Curvature and Topology

Abstract

In this final chapter, we bring together most of the tools we have developed so far to prove some significant local-to-global theorems relating curvature and topology of Riemannian manifolds. The main results are (1) the Killing–Hopf theorem, which characterizes complete, simply connected manifolds with constant sectional curvature; (2) the Cartan–Hadamard theorem, which topologically characterizes complete, simply connected manifolds with nonpositive sectional curvature; and (3) Myers’s theorem, which says that a complete manifold with Ricci curvature bounded below by a positive constant must be compact and have a finite fundamental group.