Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

On the round sphere $\mathbb S^2$ or the hyperbolic plane $\mathbb H^2$, billiard trajectories follow geodesics and reflect at a smooth strictly convex boundary with equal angles. The $\delta$-Gutkin property means that the family arriving at the boundary at the fixed angle $\delta\in(0,\pi/2)$ is invariant under this billiard map: the next chord has the same contact angle. The source motivates the question using noncircular plane curves.

Statement recorded in the supplied source.

Let $\gamma$ be a smooth closed strictly convex curve and $\delta\in(0,\pi/2)$. Say that $\gamma$ has the $\delta$-Gutkin property if the curve of incoming oriented lines meeting $\gamma$ at the constant angle $\delta$ is invariant under the Birkhoff billiard. What are the Gutkin billiards on the sphere and on the hyperbolic plane? [S2]

Short English statement

Classify the Gutkin billiard curves on the sphere and hyperbolic plane, including the possibility of noncircular curves.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200003 (AMR-051-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Misha Bialy, Problem 3, p. 3. https://arxiv.org/pdf/2110.10750v2.