|
|
| (8 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| {{sectional curvature result}}
| | #redirect [[Bonnet-Myers theorem]] |
| | |
| ==Statement==
| |
| | |
| If a [[Riemannian manifold]] has the property that there exists a positive constant that lower-bounds the sectional curvature for all tangent planes at all points, then the manifold is compact with finite fundamental group.
| |
| | |
| ==Relation with other results==
| |
| | |
| ===Cartan-Hadamard theorem===
| |
| | |
| The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.
| |
| | |
| Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.
| |