Parallel tractor extension and metrics of split $G_2$ holonomy [Ph.D. thesis]
Abstract
Nurowski showed that a maximally nonintegrable $2$-plane field on a $5$-manifold induces a natural conformal structure of signature $(2, 3)$ on that manifold. We show that in the real-analytic case, applying the Fefferman–Graham ambient construction to the conformal structures produced this way on oriented manifolds always yields metrics with pseudo-Riemannian holonomy contained in the split real form $G_2$ of the exceptional Lie group $G_2^{\Bbb C}$. We furthermore show that such metrics have holonomy equal to $G_2$ generically, in the sense that if the holonomy of such a metric is a proper subgroup of $G_2$, then the $7$-jet of the underlying $2$-plane field at any arbitrary point must be contained in some proper subvariety of the $7$-jet space there. This construction hence yields an infinite-dimensional family of metrics with holonomy equal to $G_2$. Both the containment and the genericity statements generalize results of Leistner and Nurowski.
To prove the containment, we prove the general result that any parallel tractor tensor on an $n$-dimensional, general-signature conformal structure, $n \geq 3$, admits an extension to a tensor on the ambient space parallel in a weak sense: If $n$ is odd, there is always an extension to an tensor on the ambient space parallel to infinite order, and if $n$ is even there is an extension parallel to order $\frac{1}{2} n - 1$. In particular, if $n$ is odd and the underlying data is real-analytic, then there is a bona fide parallel extension for any real-analytic ambient metric. Hammerl and Sagerschnig produced for any $2$-plane field on an oriented $5$-manifold a parallel tractor $3$-form suitably compatible with the tractor metric, so the extension result produces a parallel $3$-form on the ambient manifold compatible with the ambient metric, the existence of which is equivalent to the containment of holonomy in $G_2 < \operatorname{SO}(3, 4)$.
We also investigate parallel extension to infinite order when $n$ is even, in which case existence of such extensions is generally obstructed, and we show that the parallel extension result yields a family of necessary integrability conditions for parallel tractor tensors of any type.