Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space

Abstract

Billiard problems for the domains on the sphere and hyperbolic space bounded by the corresponding conical sections are investigated. It is shown that these discrete systems are integrable and correspond to the translations on the Jacobi varieties of certain hyperelliptic curves. The explicit formulas in terms of θ-functions are exhibited. The consideration is based on the factorization method, developed recently by J. Moser and the author.