The Cartan-Hadamard Theorem, in Riemannian geometry, concerns the global structure of a complete Riemannian manifold whose sectional curvature is everywhere non-positive. It states that the universal cover of such a manifold is diffeomorphic to ordinary Euclidean space, via the exponential map taken at any chosen point. The theorem was first proved for surfaces by Hans von Mangoldt in 1881 and independently by Jacques Hadamard in 1898, and Elie Cartan generalized it to Riemannian manifolds of any dimension in 1928.
Facts
StatementThe universal cover of a complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to Euclidean space via the exponential map at any point. 1 Classification
Statement FormCharacterization Theorem 1 Sources
1. Cartan-Hadamard theorem, Wikipedia
Lead paragraphQuote, Lead paragraph
The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point.
2. Cartan-Hadamard theorem (Wikipedia)
It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881View the Source Reader Challenges (0)
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