Conformal structures with explicit ambient metrics and conformal G_2 holonomy

Details Conformal structures with explicit ambient metrics and conformal G_2 holonomy Conformal structures with explicit ambient metrics and conformal G_2 holonomy

2007-01-30

arxiv.org/abs/math/0701891v2

Mathematics

Differential Geometry

To appear in Proceedings of the Workshop at 2006 IMA Summer Program "Symmetries and Overdetermined Systems of Partial Differen

Scientific paper

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.

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