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 anygiven bound $L$. For example, a cylinder or a torus allows tightwinding 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 geodesicforms a simple closed loop.
Q. Which surfaces $S$ admit arbitrarily long simple geodesics?
To be specific: Do ellipsoids possess such geodesics?
Update (11 May 2017).
This paper settles a version of my 2-yr-old question byproving 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).