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This content will become publicly available on June 1, 2025

Title: Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms
Abstract Let be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.  more » « less
Award ID(s):
1926686
PAR ID:
10534650
Author(s) / Creator(s):
; ;
Publisher / Repository:
London Mathematical Society
Date Published:
Journal Name:
Journal of the London Mathematical Society
Volume:
109
Issue:
6
ISSN:
0024-6107
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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