Reductions of path structures and classification of homogeneous structures in dimension three

TL;DR

This paper investigates the local geometric properties of path structures in three dimensions, showing how non-flat structures reduce to simpler forms and classifying invariant structures on Lie groups.

Contribution

It establishes a canonical reduction for non-flat path structures and classifies invariant structures on three-dimensional Lie groups, extending classical results.

Findings

Non-flat path structures reduce to Z/2Z-structures locally.

Automorphism groups of non-flat structures have maximal dimension three.

Complete classification of invariant path structures on 3D Lie groups.

Abstract

In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a Z/2Z-structure at a neighbourhood of that point (in many cases it has a canonical parallelism). A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896). We also classify the invariant path structures on three-dimensional Lie groups.