Parallel tractor extension and ambient metrics of holonomy split G_2

TL;DR

This paper investigates the holonomy of ambient metrics associated with certain conformal structures on 5-manifolds, showing it is contained in and often equal to the split real form G_2, thus expanding understanding of special geometric structures.

Contribution

It proves the holonomy is always contained in G_2 and characterizes when it equals G_2, extending previous work and introducing an ambient extension theorem for parallel tractors.

Findings

Holonomy always contained in split G_2

Holonomy equals G_2 for an open dense set of 2-plane fields

Provides an infinite-dimensional family of metrics with G_2 holonomy

Abstract

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split G_2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G_2 is established by proving an ambient extension theorem for parallel tractors for conformal structures in general signature and dimension, which is expected to be of independent interest. Parallel extension beyond the critical order in even dimensions is considered in certain cases.