There are a whole bunch of theorems in Riemannian geometry to the effect that “if the Riemannian manifold has property A of the curvature, then it has the topological property B.” Over the rest of MaBloWriMo and in the following weeks, I aim to talk about a few such results. The first one characterizes manifolds of negative curvature.

**Negative curvature **

Let be a Riemannian manifold with Riemannian metric Say that has **negative curvature** if for all ,

(Later I will interpret this in terms of the sectional curvature, which I have not yet defined.)

**Statements **

**Theorem 1 (Cartan-Hadamard)** *Let be a complete Riemannian manifold of negative curvature. Then for , the map is a covering map. In particular, if is simply connected, then it is diffeomorphic to . *

Of course, the diffeomorphism doesn’t have to preserve the Riemannian metric.

The strategy of the proof is as follows. First, we will show that the map is an immersion (though in general not injective), using the discussion yesterday about how Jacobi fields determine the differential of the exponential map. Then we will invoke

Proposition 2Let be a complete Riemannian manifold. Suppose and the map is an immersion. Then is a covering map.

The condition of the result is often stated to the effect that “ has no conjugate points to .”

To see this, we will appeal to yet another result:

Theorem 3 (Ambrose)Let be a surjective morphisms of Riemannian manifolds with complete. Suppose preserves the metric on the tangent spaces. Then is a covering map.

I will work backwards to prove these three results. (more…)