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 Connections
Has Statement Form
In Branch
Source Cartan-Hadamard theorem, Wikipedia
Principal Result Of
Source Cartan-Hadamard theorem, Wikipedia
Source Cartan-Hadamard theorem, Wikipedia
Proved By
Source Cartan-Hadamard theorem, Wikipedia
Sources
1. Cartan-Hadamard theorem, Wikipedia
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.
Principal Result Of: Élie Cartan, Lead sentence of the article on the result
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).
Principal Result Of: Jacques Hadamard, Lead sentence of the article on the result
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).
- In Branch: Differential Geometry, Lead sentence
View the Source2. Cartan-Hadamard theorem (Wikipedia)
It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.