Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

For κ = 0, 1,−1, let Mκ be the three dimensional space form of curvature κ, that is, R3, S3 and hyperbolic 3-space H3. Let Gκ be the manifold of all oriented (unparametrized) complete geodesics of Mκ, i.e., G0 and G−1 consist of oriented lines and G1 of oriented great circles. Given a strictly convex surface S of Mκ, we define an outer billiard map Bκ on Gκ. The billiard table is the set of all oriented geodesics not intersecting S, whose boundary can be naturally identified with the unit tangent bundle of S. We show that Bκ is a diffeomorphism under the stronger condition that S is quadratically convex. We prove that B1 and B−1 arise in the same manner as Tabachnikov’s original construction of the higher dimensional outer billiard on standard symplectic space ( R2n, ω ) . For that, of the two canonical Kähler structures that each of the manifolds G1 and G−1 admits, we consider the one induced by the Killing form of Iso (Mκ). We prove that B1 and B−1 are symplectomorphisms with respect to the corresponding fundamental symplectic forms. Also, we discuss a notion of holonomy for periodic points of B−1.