Statements (16)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
gptkb:Riemannian_manifolds
|
| gptkbp:field |
gptkb:Riemannian_geometry
differential geometry |
| https://www.w3.org/2000/01/rdf-schema#label |
Hopf-Rinow theorem
|
| gptkbp:implies |
geodesic completeness
|
| gptkbp:namedAfter |
gptkb:Werner_Rinow
gptkb:Heinz_Hopf |
| gptkbp:publishedIn |
gptkb:Mathematische_Annalen
|
| gptkbp:relatedTo |
compactness
geodesics metric completeness |
| gptkbp:state |
For a connected Riemannian manifold, the following are equivalent: the manifold is complete as a metric space, closed and bounded subsets are compact, and any two points can be joined by a minimizing geodesic.
|
| gptkbp:yearProved |
1931
|
| gptkbp:bfsParent |
gptkb:Riemannian_manifold
|
| gptkbp:bfsLayer |
5
|