Weyl-Ambient Geometries

TL;DR

This paper extends the Fefferman-Graham ambient construction from conformal to Weyl manifolds, introducing a Weyl-ambient metric and exploring Weyl-covariant tensors with explicit obstruction tensors.

Contribution

It generalizes the ambient construction to Weyl geometry, defining a Weyl-ambient metric and analyzing Weyl-covariant tensors and obstructions.

Findings

Weyl-ambient space induces a codimension-2 Weyl geometry.

A well-defined initial value problem for Weyl structures is established.

Explicit extended Weyl-obstruction tensors are constructed.

Abstract

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal $\mathbb{R}_+$-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.