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Curvature and Topology

  • John M. Lee
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 176)

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.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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