English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 convexsurface in $M_{\kappa }$. We define an outer billiard map $B$ on the fourdimensional 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 structureassociated with the Killing form of $\operatorname{Iso}(M_{\kappa })$. We provethat $B$ is a symplectomorphism with respect to its fundamental form and that$B$ can be obtained as an analogue to the construction of Tabachnikov of theouter billiard in $\mathbb{R}^{2n}$ defined in terms of the standard symplecticstructure. We show that $B$ does not preserve the fundamental symplectic formon $\mathcal{G}_{\kappa }$ 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 caseby introducing and discussing a notion of holonomy for periodic points.

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