Flat $(2,3,5)$-Distributions and Chazy's Equations

TL;DR

This paper links certain geometric structures called $(2,3,5)$-distributions to generalized Chazy equations, simplifying complex differential equations and providing new examples of flat distributions with specific symmetry groups.

Contribution

It reduces high-order nonlinear ODEs associated with $(2,3,5)$-distributions to generalized Chazy equations, revealing new flat distribution examples and their symmetry properties.

Findings

Reduction of 6th order ODE to 3rd order Chazy equation

Reduction of 7th order ODE to a related Chazy equation

Discovery of new flat $(2,3,5)$-distributions with specific functions

Abstract

In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions determined by a single function of the form $F(q)$, the vanishing condition for the curvature invariant is given by a 6$^{\rm th}$ order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We show that the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ order nonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat $(2,3,5)$-distributions not of the form $F(q)=q^m$. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split $G_2$ as their group of symmetries.