Riemannian Hopf–Rinow theorem
GPTKB entity
Statements (13)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
gptkb:Riemannian_manifolds
|
| gptkbp:field |
differential geometry
|
| https://www.w3.org/2000/01/rdf-schema#label |
Riemannian Hopf–Rinow theorem
|
| gptkbp:implies |
compactness of closed and bounded subsets in complete Riemannian manifolds
|
| gptkbp:namedAfter |
gptkb:Willi_Rinow
gptkb:Heinz_Hopf |
| gptkbp:publishedIn |
gptkb:Mathematische_Annalen
|
| gptkbp:state |
A Riemannian manifold is geodesically complete if and only if it is complete as a metric space.
If a Riemannian manifold is complete as a metric space, then any two points can be joined by a minimizing geodesic. |
| gptkbp:year |
1931
|
| gptkbp:bfsParent |
gptkb:Riemannian_geometry
|
| gptkbp:bfsLayer |
5
|