Ambient metrics for $n$-dimensional $pp$-waves

Details Ambient metrics for $n$-dimensional $pp$-waves Ambient metrics for $n$-dimensional $pp$-waves

2008-10-16

arxiv.org/abs/0810.2903v3

Mathematics

Differential Geometry

17 pages, in v2 footnote and references added and typos corrected, in v3 remark in the Introduction about Brinkmann's results

Scientific paper

We provide an explicit formula for the Fefferman-Graham-ambient metric of an $n$-dimensional conformal $pp$-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional $pp$-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional $pp$-waves have vanishing critical $Q$-curvature.

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