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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.

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

To be specific: Do ellipsoids possess such geodesics?

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).

NB: closed simple geodesics.

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.

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

To be specific: Do ellipsoids possess such geodesics?

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).

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.

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

To be specific: Do ellipsoids possess such geodesics?

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.

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

To be specific: Do ellipsoids possess such geodesics?

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).

NB: closed simple geodesics.

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.

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

To be specific: Do ellipsoids possess such geodesics?

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.

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

To be specific: Do ellipsoids possess such geodesics?