2022/01/22 by Gil Bor, Bor, Gil, Travis Willse +1
Mathematics · #53A04 #53A05 #53A40 #53A55 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2201.09141
openalex publication_date 2022/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature (1,1) on a contact 3-manifold. This approach is significantly simpler than the standard method of solving the full equivalence problem for path geometry. We then use these equations to give a characterization of projective path geometries in terms of their chains (the chains projected to the surface coincide with the paths) and study the chains of four examples of homogeneous path geometries. In one of these examples (horocycles in the hyperbolic planes) the projected chains are bicircular quartics.