Cartan’s incomplete classification and an explicit ambient metric of holonomy $G_2^*$

Published in European Journal of Mathematics, 2017
(springer.com | arXiv:1411.7172)

Abstract

In his 1910 "Five Variables" paper, Cartan solved the equivalence problem for the geometry of $(2, 3, 5)$ distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type $G_2$. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least $6$ (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution $\bf E$. We produce a closed form for the Fefferman–Graham ambient metric $\widetilde{g}_{\bf E}$ of the conformal class induced by (a real form of) $E$, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of $\widetilde{g}_{\bf E}$ is the exceptional group $G_2^*$ and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for $\widetilde{g}_{\bf E}$.