Hopf–Rinow theorem

E679322

mathematical theorem theorem in Riemannian geometry

The Hopf–Rinow theorem is a fundamental result in Riemannian geometry that characterizes when a Riemannian manifold is geodesically complete, relating metric completeness, compactness of closed and bounded sets, and the existence of minimizing geodesics between points.

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Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in Riemannian geometry ⓘ
appearsIn	textbooks on Riemannian geometry ⓘ textbooks on differential geometry ⓘ
appliesTo	connected Riemannian manifold ⓘ finite-dimensional Riemannian manifold ⓘ
assumes	Riemannian metric is smooth ⓘ finite-dimensional manifold ⓘ
characterizes	geodesic completeness of a Riemannian manifold ⓘ
concerns	existence of length-minimizing curves ⓘ properness of the distance function ⓘ
equivalenceCondition	any two points can be joined by a minimizing geodesic ⓘ closed and bounded subsets are compact ⓘ the exponential map at any point is defined on the whole tangent space ⓘ the manifold is complete as a metric space ⓘ the manifold is geodesically complete ⓘ
field	Riemannian geometry ⓘ differential geometry ⓘ
generalizationOf	classical results on completeness in metric spaces ⓘ
hasVersion	formulation for length metric spaces ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	closed and bounded subsets of a complete Riemannian manifold are compact ⓘ geodesic completeness implies existence of minimizing geodesics between points ⓘ metric completeness implies geodesic completeness ⓘ
namedAfter	Heinz Hopf NERFINISHED ⓘ Willi Rinow NERFINISHED ⓘ
relatedConcept	Cauchy sequence ⓘ Riemannian distance ⓘ complete metric space ⓘ exponential map ⓘ geodesic ⓘ length space ⓘ minimizing geodesic ⓘ proper metric space ⓘ
relates	metric completeness and compactness of closed and bounded sets ⓘ metric completeness and existence of minimizing geodesics ⓘ metric completeness and geodesic completeness ⓘ
states	for a connected Riemannian manifold the following conditions are equivalent ⓘ
subject	Riemannian manifold ⓘ geodesic completeness ⓘ geodesics ⓘ length spaces ⓘ metric completeness ⓘ
usedIn	comparison geometry ⓘ geometric analysis ⓘ global Riemannian geometry ⓘ study of completeness of Riemannian metrics ⓘ

Referenced by (1)

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Heinz Hopf → notableWork → Hopf–Rinow theorem ⓘ