KnowraHopf–Rinow theoremLinked fromLinked fromThe 3 pages that link to Hopf–Rinow theorem, each with the reason it gives.All 3Related 3Exponential mapRelated: It connects global completeness with geodesics reaching points and the exponential map’s domain.Riemannian distanceRelated: It guarantees minimizing geodesics between points when the manifold is complete.Gauss's lemma (differential geometry)Related: Combined with Gauss's lemma, it underlies the characterization of distance by minimizing geodesics.