CartanHadamard

CartanHadamard refers to the Cartan–Hadamard theorem in differential geometry, named for Élie Cartan and Jacques Hadamard. The theorem concerns complete, simply connected Riemannian manifolds with nonpositive sectional curvature, and it is central to understanding global geometric structure under curvature constraints. Manifolds satisfying this condition are often called Cartan–Hadamard manifolds or Hadamard spaces.

The core statement is that if M is a complete simply connected Riemannian manifold with nonpositive sectional curvature, then for any point p in M the exponential map from the tangent space T_pM to M is a diffeomorphism. Consequently, M is diffeomorphic to Euclidean space of the same dimension. A key geometric consequence is that between any two points there exists a unique geodesic, and this geodesic minimizes distance. In addition, the distance function exhibits convexity along geodesics, and geodesic triangles in M are thinner than their Euclidean counterparts.

Implications of the theorem include strong global control over the geometry and topology of such spaces: they are contractible, simply connected, and have no conjugate points. Examples include hyperbolic spaces H^n and Euclidean space R^n, as well as more general complete simply connected manifolds with K ≤ 0. The Cartan–Hadamard framework also informs areas such as geometric group theory, where nonpositively curved spaces provide canonical models for analysis of group actions. The theorem was proved independently by Cartan and Hadamard in the early 20th century.