Fefferman-Graham ambient metrics of Patterson-Walker metrics

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Authors

HAMMERL Matthias SAGERSCHNIG Katja ŠILHAN Josef TAGHAVI-CHABERT Arman ŽÁDNÍK Vojtěch

Year of publication 2018
Type Article in Periodical
Magazine / Source BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
MU Faculty or unit

Faculty of Science

Citation
Web https://arxiv.org/abs/1608.06875
Doi http://dx.doi.org/10.1112/blms.12136
Keywords projective structure; conformal structure; ambient metric
Description Given an n-dimensional manifold N with an affine connection D, we show that the associated Patterson-Walker metric g on TN admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which the ambient metric exists to all orders and can be realised in a natural and explicit way. In particular, it follows that Patterson-Walker metrics have vanishing Fefferman-Graham obstruction tensors. As an application of the concrete ambient metric realisation we show in addition that Patterson-Walker metrics have vanishing Q-curvature. We further show that the relationship between the geometric constructions mentioned above is very close: the explicit Fefferman-Graham ambient metric is itself a Patterson-Walker metric.
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