Cartan–Hadamard theorem

The Cartan–Hadamard theorem states that a complete, simply connected Riemannian manifold with nonpositive sectional curvature is diffeomorphic to each tangent space via the exponential map.

Exponential map: A map from a tangent space to a manifold that sends each vector to the endpoint of its geodesic at unit time. The theorem's central map is the exponential map at each base point.
Riemannian manifold: A smooth manifold equipped with smoothly varying inner products on its tangent spaces. The theorem concerns global geometry built from a Riemannian metric.
Cartan–Hadamard manifold: A complete simply connected Riemannian manifold whose sectional curvature is nonpositive everywhere. Such manifolds inherit the theorem's global tangent-space model.
Sphere: The set of points at a fixed distance from a center in Euclidean space, with its standard round metric when treated as a manifold. Positive curvature and multiple geodesics prevent the theorem's global exponential-map conclusion.
Sectional curvature: A Riemannian curvature measure assigned to each two-dimensional tangent plane. Nonpositive sectional curvature is the local geometric condition in the theorem.
Simple connectivity: The property that a path-connected space has no noncontractible loops. It eliminates nontrivial covering transformations that could identify points in the tangent-space image.
Unique geodesic: A geodesic segment is unique when exactly one geodesic joins its two endpoints. Any two points in a Hadamard manifold are joined by a unique geodesic segment.
Flat torus: A quotient of Euclidean space by a lattice, carrying a flat Riemannian metric. It is complete and flat but not simply connected, so its exponential map is not globally one-to-one.
Geodesic completeness: The property that every geodesic extends for all real values of its affine parameter. Completeness ensures geodesics defining the exponential map exist at every tangent vector.
Completeness (Riemannian geometry): A Riemannian metric is complete when its induced distance makes the manifold a complete metric space. Metric completeness is equivalent to geodesic completeness for connected Riemannian manifolds.

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