Conformal Structures with Explicit Ambient Metrics and Conformal G 2 Holonomy

Abstract

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in [7]. 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 G2. 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 [2], can be written in an explicit form. We propose to study the relations between the conformal G2-holonomy of metrics [g] and the possible pseudo-Riemannian G2-holonomy of the corresponding ambient metrics.