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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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