How to show a simply-connected negatively-curved n-dimensional Riemannian manifold is homeomorphic to $R^n$?

Question

I saw this word in a paper , but there is not proof . And a simple hint is given that for fixing a poing $0\in M$ , mapping the $T_0M$ noto $M$ via exponential map. But it is not enough to say it is homeomorphic . How to show it ?

Answer 1

The homeomorphism is given by mapping point $rv \in T_0M$, where $ r \in \Bbb R^{+}$ and $\|v\| = 1$, to the point to $\exp_0(rv) \in M$.

You still have to show that this map is injective, but that's what the negative curvature is for: you use it to estimate the distance to the nearest "focal point" of $\exp$, which turns out to be infinitely far away.