Address: School of Mathematics and Statistics, Nanjing University of Information Science and Technology, 219 Ningliu Road, Nanjing 210044, China
E-mail: 100093@nuist.edu.cn
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 $\mathfrak{g}_2$.
AMSclassification: primary 58A15; secondary 58A17, 34A05, 34A34.
Keywords: (2,3,5)-distributions, generalised Chazy equation.
DOI: 10.5817/AM2025-1-9