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 \( \mathbf {R} ^{n} \). 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 \( \mathbf {R} ^{n} \).

Examples

The Euclidean space Rn with its usual metric is a Cartan-Hadamard manifold with constant sectional curvature equal to 0.

Standard n-dimensional hyperbolic space Hn is a Cartan-Hadamard manifold with constant sectional curvature equal to −1.

Properties

- In Cartan-Hadamard Manifolds, the map
*exp*_{p}mapping*TM*_{p}to*M*is a covering map for all*p*in*M*.

See also

Cartan–Hadamard theorem

Hadamard space

Cartan–Hadamard conjecture

References

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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Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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