I am reading about Hopf-Rinow theorem using the Jonh M. Lee "Riemannian manifolds: an introduction to curvature",page 108, Theorem 6.13. My doubts are the next ones:
- In the book, there is an exercise asking to prove that $\mathbb{H}_{\mathbb{R}}^n$, the hiperbolic space, is complete. How can I do this? I know I could just try to compute the geodesics of the space, but I am trying to deduce this from Hopf-Rinow theorem.
- In the statement of the main theorem, one of the hypothesis is for $M$ being connected. I am looking for an example where the completeness fails due to $M$ being not connected, and where is used this hypothesis of being connected in the proof of Hopf-Rinow theorem.
Best Answer
For the hiperbolic space, since every homogeneous Riemannian connected is complete, if you prove that $\mathbb{H}_{\mathbb{R}}^n$ is homogeneous, it's all done. For that, just check $O_{+}(n,1)$ acts transitively on the set of orthonormal bases on $\mathbb{H}_{\mathbb{R}}^n$.
As they told you, the connectedness question has been developed in Connectedness and Hopf-Rinow Theorem .