On curves with the Poritsky property

Abstract

Reflection in planar billiard acts on oriented lines. For a given closed convex planar curve \(\gamma \), the string construction yields a one-parameter family \(\Gamma _p\) of nested billiard tables containing \(\gamma \) for which \(\gamma \) is a caustic: the reflection from \(\Gamma _p\) sends each tangent line to \(\gamma \) to a line tangent to \(\gamma \). The reflections from \(\Gamma _p\) act on the corresponding tangency points, inducing a family of string diffeomorphisms \(\mathcal T_p:\gamma \rightarrow \gamma \). We say that \(\gamma \) has the string Poritsky property, if it admits a parameter t (called the Poritsky string length) in which all the transformations \(\mathcal T_p\) with small p are translations \(t\mapsto t+c_p\). These definitions also make sense for germs of curves \(\gamma \). The Poritsky property is closely related to the famous Birkhoff Conjecture. Each conic has the string Poritsky property. Conversely, each germ of planar curve with the Poritsky property is a conic (Poritsky, 1950). In the present paper, we extend this result of Poritsky to curves on surfaces of constant curvature and to outer billiards on all these surfaces. For curves with the Poritsky property on a surface with arbitrary Riemannian metric, we prove the following two results: 1) the Poritsky string length coincides with Lazutkin parameter up to additive and multiplicative constants; (2) a germ of \(C^5\)-smooth curve with the Poritsky property is uniquely determined by its 4-jet. In the Euclidean case, the latter statement follows from the above-mentioned Poritsky’s result on conics.