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 ) {\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 R n . {\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 R n . {\displaystyle \mathbb {R} ^{n}.}
Examples The Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with its usual metric is a Cartan–Hadamard manifold with constant sectional curvature equal to 0. {\displaystyle 0.}
Standard n {\displaystyle n} -dimensional hyperbolic space H n {\displaystyle \mathbb {H} ^{n}} is a Cartan–Hadamard manifold with constant sectional curvature equal to − 1. {\displaystyle -1.}
Properties In Cartan-Hadamard manifolds, the map exp p : T M p → M {\displaystyle \exp _{p}:\operatorname {T} M_{p}\to M} is a diffeomorphism for all p ∈ M . {\displaystyle p\in M.}
See also Cartan–Hadamard conjecture Cartan–Hadamard theorem – On the structure of complete Riemannian manifolds of non-positive sectional curvature Hadamard space – Non-linear generalization of a Hilbert space
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