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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?

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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.

Thomas
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  • Thank you for your answer. Can you briefly give an example of a complete, but not compact, surface satisfying the conditions? –  May 03 '16 at 18:38
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    Actually no, not without thinking about it for a while. I'd try to start looking for one by looking at a simple function $z=z(x)$ on, say $(1,2)$ (like a scaled and translated $\cosh$) which is positive and unbounded near the boundary of the interval and would rotate it around the $z$ axis in $x, y, z$ space. – Thomas May 03 '16 at 18:45
  • @mysteriousmathstudent I added a sketch to my answer – Thomas May 03 '16 at 18:52