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. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space ${ℝ}^n.$ 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 ${ℝ}^n.$

Examples

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

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

Properties

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

See also

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