February 2020 Explicit ambient metrics and holonomy
Ian M. Anderson, Thomas Leistner, Paweł Nurowski
Author Affiliations +
J. Differential Geom. 114(2): 193-242 (February 2020). DOI: 10.4310/jdg/1580526015


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 $\operatorname{Spin}(4, 3)$.

Funding Statement

Support for this research was provided by National Science Foundation (ACI 1148331SI2-SSE); by the Australian Research Council (FT110100429 and DP120104582); the PolishMinistry of Research and Higher Education (NN201 607540 and NN202 104838); and by the Polish National Science Center (NCN) via DEC- 2013/09/B/ST1/01799.


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Ian M. Anderson. Thomas Leistner. Paweł Nurowski. "Explicit ambient metrics and holonomy." J. Differential Geom. 114 (2) 193 - 242, February 2020. https://doi.org/10.4310/jdg/1580526015


Received: 12 May 2015; Published: February 2020
First available in Project Euclid: 1 February 2020

MathSciNet: MR4058962
zbMATH: 07163291
Digital Object Identifier: 10.4310/jdg/1580526015

Primary: 53C29
Secondary: 53A30

Keywords: conformal structures , exceptional holonomy , Fefferman–Graham ambient metric , generic distributions , pp-waves

Rights: Copyright © 2020 Lehigh University


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Vol.114 • No. 2 • February 2020
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