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:

• 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
• Any complete simply connected nonpositively curved manifold is diffeomorphic to $\mathbb {R} ^{n}$ (such a manifold is termed a CH-manifold)

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.

Proof

Main ingredient of proof

The main ingredient is to show that the exponential map from the tangent space at any point, to the manifold, is a well-defined covering map.

Proof details

First, the following observations:

• Since the manifold is complete, the exponential map from the tangent space at any point, to the whole manifold, is well-defined and surjective (we here invoke the Hopf-Rinow theorem)
• Since the manifold is nonpositively curved, it is conjugate-free. In other words, no two points in it are conjugate. In other words, given any two points, any two geodesics joining the two points are "far away".
• We know that the map $\exp _{p}:T_{p}M\to M$ looks like a covering map at any point $q$ in the manifold not conjugate to $p$.

Piecing together these facts, the exponential map is a covering map from $\mathbb {R} ^{n}$ to the manifold.

This proves the result.