Rigidity on horocycles and hypercycles

2024/05/27 by Lo, Cheikh, Sane, Abdoul Karim · 1 citation
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2405.17598

Abstract

We show that a bijection f:ℍ²→ℍ² of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics to all curves with constant curvature in ℍ². We go beyond by showing that every abstract automorphism of the geodesic graph (respectively horocycles and hypercycles graphs) is induced by an earthquake map (respectively an isometry) of ℍ². This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles.

Cited by

• Geodesic-preserving bijections of the Thurston geometries

Related

• Weaving Geodesics and New Phenomena in Horocyclic Dynamics
• Fundamental theorem of hyperbolic geometry without the injectivity assumption
• Bi-geodesic mappings between hyperbolic surfaces with boundary
• Slice rigidity property of holomorphic maps Kobayashi-isometrically\n preserving complex geodesics
• Packable hyperbolic surfaces with symmetries