A Cartan-theoretic classification of multiply-transitive $(2,3,5)$-distributions

TL;DR

This paper classifies multiply-transitive (2,3,5)-distributions using modern Cartan geometry, providing complete local classifications and analyzing their curvature and holonomy, with applications to rolling sphere models.

Contribution

It offers a modern Cartan-geometric classification of multiply-transitive (2,3,5)-distributions, including curvature and holonomy analysis, and proves the exceptional 3:1 ratio case.

Findings

Complete local classifications in complex and real settings.

Determination of full curvature and infinitesimal holonomy.

Proof of the exceptional 3:1 ratio for rolling spheres.

Abstract

In his 1910 paper, \'Elie Cartan gave a tour-de-force solution to the (local) equivalence problem for generic rank 2 distributions on 5-manifolds, i.e. $(2,3,5)$-distributions. From a modern perspective, these structures admit equivalent descriptions as (regular, normal) parabolic geometries modelled on a quotient of $G_2$, but this is not transparent from his article: indeed, the Cartan "connection" of 1910 is not a "Cartan connection" in the modern sense. We revisit the classification of multiply-transitive $(2,3,5)$-distributions from a modern Cartan-geometric perspective, incorporating $G_2$ structure theory throughout, obtaining: (i) the complete (local) classifications in the complex and real settings, phrased "Cartan-theoretically", and (ii) the full curvature and infinitesimal holonomy of all these models. Moreover, we Cartan-theoretically prove exceptionality of the $3:1$ ratio for two 2-spheres rolling on each other without twisting or slipping, yielding a $(2,3,5)$-distribution with symmetry the Lie algebra of the split real form of $G_2$.