Curvature, sphere theorems, and the Ricci flow

Abstract

In 1926, Hopf proved that any compact, simply connected Riemannian manifold with constant curvature 1 is isometric to the standard sphere. Motivated by this result, Hopf posed the question whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere. In the first part of this paper, we provide a background discussion, aimed at nonexperts, 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. © 2010 American Mathematical Society.