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....

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    $\begingroup$ $\mathbb{R}^{n}$ is a complete metric space with the standard metric. Completeness depends on metric. $\endgroup$ – Pedro Oct 2 '14 at 16:12

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 $n=1$?


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