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

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