In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to infinity in finite time."
(To be clear, $g$ is the round metric on the sphere being pulled back to $\mathbb{R}^n$ via stereographic projection.)
This is supposed to be in contrast to Hopf-Rinow, but I'm completely confused: $\mathbb{R}^n$ IS a complete metric space. So this would be a counterexample to Hopf-Rinow, which is absurd....
A space is geodesically complete if all geodesics exists for all time. With the standard metric on $\mathbb{R}^n$, the geodesics are exactly straight lines parameterized by constant speed, and have a full domain $\mathbb{R}$, or are constant curves. The geodesics of the stereographic projection metric reach infinity in finite time.
It is also nice to see that if $n>2$ the points $(-x,0,\ldots,0)$ and $(x,0,\ldots,0)$ get close to each other if $x\rightarrow \infty$. Thus these spaces are also metrically incomplete.
What happens for $n=1$?