Cartan–Hadamard theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem


00:01:03 1 Riemannian geometry
00:02:00 2 Metric geometry
00:03:57 2.1 Generalization to locally convex spaces
00:04:26 3 Significance
00:05:34 4 See also
00:06:18 5 References



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SUMMARY
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In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898.  Élie Cartan generalized the theorem to Riemannian manifolds in 1928 (Helgason 1978; do Carmo 1992; Kobayashi & Nomizu 1969). The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by Ballmann (1990) for metric spaces of non-positive curvature and by Alexander & Bishop (1990) for general locally convex metric spaces.