Explicit ambient metrics and holonomy

Abstract

We present three large classes of examples of conformal structures whose Fefferman–Graham ambient metrics can be found explicitly. Our method for constructing these examples rests upon a set of sufficiency conditions under which the Fefferman–Graham equations are assured to reduce to a system of inhomogeneous linear partial differential equations. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic co-rank $3$ distributions in dimensions $5$ and $6$. Our examples illustrate various aspects of the ambient metric construction.

The holonomy algebras of our ambient metrics are studied in detail. In particular, we exhibit a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\operatorname{G}_2$ as well as ambient metrics with holonomy contained in $\operatorname{Spin}(4, 3)$.