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.[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.