Local equivalence of some maximally symmetric $(2,3,5)$-distributions

TL;DR

This paper demonstrates the local equivalence between maximally symmetric (2,3,5)-distributions and the flat Cartan distribution using complex parametrisation, providing explicit vector fields for the associated Lie algebra.

Contribution

It introduces a coordinate transformation linking the maximally symmetric rolling distribution to the flat Cartan distribution, clarifying their local equivalence and Lie algebra structure.

Findings

Established local equivalence between distributions

Provided explicit vector fields generating the Lie algebra

Connected the distribution to the split real form of rak{g}_2

Abstract

Using a complex parametrisation of $su(2)$, we show a change of coordinates that maps the maximally symmetric rolling $(2,3,5)$-distribution to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric rolling model and the flat Cartan or Hilbert-Cartan distribution. For the maximally symmetric rolling distribution, we write down the vector fields that bracket-generate to give the split real form of the Lie algebra of $\frak{g}_2$, with two of the vector fields in the bracket-generating set given by the span of the rolling distribution.