It is known that there are at least three simple, closed geodesics on the surface of any smooth convex body $K$ in $\mathbb{R}^3$, the Lusternik-Schnirelmann Theorem (see links below for references). For $\gamma_1$ and $\gamma_2$ two such geodesics, let $\alpha_\max( \gamma_1, \gamma_2 )$ be the largest angle among all the crossings of $\gamma_1$ and $\gamma_2$. My question is:

Q. Is there a lower bound on $\alpha_\max$, over all simple, closed geodesics on $K$, over all $K$? Or is it possible that, for some $K$, all crossing angles of all geodesic pairs can be arbitrarily small?


     Image from the article Geodesics on an ellipsoid.

  • $\begingroup$ @dodd: One can ask my question for the torus or for other surfaces, but I am specifically interested in convex surfaces, where the L-S theorem guarantees a few simple, closed geodesics. $\endgroup$ – Joseph O'Rourke Dec 7 '20 at 21:15
  • $\begingroup$ If you have only a few (finitely many) closed geodesics, there is a low bound of the angles. $\endgroup$ – Mark Sapir Dec 7 '20 at 21:19
  • $\begingroup$ @dodd: I'm afraid I don't understand your remark. What I'm seeking is a lower bound over all possible convex surfaces. I want to say: there must exist a pair of simple, closed geodesics that cross at angle $\ge \alpha$, where $\alpha$ holds over all convex surfaces. $\endgroup$ – Joseph O'Rourke Dec 7 '20 at 22:07
  • 2
    $\begingroup$ That is not what you asked. In your question $K$ is fixed. $\endgroup$ – Mark Sapir Dec 7 '20 at 22:16
  • $\begingroup$ @dodd: Corrected; thank you! $\endgroup$ – Joseph O'Rourke Dec 8 '20 at 0:09

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