Cartan's incomplete classification and an explicit ambient metric of holonomy $\mathrm{G}_2^*$

TL;DR

This paper explicitly constructs an ambient metric with holonomy group G2* related to Cartan's classification of (2,3,5) distributions, correcting previous gaps and providing new insights into conformal geometry and exceptional holonomy.

Contribution

It provides a closed-form ambient metric for a specific distribution, demonstrating the G2* holonomy, and corrects earlier classification gaps in Cartan's work.

Findings

Explicit ambient metric for distribution E

Holonomy group G2* established

New observations on ambient metrics of conformal structures

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 $\textrm{G}_2$. 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) $\bf E$, 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 $\textrm{G}_2^*$ 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 $\widetilde{g}_{\bf E}$.