Two constructions in the path geometry of surfaces: chains and dancing

Wojciech Kryński, Omid Makhmali

Published 2023-03-15Version 1

Given a path geometry on a surface $M$, there are two constructions via which the 3-dimensional projectivized tangent bundle $\mathbb{P}TM$ can be endowed with a (generalized) path geometry. One of these constructions is given by the well-known class of chains. The other construction is referred to as the dancing construction. In this article we provide necessary and sufficient conditions that determine whether a path geometry in dimension three arises via one of these two constructions.