Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

Abstract

We consider a convex curve ${\hat{I}}^3$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by ${\hat{I}}^3$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $\hat{I}“$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.

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