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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.

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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