Cartan\u2019s incomplete classification and an explicit ambient metric of holonomy mathrm{G}_2^*

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 mathrm{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 mathbf {E}. We produce a closed form for the Fefferman–Graham ambient metric {smash {{smash {widetilde{g}}}}}_{mathbf {E}} of the conformal class induced by (a real form of) mathbf {E}, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of smash {{smash {widetilde{g}}}_{mathbf {E}}} is the exceptional group {smash {mathrm{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 smash {{smash {widetilde{g}}}_{mathbf {E}}}.