Parallel tractor extension and ambient metrics of holonomy split \(G_2\). (English) Zbl 1268.53075

Graham, C. Robin; Willse, Travis

J. Differ. Geom. 92, No. 3, 463-506 (2012).

Authors’ abstract: The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic \(2\)-plane fields on oriented \(5\)-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 in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.

Reviewer: Witold Mozgawa (Lublin)

MSC:

53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C29	Issues of holonomy in differential geometry
53C10	\(G\)-structures

Keywords:

\(G_2\); holonomy; ambient metric; Weyl tensor; Cotton tensor; tractor; conformal; parallel extension