# Non complete connected Riemannian manifold

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).

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".