Chains of path geometries on surfaces: theory and examples

TL;DR

This paper develops a simplified method to derive chain equations for path geometries on surfaces, characterizes projective path geometries via chains, and explores examples including hyperbolic horocycles with bicircular quartic chains.

Contribution

It introduces a new, simpler approach to derive chain equations for path geometries using sub-Riemannian structures, and applies this to characterize projective geometries and analyze examples.

Findings

Derived chain equations for path geometries on surfaces.

Characterized projective path geometries through their chains.

Identified bicircular quartic chains in hyperbolic horocycles.

Abstract

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.