Local equivalence of some maximally symmetric rolling distributions and $SU(2)$ Pfaffian systems

TL;DR

This paper explores the geometric structures of certain rank 2 distributions in five dimensions, showing how they relate to flat metrics and $SU(2)$ symmetries, with implications for understanding rolling distributions and their symmetries.

Contribution

It provides explicit coordinate transformations linking Nurowski's conformal structures to flat metrics for maximally symmetric $(2,3,5)$-distributions and analyzes $SU(2)$-symmetric Pfaffian systems.

Findings

Explicit coordinate change to flat Engel metric for hyperboloid cases

Identification of $SU(2)$ symmetry in specific Pfaffian systems

Discussion of complexifications of rolling distributions

Abstract

We give a description of Nurowski's conformal structure for some examples of bracket-generating rank 2 distributions in dimension 5, aka $(2,3,5)$-distributions, namely the An-Nurowski circle twistor distribution for pairs of surfaces of constant Gauss curvature rolling without slipping or twisting over each other. In the case of hyperboloid surfaces whose curvature ratios give maximally symmetric $(2,3,5)$-distributions, we find the change of coordinates that map the conformal structure to the flat metric of Engel. We also consider a rank $3$ Pfaffian system in dimension 5 with $SU(2)$ symmetry obtained by rotating two of the 1-forms in the Pfaffian system of the spheres rolling distribution, and discuss complexifications of such distributions.