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

Here are some equivalent formulations:

1. 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.
2. Any complete simply connected nonpositively curved manifold is diffeomorphic to $\mathbb {R} ^{n}$ (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

Bonnet-Myers theorem

Further information: Bonnet-Myers theorem

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

Facts used

1. Hopf-Rinow theorem
2. Nonpositively curved implies conjugate-free
3. Local isometry of complete Riemannian manifolds is covering map