Hopf–Rinow theorem
E679322
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hopf–Rinow theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7648311 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hopf–Rinow theorem Context triple: [Heinz Hopf, notableWork, Hopf–Rinow theorem]
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A.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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B.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
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C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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D.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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E.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hopf–Rinow theorem Target entity description: 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.
-
A.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
-
B.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
-
C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
D.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
E.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
- F. None of above. chosen
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
ⓘ
Willi Rinow ⓘ |
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hopf–Rinow theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.