# 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.