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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.

Three-Period Orbits in Billiards on the Surfaces of Constant Curvature