT #math.MG

paper · pdf · doi:10.48550/arxiv.1001.2278

This is an invited contribution for the Bulletin of the AMS

arxiv created 2010/05/31 · arxiv updated 2010/06/01

Abstract

This is a survey paper focusing on the interplay between the curvature and topology of a Riemannian manifold. The first part of the paper provides a background discussion, aimed at non-experts, of Hopf's pinching problem and the Sphere Theorem. In the second part, we sketch the proof of the Differentiable Sphere Theorem, and discuss various related results. These results employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton's Ricci flow.

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• Einstein manifolds under cone conditions for the curvature operator of the second kind

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