# Angles between simple, closed geodesics on convex surface

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.

• @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. – Joseph O'Rourke Dec 7 '20 at 21:15
• If you have only a few (finitely many) closed geodesics, there is a low bound of the angles. – Mark Sapir Dec 7 '20 at 21:19
• @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. – Joseph O'Rourke Dec 7 '20 at 22:07
• That is not what you asked. In your question $K$ is fixed. – Mark Sapir Dec 7 '20 at 22:16
• @dodd: Corrected; thank you! – Joseph O'Rourke Dec 8 '20 at 0:09