Ambient connections realising conformal Tractor holonomy

TL;DR

This paper introduces ambient connections on conformal manifolds to realize conformal tractor holonomy as affine holonomy, providing explicit examples linked to C-spaces and Einstein manifolds, and connecting to known ambient metrics.

Contribution

It defines ambient connections with specific properties that realize conformal tractor holonomy as affine holonomy, extending the ambient metric framework to broader contexts.

Findings

Constructed ambient connections for conformal manifolds.

Realized conformal tractor holonomy as affine holonomy.

Connected ambient connections to C-spaces and Einstein manifolds.

Abstract

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus for a $C$-space this example leads to an ambient metric in the weaker sense of \v{C}ap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham.