Title: Only quadrics have pseudo-caustics -- on caustics of Riemannian, pseudo-Euclidean and projective billiards in higher dimensions Show Chinese title

Title: 只有二次曲面才有伪焦散线——高维黎曼、伪欧几里得和射影弹道的焦散线

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. Show Chinese abstract

Abstract: 本文研究具有广义反射定律的台球模型，即所谓的投影台球。它们统一了各种定律，包括欧几里得、伪欧几里得或黎曼度量中的经典定律。它们由Sergey Tabachnikov引入并研究。本文在至少三维的情况下研究这些模型，并关注焦散线的存在性和性质，即任何光线轨迹在连续反射后仍保持相切的超曲面。然后，它将研究限制在黎曼台球的特殊情况，其测地线由直线支撑，以及伪欧几里得台球。更准确地说，本文1）描述了仅可能具有二次曲面焦散线的投影台球类；2）将此结果转化为黎曼台球；3）将其应用于显示，如果一个伪欧几里得台球有焦散线，则两者都是伪共焦点二次曲面。