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

Abstract

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 Kähler structure associated with the Killing form of $\mathrm{Iso}(M_{\kappa })$. 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.

We corrected some typos and two signs in the last display in the proof of Proposition 1.3