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

TL;DR

This work defines the outer billiard map $B$ on the space $\mathcal{G}_{\kappa}$ of oriented geodesics in the 3D space forms $M_{\kappa}$, using the exterior of a smooth strictly convex surface $S$ as the billiard table. It establishes that $B$ is a diffeomorphism when $S$ is quadratically convex and builds a rich geometric framework by endowing $\mathcal{G}_{\kappa}$ with canonical Kähler structures $(g_K,\mathcal{J})$ and $(g_{\times},\mathcal{J})$, linking $B$ to Tabachnikov's construction via $(g_K,\mathcal{J})$ for $\kappa=\pm1$. It proves that for $\kappa=1,-1$, $B$ is a symplectomorphism with respect to the fundamental form $\omega_K$, while $B$ does not preserve $\omega_{\times}$; in the Euclidean setting a Poisson structure governs parallel leaves and yields area-preserving restrictions. In the hyperbolic case, the authors introduce a holonomy notion for periodic orbits and demonstrate the existence of periodic points with nonzero holonomy, indicating rich and largely open dynamical behavior. Overall, the paper generalizes outer billiard dynamics to curved, higher-dimensional geodesic spaces and reveals deep connections between convexity, Jacobi-field analysis, and symplectic geometry.

Abstract

Let $M_{κ}$ be the three-dimensional space form of constant curvature $κ=0,1,-1$, 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 $M_{κ}$. We define an outer billiard map $B$ on the four dimensional space $\mathcal{G}_{κ}$ of oriented complete geodesics of $M_{κ}$, for which the billiard table is the subset of $\mathcal{G}_{κ}$ consisting of all oriented geodesics not intersecting $S$. We show that $B$ is a diffeomorphism when $S$ is quadratically convex. For $κ=1,-1$, $\mathcal{G}_{κ}$ has a Kähler structure associated with the Killing form of $\operatorname{Iso}(M_{κ})$. 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 $\mathcal{G}_{κ}$ associated with the cross product on $M_{κ}$, for $κ=0,1,-1$. We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Figures (7)

• Figure 1: The outer billiard map in the plane
• Figure 2: The geodesic $\ell$ is in the outer billiard correspondence with $\ell^{\prime }$ and $\ell^{\prime \prime }$
• Figure 3: The outer billiard map on $\mathcal{L}_{\kappa }$ associated with $S$
• Figure 4: The linear transformation $\mathcal{J}_\ell$ maps the green variation of geodesics to the red variation of geodesics
• Figure 5: The parallel transport of $u$ along $\gamma _{iu}$ between $0$ and $t$

Theorems & Definitions (33)

• Proposition 1.1
• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Proposition 1.9
• proof : Proof of Proposition \ref{['esfera']}