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 [math]\displaystyle{ (M, g) }[/math] 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 [math]\displaystyle{ \mathbb{R}^n. }[/math] 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 [math]\displaystyle{ \mathbb{R}^n. }[/math]

Examples

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

Standard [math]\displaystyle{ n }[/math]-dimensional hyperbolic space [math]\displaystyle{ \mathbb{H}^n }[/math] is a Cartan–Hadamard manifold with constant sectional curvature equal to [math]\displaystyle{ -1. }[/math]

Properties

In Cartan-Hadamard manifolds, the map [math]\displaystyle{ \exp_p : \operatorname{T}M_p \to M }[/math] is a diffeomorphism for all [math]\displaystyle{ p \in M. }[/math]

See also

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