Three-period orbits in billiards on the surfaces of constant curvature

An approach due to Wojtkovski [15], based on the Jacobi fields, is applied to study sets of three-period orbits in billiards on hyperbolic plane and on two-dimensional sphere. It is found that the set of three-period orbits in billiards on hyperbolic plane, as in the planar case, has zero measure. For the sphere, a new proof of Baryshnikov's theorem is obtained that states that three-period orbits can form a set of positive measure if and only if a certain natural condition on the orbit length is satisfied.