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In John M. Lee "Riemannian manfidolds: an introduction to curvature", page 108, as a previous motivation for Hopf-Rinow theorem, there is an example of non geodesically complete manifold:

"$\mathbb{R}^n$ with the metric $(\sigma^{-1})^{*}\mathring{g}$ obtained from the sphere by stereographic projection"

In the book says that there are geodesics that escape to infinity in finite time.

I am trying to prove it for the case $n=2$. I was intending to compute explicity the pull back via the inverse of stereographic projection (https://en.wikipedia.org/wiki/Stereographic_projection#First_formulation) of the round metric in the sphere, but I did not get anywhere. Do you know how to prove it? (Without using Hopf-Rinow theorem, I want to see this as a motivation for that theorem).

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Let $\mathbb{S}^n$ be the $n$ dimensional sphere, $N = (0,\ldots,0,1)$ its north pole and $S$ its south pole. Let $g$ be the usual round metric on $\mathbb{S}^n$. Then $\left(\mathbb{S}^n \setminus\{N\},g\right)$ is a Riemannian manifold.

Suppose $v$ is a unit tangent vector at $S$. Define $$ \forall t \in \left(-\pi,\pi\right),~\gamma(t) = \cos t \cdot S + \sin t \cdot v $$ Then $\gamma$ is a geodesic. It goes "out" of $\mathbb{S}^n\setminus\{N\}$ in finite time (at $t = \pi$).

Now, let $f : \mathbb{S}^n\setminus\{N\} \overset{\sim}{\to} \mathbb{R}^{n+1}$ be the stereographic projection. It is a diffeomorphism, which allow us to push forward the metric $g$ on $\mathbb{R}^{n+1}$, say $h = f_*g$ (or $h = (f^{-1})^*g$), so that it becomes an isometry.

Being an isometry, $f$ maps geodesics to geodesics, and thus, $f\circ \gamma : \left(-\pi,\pi\right) \to \mathbb{R}^{n+1}$ is a geodesic. It goes to infinity in finite time ($t = \pi$).

Remark that if $(M,g)$ is a connected Riemannian manifold and if $x \in M$, then $(M\setminus\{x\},g)$ cannot be complete. Any geodesic of $M$ passing through $x$ can be viewed, by a sort of local stereographic projection, as a geodesic in $M\setminus\{x\}$ going out of $M\setminus\{x\}$ in finite time. Here, the stereographic projection identify $x$ with "infinity".

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