: Mathematics: Hadamard manifold

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topology > differential topology > differentiable manifold > Riemannian manifold > Hadamard manifold

geometry > differential geometry > Riemannian geometry > Riemannian manifold > Hadamard manifold

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 ${\\displaystyle (M,g)}$ 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 ${\\displaystyle \\mathbb {R} ^{n}.}$ . 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 ${\\displaystyle \\mathbb {R} ^{n}.}$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Hadamard_manifold)

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