Geodesic-preserving bijections of the Thurston geometries

2025/10/29 by Dickmann, Ryan, Lideros, Palani, Narayanan, Akash
#51M10 #57M50 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2510.25585

Abstract

We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder \mathbbS¹ × ℝ and the bijections of the hyperbolic plane ℍ² that preserve constant curvature curves.

Citations

• Bi-geodesic mappings between hyperbolic surfaces with boundary
• Rigidity on horocycles and hypercycles
• Complete totally geodesic subsets of the complex hyperbolic plane: an elementary classification
• The fundamental theorem of affine geometry on tori
• The Fundamental Theorems of Affine and Projective Geometry Revisited
• Fundamental theorem of hyperbolic geometry without the injectivity assumption
• The Geometries of 3-Manifolds

Related

• Classical Notions and Problems in Thurston Geometries
• Bijections on strictly convex sets and closed convex projective surfaces that preserve complete geodesics
• Rigidity on horocycles and hypercycles
• Submersions and curves of constant geodesic curvature
• The infinitesimal and global Thurston geometry of Teichmüller space