Highly symmetric $2$-plane fields on $5$-manifolds and $5$-dimensional Heisenberg group holonomy

Published in Differential Geometry and its Applications, 2014
(sciencedirect.com | arXiv:1302.7163)

Abstract

Nurowski showed that any generic $2$-plane field $D$ on a $5$-manifold $M$ determines a natural conformal structure $c_{\bf D}$ on $M$; these conformal structures are exactly those (on oriented $M$) whose normal conformal holonomy is contained in the (split, real) simple Lie group $\mathrm{G}_2$. Graham and Willse showed that for real-analytic $\bf D$ the same holds for the holonomy of the real-analytic Fefferman–Graham ambient metric of $c_{\bf D}$, and that both holonomy groups are equal to $\mathrm{G}_2$ for almost all $\bf D$. We investigate here independently interesting $2$-plane fields for which the associated holonomy groups are a proper subgroup of $\mathrm{G}_2$.

Cartan solved the local equivalence problem for $2$-plane fields $\bf D$ and constructed the fundamental curvature tensor $A$ for these objects. He furthermore claimed to describe locally all $\bf D$ whose infinitesimal symmetry algebra has rank at least $6$ and gave a local quasi-normal form, depending on a single function of one variable, for those that furthermore satisfy a natural degeneracy condition on $A$, but Doubrov and Govorov recently rediscovered a counterexample to Cartanʼs claim. We show that for all $\bf D$ given by Cartanʼs alleged quasi-normal form, the conformal structures $\bf c_D$ induced via Nurowskiʼs construction are almost Einstein, that we can write their ambient metrics explicitly, and that the holonomy groups associated to $c_{\bf D}$ are always the $5$-dimensional Heisenberg group, which here acts indecomposably but not irreducibly. (Not all of these properties hold, however, for Doubrov and Govorovʼs counterexample.) We also show that the similar results hold for the related class of $2$-plane fields defined on suitable jet spaces by ordinary differential equations $z^\prime=F(y^\prime{}^\prime(x))$ satisfying a simple genericity condition.