On curves with Poritsky property

Abstract

For a given closed convex planar curve $\gamma$ with smooth boundary and a given $p>0$, the string construction yields a family of nested billiards $\Gamma_p$ for which $\gamma$ is a caustic. The action of the corresponding reflections $T_p$ on the tangent lines to $\gamma$ induces their actions on the tangency points: a family of string diffeomorphisms $\mathcal T_p:\gamma\to\gamma$. We say that $\gamma$ has string Poritsky property, if it admits a parameter $t$ (called 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 $\gamma$. Poritsky property is closely related to the famous Birkhoff Conjecture. It is classically known that each conic has string Poritsky property. In 1950 H.Poritsky proved the converse: each germ of planar curve with Poritsky property is a conic. In the present paper we extend this Poritsky's result to germs of curves on simply connected complete surfaces with Riemannian metric of constant curvature and to outer billiards on all these surfaces. In the general case of curves with Poritsky property on any two-dimensional surface with Riemannian metric we prove the two following results: 1) the Poritsky string length coincides with Lazutkin parameter, introduced by V.F.Lazutkin in 1973, up to additive and multiplicative constants; 2) a germ of $C^5$-smooth curve with Poritsky property is uniquely determined by its 4-th jet. In the Euclidean case the latter statement follows from the above-mentioned Poritsky's result.