Explicit ambient metrics and holonomy. (English) Zbl 1437.53038

Anderson, Ian M.; Leistner, Thomas; Nurowski, Paweł

J. Differ. Geom. 114, No. 2, 193-242 (2020).

The paper under review is concerned with conformal geometry, in particular the program of Fefferman-Graham which proposes to recover the conformal invariants of a given conformal manifold by the semi-Riemannian invariants of an ambient semi-Riemannian manifold. Recall that explicit constructions of ambient manifolds as such have been given for manifolds endowed with Einstein metrics.
This work provides three large classes of examples of ambient manifolds, which do not come from Einstein metrics. They are based on earlier work by the authors, where they set up conditions ensuring the linearity and inhomogeneity of the Fefferman-Graham system of PDEs involved. A common feature of all the constructions is that the Shouten tensor arising at the singularities is 2-step nilpotent. The holonomy of these examples is studied thoroughly as well.
From the abstract: 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 \(G_2\) as well as ambient metrics with holonomy contained in \(\mathrm{Spin}(4, 3)\).

Reviewer: Iakovos Androulidakis (Athína)

MSC:

53C29	Issues of holonomy in differential geometry
53C18	Conformal structures on manifolds
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

conformal structures; Fefferman-Graham ambient metric; exceptional holonomy; generic distributions; pp-waves