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.
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Lee, J.M. (2018). Curvature and Topology. In: Introduction to Riemannian Manifolds. Graduate Texts in Mathematics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-91755-9_12
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DOI: https://doi.org/10.1007/978-3-319-91755-9_12
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-91755-9
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