Let $S$ be a surface embedded in $\mathbb{R}^3$.
A *simple* geodesic on $S$ is one that does not self-intersect.
Some surfaces have simple geodesics whose length exceeds any
given bound $L$. For example, a cylinder or a torus allows tight
winding geodesics that are arbitrarily long before they cross themselves.
But a sphere, or a Zoll surface,
does not admit arbitrarily long simple geodesics, because every geodesic
forms a simple closed loop.

. Which surfaces $S$ admit arbitrarily long simple geodesics?Q

To be specific: Do ellipsoids possess such geodesics?

**Update**(

*11 May 2017*).

This paper settles a version of my 2-yr-old question by
proving that "if the surface of a convex body $K$ contains arbitrary long ** closed** simple geodesics, then $K$ is an isosceles tetrahedron":

Akopyan, Arseniy, and Anton Petrunin. "Long geodesics on convex surfaces." arXiv preprint arXiv:1702.05172 (2017).

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