Questions about Hopf-Rinow theorem

Question

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:

1. 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.
2. 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.

Answer 1

1. 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 $\mathbb{H}_{\mathbb{R}}^n$.

2. As they told you, the connectedness question has been developed in Connectedness and Hopf-Rinow Theorem .

Answer 2

1. Well, as the Hopf-Rinow theorem suggests, it would suffice to show that $\mathbb{H}^n$ is complete as a metric space; if you want to do this by hand, you have to take a Cauchy sequence in the hyperbolic space, and show that it converges to a point in $\mathbb{H}^n$ (endowed with the hyperbolic metric, of course; perhaps using the upper half-plane model is a good idea for this).

2. To see why you need this, the easy answer is that even defining what geodesically complete means in the disconnected case is somewhat strange; for instance, in the context of Riemannian manifolds, your distance function is given by the infimum of all curves joining two given points. If no such curve exists, what is the distance in this case?

As far as I know there are some alternative definitions for these cases, but they are far less interesting than the usual connected cases.