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Concept
Cartan–Hadamard theorem
Summary
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 (; ; ). The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by for metric spaces of non-positive curvature and by for general locally convex metric spaces.
The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to Rn. In fact, for complete manifolds of non-positive curvature, the exponential map based at any point of the manifold is a covering map.
The theorem holds also for Hilbert manifolds in the sense that the exponential map of a non-positively curved geodesically complete connected manifold is a covering map (; ). Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point.
In metric geometry, the Cartan–Hadamard theorem is the statement that the universal cover of a connected non-positively curved complete metric space X is a Hadamard space. In particular, if X is simply connected then it is a geodesic space in the sense that any two points are connected by a unique minimizing geodesic, and hence contractible.
A metric space X is said to be non-positively curved if every point p has a neighborhood U in which any two points are joined by a geodesic, and for any point z in U and constant speed geodesic γ in U, one has
This inequality may be usefully thought of in terms of a geodesic triangle Δ = zγ(0)γ(1).
Official source
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