Hadamard manifold

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In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold 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

Examples[edit]

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

Standard -dimensional hyperbolic space is a Cartan-Hadamard manifold with constant sectional curvature equal to

Properties[edit]

In Cartan-Hadamard Manifolds, the map is a covering map for all

See also[edit]

References[edit]

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