{"paper":{"title":"Outer billiards in the spaces of oriented geodesics of the three dimensional space forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.DS","authors_text":"Marcos Salvai, Michael Harrison, Yamile Godoy","submitted_at":"2021-10-04T19:24:40Z","abstract_excerpt":"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.\n  F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.01679","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.01679/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}