Open Access
November 2012 Parallel tractor extension and ambient metrics of holonomy split $G_2$
C. Robin Graham, Travis Willse
J. Differential Geom. 92(3): 463-506 (November 2012). DOI: 10.4310/jdg/1354110197


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.


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C. Robin Graham. Travis Willse. "Parallel tractor extension and ambient metrics of holonomy split $G_2$." J. Differential Geom. 92 (3) 463 - 506, November 2012.


Published: November 2012
First available in Project Euclid: 28 November 2012

zbMATH: 1268.53075
MathSciNet: MR3005060
Digital Object Identifier: 10.4310/jdg/1354110197

Rights: Copyright © 2012 Lehigh University

Vol.92 • No. 3 • November 2012
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