Only quadrics have pseudo-caustics -- on caustics of Riemannian, pseudo-Euclidean and projective billiards in higher dimensions

Abstract

This paper studies billiard models with a generalized law of reflection, the so-called projective billiards. They unify various laws, including the classical one in a Euclidean, pseudo-Euclidean or Riemannian metric. They were introduced and studied by Sergey Tabachnikov. The paper studies these models in dimension at least 3 and focuses on the existece and properties of caustics, i.e. hypersurfaces to which any light trajectory remains tangent to after successive reflections. It restricts then the study to the particular case of Riemannian billiards whose geodesics are supported by lines, and of pseudo-Euclidean billiards. More precisely, this article 1) describes the class of projective billiards whose only possible caustics are quadrics; 2) translates this result to Riemannian billiards; 3) applies it to show that if a pseudo-Euclidean billiard has a caustic, then both are pseudo-confocal quadrics.