Outer billiards in the spaces of oriented geodesics of the three-dimensional space forms

Let M kappa$M_{\kappa }$ be the three-dimensional space form of constant curvature kappa=0,1,-1$\kappa =0,1,-1$, that is, Euclidean space R3$\mathbb {R}<^>{3}$, the sphere S3$S<^>{3}$, or hyperbolic space H3$H<^>{3}$. Let S$S$ be a smooth, closed, strictly convex surface in M kappa$M_{\kappa }$. We define an outer billiard map B$B$ on the four-dimensional space G kappa$\mathcal {G}_{\kappa }$ of oriented complete geodesics of M kappa$M_{\kappa }$, for which the billiard table is the subset of G kappa$\mathcal {G} _{\kappa }$ consisting of all oriented geodesics not intersecting S$S$. We show that B$B$ is a diffeomorphism when S$S$ is quadratically convex. For kappa=1,-1$\kappa =1,-1$, G kappa$\mathcal {G}_{\kappa }$ has a K & auml;hler structure associated with the Killing form of Iso(M kappa)$\operatorname{Iso}(M_{\kappa })$. We prove that B$B$ is a symplectomorphism with respect to its fundamental form and that B$B$ can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in R2n$\mathbb {R}<^>{2n}$ defined in terms of the standard symplectic structure. We show that B$B$ does not preserve the fundamental symplectic form on G kappa$\mathcal {G}_{\kappa }$ associated with the cross product on M kappa$M_{\kappa }$, for kappa=0,1,-1$\kappa =0,1,-1$. We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.