Ambient metrics with exceptional holonomy (Q2915220)

The authors describe here an explicit local construction in coordinates for a family (in 8 parameters) of semi-Riemannian metrics in dimension 7 of signature \((3,4)\) with exceptional non-compact holonomy \(G_{2(2)}\) (of split real form) sitting irreducibly in \(SO(3,4)\).NEWLINENEWLINEThe construction starts with a conformal metric (with parameters) on a 5-manifold of signature (2,3), whose normal Cartan connection has holonomy in \(G_{2(2)}\). (Those conformal structures arise naturally from generic \(2\)-distributions and are related to certain classical ODE's.)NEWLINENEWLINEThen the classical Fefferman-Graham ambient metric construction of conformal geometry is applied which gives rise to metrics of signature \((3,4)\) in dimension \(7\). First, the authors show that the ambient metrics in their case admit non-null parallel spinors, which implies that the ambient holonomy is contained in \(G_{2(2)}\). In order to prove the equality with \(G_{2(2)}\) they test the existence of Einstein metrics and metrics with a certain null condition for the Ricci tensor in the underlying conformal class, respectively. If certain parameters of the family do not vanish such metrics do not occur and the ambient holonomy must be \(G_{2(2)}\).