Misplaced Pages

In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold ${\displaystyle (M,g)}$ that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space ${\displaystyle \mathbb {R} ^{n}.}$ 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 ${\displaystyle \mathbb {R} ^{n}.}$

Examples

The Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with its usual metric is a Cartan–Hadamard manifold with constant sectional curvature equal to ${\displaystyle 0.}$

Standard ${\displaystyle n}$-dimensional hyperbolic space ${\displaystyle \mathbb {H} ^{n}}$ is a Cartan–Hadamard manifold with constant sectional curvature equal to ${\displaystyle -1.}$

Properties

In Cartan-Hadamard manifolds, the map ${\displaystyle \exp _{p}:\operatorname {T} M_{p}\to M}$ is a diffeomorphism for all ${\displaystyle p\in M.}$

See also

• Cartan–Hadamard conjecture
• Cartan–Hadamard theorem – On the structure of complete Riemannian manifolds of non-positive sectional curvature
• Hadamard space – geodesically complete metric space of non-positive curvaturePages displaying wikidata descriptions as a fallback

References

1. Li, Peter (2012). Geometric Analysis. Cambridge University Press. p. 381. doi:10.1017/CBO9781139105798. ISBN 9781107020641.
2. Lang, Serge (1989). Fundamentals of Differential Geometry, Volume 160. Springer. pp. 252–253. ISBN 9780387985930.

This topology-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Riemannian manifolds
• Topology stubs