Vol. 290, No. 2, 2017

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The ambient obstruction tensor and conformal holonomy

Thomas Leistner and Andree Lischewski

Vol. 290 (2017), No. 2, 403–436

For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurowski and Bryant.

Fefferman–Graham ambient metric, obstruction tensor, conformal holonomy, exceptional conformal structures, normal conformal Killing forms
Mathematical Subject Classification 2010
Primary: 53A30
Secondary: 53C29
Received: 20 January 2016
Revised: 27 March 2017
Accepted: 27 March 2017
Published: 25 July 2017
Thomas Leistner
School of Mathematical Sciences
University of Adelaide
Adelaide SA
Andree Lischewski
Institute of Mathematics
Humboldt University Berlin