Cartan–Hadamard theorem

Cartan-Hadamard theorem
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature.wikipedia
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Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature.

Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
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 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 R n.
In 1928, Élie Cartan proved the Cartan–Hadamard theorem: if M is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space.

Hadamard space

Bruhat-Tits fixed point theoremCAT(0) spacesHadamard manifold
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.
The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem.

Glossary of Riemannian and metric geometry

injectivity radiusgeodesic metric spaceproper
Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to R n via the exponential map; for metric spaces, the statement that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense of Alexandrov is a (globally) CAT(0) space.

Aspherical space

asphericalaspherical manifoldasphericity
The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is aspherical.

Hadamard manifold

Cartan-Hadamard manifoldsCartan–Hadamard manifolds
By Cartan–Hadamard theorem all Cartan–Hadamard manifold are diffeomorphic to the Euclidean space.

CAT(k) space

CAT(0)CAT(0) spaceCAT(''k'') space
This condition, called the CAT(0) condition is an abstract form of Toponogov's triangle comparison theorem.
* Cartan–Hadamard theorem

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
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 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 R n.

Euclidean space

EuclideanspaceEuclidean vector space
The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point.

Exponential map (Riemannian geometry)

exponential mapexponential map of this Riemannian metricExponential map, Riemannian geometry
The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point.

Hans Carl Friedrich von Mangoldt

von MangoldtHans von Mangoldtvon Mangoldt, Hans Carl Friedrich
It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898.

Surface (topology)

surfaceclosed surfacesurfaces
It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898.

Jacques Hadamard

HadamardHadamard, JacquesHadamard, Jacques Solomon
It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898.

Élie Cartan

CartanÉlie Joseph CartanE. Cartan
Élie Cartan generalized the theorem to Riemannian manifolds in 1928.

Mikhail Leonidovich Gromov

Mikhail GromovGromovMikhael Gromov
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.

Covering space

universal covercovering mapuniversal covering space
The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. 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 R n.

Connected space

connectedconnected componentpath-connected
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 R n.

Complete metric space

completecompletioncompleteness
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 R n.

Diffeomorphism

diffeomorphicdiffeomorphismsdiffeomorphism group
The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. 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 R n.

Hilbert manifold

Hilbert bundle
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.

Tangent space

tangent planetangenttangent vector
Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point.

Metric space

metricmetric spacesmetric geometry
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. 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.

Connectedness

connectedconnectivitypath connected
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.

Simply connected space

simply connectedsimply-connectedmultiply connected
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.

Contractible space

contractiblelocally contractiblecontractibility
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.