Complete manifold in the Lee's book.

Question

In the completeness charpter of book "Riemannian manifold: an introduction to curvature", he constructed two examples (at page 108) of non geodesically complete manifold. 1) any proper open subset of $\mathbb{R}^n$ with standard metric is not geodesically complete, because the geodesics reach the boundary in finite time, so what happens to closed subset? I guess it is complete otherwise it violates the Hopf-Rinow theorem, but what the difference between open and closed set. 2) $\mathbb{R}^n$ with the pullback metric from $\mathbb{S}^n$ by stereographic projection, there are geodesics that escape to infinity in finite time, honestly speaking, I have no idea what it means. Could someone explain a little bit about this. Much appreciated.

Answer 1

Remember that if you put a metric on $\mathbb{R}^n$ making a pullback of the metric in $\mathbb{S}^n \subset \mathbb{R}^{n+1}$ by the stereographic projection $\sigma$, then because $\sigma$ is a diffeomorphism (from $\mathbb{S}^n$ minus the North Pole to $\mathbb{R}^n$), it becomes an isometry. Now, isometries take geodesics to geodesics.
Just take a geodesic through the North Pole $N=(0,0, \dots, 0,1)$ (a great circle which is the intersection of $\mathbb{S}^n$ with a plane $P$ through the origin in $\mathbb{R}^{n+1}$ and $N$). Take the piece of it that goes from the South Pole $S=(0,0, \dots, 0,-1)$ to $N$ and parametrize it by acrlength, if you want. This geodesic goes to a straight line through the origin in $\mathbb{R}^n$, and since the one in $\mathbb{S}^n$ is parametrized by arclenght, this straight line $\gamma=\gamma(t)$ is also parametrized by arclenght and it satisfies $\lim_{t\to \pi} \gamma(t)=\infty$. Its lenght between the parameters $0$ and $\pi$ is $\infty$ in the topological sense that in time close $\pi$ it's out of every compact.
I think that's what Lee meant.