Outer billiards on spaces of oriented geodesics

GODOY, YAMILE; HARRISON, MICHAEL; SALVAI, MARCOS

Presentación virtual

Congreso; Differential Geometry, Billiards, and Geometric Optics. Congreso híbrido; 2021

Centre International de Rencontres Mathématiques, Marsella

For k = 0,1,-1, let M_k be the three dimensional space form of curvature k, that is, R^3, S^3 and hyperbolic 3-space H^3. Let G_k be the manifold of all oriented (unparametrized) complete geodesics of M_k, i.e., G_0 and G_{-1} consist of oriented lines and G_1 of oriented great circles.Given a strictly convex surface S of M_k, we define an outer billiard map B_k on G_k. 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 B_k is a diffeomorphism under the stronger condition that S is quadratically convex.We prove that B_1 and B_{-1} arise in the same manner as Tabachnikov´s original construction of the higher dimensional outer billiard on standard symplectic space ( R^{2n} , omega ). For that, of the two canonical Kähler structures that each of the manifolds G_1 and G_{-1} admits, we consider the one induced by the Killing form of Iso( M_k). We prove that B_1 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}.