I agree completely, Igor - it's clearly the geometry, rather than the topology, that's important.

Now I'm just confused about the quantifiers. Is the question "does every surface in R^3 have a closed geodesic?" or "does every surface in R^3 have an infinite geodesic?", or even "does every surface in R^3 have an infinite non-self-intersecting geodesic?"?

I presume you want your surface to be compact? Also, when you say "the ant will walk along the curvature of the surface", do you mean that it follows a local geodesic?

Right! I felt this was an important part of the original question, and deserved a little emphasis.

While I agree with you and Richard, I think Ilya's right to want the opinion of someone who's actually sat on a hiring committee. (Which I presume no postdoc has done.)

To be precise, I suppose I mean a proof that doesn't involve homotopy lifting.

Having just taught this result, I definitely agree with Qiaochu! Sam, if you can write down a "tautological" proof of this fact, I'd like to see it!

I don't know about E8xR, but crossing with R sometimes makes things better. For instance, the Cannon-Edwards Theorem asserts that the double suspension of a homology sphere is homeomorphic to a sphere. It follows that although the cone C on a homology sphere is not a manifold, Cx(R^2) is a manifold, remarkably. On the the other hand, this gives a way of building "bad" triangulations of manifolds.