Fefferman-Graham ambient metrics of Patterson-Walker metrics

TL;DR

This paper constructs explicit Fefferman-Graham ambient metrics for Patterson-Walker metrics derived from affine connections, revealing new conformal structures with vanishing obstructions and Q-curvature.

Contribution

It provides a global, explicit ambient metric construction for Patterson-Walker metrics, expanding the class of conformal structures with known ambient metrics.

Findings

Patterson-Walker metrics admit explicit ambient metrics to all orders.

These metrics have vanishing Fefferman-Graham obstruction tensors.

Patterson-Walker metrics also have vanishing Q-curvature.

Abstract

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 realized 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 realization we show in addition that Patterson-Walker metrics have vanishing Q-curvature.