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

This papers studies properties of caustics of different billiard models in dimension d, with \ (d 3\). Namely, we consider the cases where the law of reflection is defined by 1) a Riemannian metric projectively equivalent to the Euclidean one; 2) a constant non-degenerate quadratic form (pseudo-Euclidean billiards) ; 3) a smooth field of transverse lines to the boundary defining a law of reflection (projective billiards). Cases 1) and 2) are particular cases of 3). The paper gives a necessary and sufficient condition so that if such billiards have a caustic then the latter is a quadric. In the case of pseudo-Euclidean billiards, we even show that the only billiards having a caustic are the quadrics, for which the caustics are pseudo-confocal quadrics.