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

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 ${\text{G}}_2$. 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 ${\tilde{g}}_{\mathbf{E}}$ 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 ${\tilde{g}}_{\mathbf{E}}$ is the exceptional group ${\text{G}}_2^{\ast }$ 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 ${\tilde{g}}_{\mathbf{E}}$.

arXiv comment16 pages. The proof of Proposition 3 has been simplified (the original is now sketched in Remark 4). Section 4 has been significantly revised for clarity. The proofs of Propositions 7 and 11 are now more explicit. Remark 12, which contains a proof that the main example given here is not among thethose given in arXiv:1501.00852, was added. Some inessential material was cut for brevity