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 $$\exp_p : \operatorname{T}M_p \to M$$ is a diffeomorphism for all $$p \in M.$$