Baker's Dozen — Chains of null geodesics

Definitions and mathematical statement

For $a,b,c>0$ consider \[ E=\{(x,y,z):x^2/a+y^2/b+z^2/c=1\},\qquad g=dx^2+dy^2-dz^2. \] The restriction of $g$ to the tangent planes of $E$ degenerates on two closed curves, called tropics, given by $z=\pm c\sqrt{x^2/a^2+y^2/b^2}$. Between them the induced metric has signature $(1,1)$. At every point of this equatorial annulus there are two null tangent lines, characterized by $g(v,v)=0$; their integral curves are the two families of unparametrized null geodesics. Label these families left and right consistently around the annulus.

Starting on a tropic, follow one family to the other tropic, switch families, and repeat. Each tropic-to-tropic arc counts as one step. An $(n,r)$-chain returns to its initial point and outgoing family after $n$ steps, and has winding number $r$ about the equator. Winding is the degree of projection to the angular coordinate in the $(x,y)$-plane; returning to the same tropic forces $n$ to be even.

For prescribed $n$ and $r$, find necessary and sufficient conditions on $(a,b,c)$ for such a chain to exist. A criterion must test these parameters, for example through explicit equations or periods, rather than merely rephrasing closure of the trajectory.

Short English statement

Find an explicit closure criterion for alternating lightlike geodesics on a Minkowski ellipsoid, specifying both the number of steps and the winding.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 600008 (AMR-005-0008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tabachnikov - A Baker's Dozen of Problems (2015), Problem 4. Original source indexed by AMR-005-0008. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/.