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

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