Conformal structures with explicit ambient metrics and conformal G_2 holonomy
Abstract
Description
Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be contained in the split-real-form of the exceptional group G_2. In this note we show that for special 2-plane fields on 5-manifolds the conformal classes [g] have the Fefferman-Graham ambient metrics which, contrary to the general Fefferman-Graham metrics given as a formal power series, can be written in an explicit form. We propose to study the relations between the conformal G_2-holonomy of metrics [g] and the possible pseudo-Riemannian G_2-holonomy of the corresponding ambient metrics.
To appear in Proceedings of the Workshop at 2006 IMA Summer Program "Symmetries and Overdetermined Systems of Partial Differential Equations", Minneapolis, July 2006
To appear in Proceedings of the Workshop at 2006 IMA Summer Program "Symmetries and Overdetermined Systems of Partial Differential Equations", Minneapolis, July 2006