Fefferman–Graham ambient metrics of Patterson–Walker metrics

Given an n‐dimensional manifold N with an affine connection D, we show that the associated Patterson–Walker metric g on T∗N 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.