This result relates information on curvature to information on topology of a manifold
This result is valid in all dimensions
Here are some equivalent formulations:
- Any complete nonpositively curved manifold, viz., any manifold which has nonpositive sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space, In fact, the exponential map at any point is a covering map.
- Any complete simply connected nonpositively curved manifold is diffeomorphic to (such a manifold is termed a CH-manifold). In fact, the exponential map at any point is a diffeomorphism.
The equivalence follows from the fact that the universal cover of a Riemannian manifold can be given the pullback metric, in which case the range of values taken by the sectional curvature is the same for both spaces.
Relation with other results
Further information: Bonnet-Myers theorem