Explicit ambient metrics and holonomy

Abstract

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit Fefferman-Graham ambient metrics provide a large class of metrics with holonomy equal to the exceptional non-compact Lie group ${\mathbf{G}}_2$ as well as ambient metrics with holonomy contained in $\mathbf{Spin}(4,3)$.

30 pages; in Section 4 of v2 the results about ambient metrics for Bryant's conformal structures are significantly generalized

Cited by in corpus (3)

• Local Type I Metrics with Holonomy in ${\mathrm{G}}_2^{\ast }$
• Fefferman-Graham ambient metrics of Patterson-Walker metrics
• The ambient obstruction tensor and conformal holonomy