Statements (21)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
complete Riemannian manifolds
manifolds with non-positive sectional curvature simply connected manifolds |
| gptkbp:consequence |
exponential map at any point is a covering map
no closed geodesics in such manifolds |
| gptkbp:field |
gptkb:Riemannian_geometry
differential geometry |
| https://www.w3.org/2000/01/rdf-schema#label |
Cartan-Hadamard theorem
|
| gptkbp:implies |
universal covering space of such a manifold is diffeomorphic to Euclidean space
|
| gptkbp:namedAfter |
gptkb:Jacques_Hadamard
gptkb:Élie_Cartan |
| gptkbp:relatedTo |
gptkb:Hadamard's_theorem
gptkb:Cartan's_theorem |
| gptkbp:sentence |
A complete, simply connected Riemannian manifold of non-positive sectional curvature is diffeomorphic to Euclidean space.
|
| gptkbp:usedIn |
topology of manifolds
global differential geometry study of geodesics |
| gptkbp:yearProposed |
early 20th century
|
| gptkbp:bfsParent |
gptkb:Riemannian_manifold
|
| gptkbp:bfsLayer |
5
|