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

TL;DR

This paper demonstrates the local equivalence between certain maximally symmetric (2,3,5)-distributions derived from solutions to specific generalized Chazy equations and the flat Cartan distribution, revealing deep geometric connections.

Contribution

It provides explicit coordinate transformations linking these distributions and characterizes their Lie algebra structure, advancing understanding of their geometric and algebraic properties.

Findings

Established local equivalence between distributions and flat Cartan distribution.

Derived vector fields parametrized by solutions to specific Chazy equations.

Identified the split real form of rak{g}_2$ as the symmetry algebra.

Abstract

We show the change of coordinates that maps the maximally symmetric $(2,3,5)$-distribution given by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-generate to give the split real form of $\frak{g}_2$.