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 ${\mathbb{ℝ}}^n.$ 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 ${\mathbb{ℝ}}^n.$

Examples

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

Standard $n$-dimensional hyperbolic space ${\mathbb{ℍ}}^n$ 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

• Cartan–Hadamard conjecture
• Cartan–Hadamard theorem – On the structure of complete Riemannian manifolds of non-positive sectional curvature
• Hadamard space