I am stuck on the following problem, which was given as homework.
What is an example of a 2-dimensional surface in $\mathbb{R}^3$ such that it's not possible to find a unit-speed geodesic $\sigma: (-\infty, \infty) \to \mathbb{R}$ satisfying $\sigma(0)=p$ and $d(\sigma(t),p)=|t|$ for all $t\in \mathbb{R}$?
I think it might be some surface of revolution but I can't really picture it. Any suggestions?
The crucial point is the relation $d(\sigma(t),p) = p$, which means that $\sigma$ is globally minimizing.
This fails to be true for every curve if the surface $M$ is bounded, since then the distance is bounded, too.
(Regarding the question in your comment: if $M$ is complete but not compact the question is more difficult, because the task is to show that no geodesic is globally minimizing).
Edit: here is a simple example of a complete, unbounded surface: look at a surface which consists of an infinite half cylinder with the $z$-axis as center and with a hemisphere attached at the bottom (like a cigar of infinite length in one direction) (and smoothed out along the curve where they are glued together). Whenever you look at two points on the surface at the same $z$ -level, the mininimizing geodesic will just wind around the cylinder and will fail to be minimizing after half a turn. But also if you look at geodesics which don't have a fixed $z$ coordinate the will go up and down with respect to the $z$-direction, since the surface is bounded from below in that direction, so eventually they will fail to be minimizing, since a curve with fixed $z$ coordinat will be shorter.