Cartan-Hadamard theorem

This article describes a result related to the sectional curvature of a Riemannian manifold

This result relates information on curvature to information on topology of a manifold

This article makes a prediction about the universal cover of a manifold based on given data at the level of a:Riemannian manifold

This result is valid in all dimensions

Statement

Any negatively curved manifold, viz any manifold which has negative sectional curvature everywhere, has the property that its universal cover is diffeomorphic to real Euclidean space.

Relation with other results

Bonnet-Myers theorem

Further information: Bonnet-Myers theorem

The Bonnet-Myers theorem states that the universal cover of a manifold with Ricci curvature bounded below by a positive number, is compact.