The ambient obstruction tensor and the conformal deformation complex

A. Rod Gover and Lawrence J. Peterson

Vol. 226 (2006), No. 2, 309–351

DOI: 10.2140/pjm.2006.226.309

Abstract

We construct here a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the (Fefferman–Graham) ambient obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence-free and vanishes identically for conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman–Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian-type operators that generalise the conformal Laplacians of Graham–Jenne–Mason–Sparling. We give an algorithm for calculating explicit formulae for these operators, and this is applied to give formulae for the obstruction tensor in dimensions 6 and 8. As background to these issues, we give an explicit construction of the deformation complex in dimensions n ≥ 4, construct two related (detour) complexes, and establish essential properties of the operators in these.

Keywords

ambient metric, conformal deformations, conformal geometry, detour complexes, differential complexes

Mathematical Subject Classification 2000

Primary: 53A55

Secondary: 22E70, 53A30, 58J10

Milestones

Received: 4 November 2004

Revised: 21 March 2005

Accepted: 17 May 2005

Published: 1 August 2006