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Finite-reflection property for cones over general submanifolds

Characterize the class of submanifolds Γ ⊂ R^{n−1} for which every billiard trajectory in the cone K over Γ undergoes only finitely many reflections, and determine whether the positive-definiteness assumption on the second fundamental form can be relaxed (e.g., to semidefinite or indefinite) and whether non-spherical topologies such as tori are allowed.

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Background

For cones over C3 strictly convex hypersurfaces with positive definite second fundamental form, every trajectory has finitely many reflections, though no uniform bound is possible.

The authors ask to extend this finite-reflection property beyond positive definite curvature and spherical topology, exploring broader geometric classes of generating submanifolds.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (3) For which submanifolds $\Gamma\subset{\mathbb R}{n-1}$ any trajectory inside the cone $K$ over $\Gamma$ has finite number of reflections? Can one relax the condition of positive definiteness of the second fundamental form in Theorem \ref{cone-bounded}? For example, can the second fundamental form of $\Gamma$ be positive semi-definite, or indefinite? Can $\Gamma$ be of a topological type different from sphere, for example, of a torus?

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Wire and cone billiards, item (3)