The ambient obstruction tensor and conformal holonomy

Thomas Leistner and Andree Lischewski

Vol. 290 (2017), No. 2, 403–436

DOI: 10.2140/pjm.2017.290.403

Abstract

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.

Keywords

Fefferman–Graham ambient metric, obstruction tensor, conformal holonomy, exceptional conformal structures, normal conformal Killing forms

Mathematical Subject Classification 2010

Primary: 53A30

Secondary: 53C29

Milestones

Received: 20 January 2016

Revised: 27 March 2017

Accepted: 27 March 2017

Published: 25 July 2017