The ambient obstruction tensor and conformal holonomy

TL;DR

This paper explores the relationship between the ambient obstruction tensor and conformal holonomy, revealing new insights into conformal structures and their geometric properties, especially in special dimensions.

Contribution

It introduces a novel relation connecting the ambient obstruction tensor with conformal holonomy and defines the conformal holonomy distribution, linking it to special conformal structures.

Findings

Vanishing and rank conditions for the obstruction tensor

Relation between holonomy distribution integrability and exceptional conformal structures

Applications to conformal structures with twistor spinors or Killing forms

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.