In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold
that is complete and simply connected and has everywhere non-positive sectional curvature.
[1][2] By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space
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.
with its usual metric is a Cartan–Hadamard manifold with constant sectional curvature equal to
is a Cartan–Hadamard manifold with constant sectional curvature equal to
In Cartan-Hadamard manifolds, the map
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