Hopf–Rinow theorem

The Hopf–Rinow theorem states that, on a connected Riemannian manifold, geodesic completeness, metric completeness, and compactness of closed bounded sets are equivalent.

Geodesic completeness: A Riemannian manifold is geodesically complete when every geodesic extends for all real parameter values. This is one of the theorem’s three equivalent conditions.
Riemannian manifold: A smooth manifold equipped with smoothly varying inner products on its tangent spaces. The theorem concerns the distance and geodesics induced by this structure.
Heinz Hopf: Heinz Hopf was a German-Swiss mathematician whose work shaped topology and differential geometry. His early work contributed to the result now bearing his name.
Shortest path problem: The shortest path problem seeks a curve of minimum length between specified points in a metric or geometric space. The theorem guarantees a minimizing geodesic between any two points under completeness.
Punctured Euclidean plane: The punctured Euclidean plane is the plane with one point removed, carrying the usual Euclidean metric restricted to it. It is not metrically complete, and some pairs lack a minimizing path within the space.
Metric completeness: A metric space is complete when every Cauchy sequence converges to a point in that space. The theorem equates completeness of the Riemannian distance with geodesic completeness.
Geodesic: A curve that locally extremizes length and has zero covariant acceleration. Geodesic completeness asks whether every such curve extends indefinitely.
Willi Rinow: Willi Rinow was a German mathematician known for contributions to differential geometry. His independent work established a form of the theorem.
Proper metric space: A metric space is proper when every closed bounded subset is compact. Hopf–Rinow shows that complete connected Riemannian manifolds are proper.
Open interval: An open interval is a real interval that excludes its finite endpoints. With its usual metric, it is incomplete and bounded without being compact.

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Gauss's lemma (differential geometry)Related: Combined with Gauss's lemma, it underlies the characterization of distance by minimizing geodesics.