# Hadamard manifold

In mathematics, a **Hadamard manifold**, named after Jacques Hadamard — more often called a **Cartan–Hadamard manifold**, after Élie Cartan — is a Riemannian manifold (*M*, *g*) that is complete and simply connected and has everywhere non-positive sectional curvature.^{[1]}^{[2]} By Cartan–Hadamard theorem all Cartan–Hadamard manifold are diffeomorphic to the Euclidean space . Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of .

## ExamplesEdit

- The Euclidean space
**R**with its usual metric is a Cartan-Hadamard manifold with constant sectional curvature equal to 0.^{n} - Standard
*n*-dimensional hyperbolic space**H**^{n}is a Cartan-Hadamard manifold with constant sectional curvature equal to −1.

## See alsoEdit

## ReferencesEdit

**^**Li, Peter (2012).*Geometric Analysis*. Cambridge University Press. p. 381. ISBN 9781107020641.**^**Lang, Serge (1989).*Fundamentals of Differential Geometry, Volume 160*. Springer. pp. 252–253. ISBN 9780387985930.

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