Hadamard manifold

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

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

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

• Cartan–Hadamard conjecture
• Cartan–Hadamard theorem – On the structure of complete Riemannian manifolds of non-positive sectional curvature
• Hadamard space – geodesically complete metric space of non-positive curvaturePages displaying wikidata descriptions as a fallback