Conformal Holonomy Equals Ambient Holonomy

TL;DR

This paper establishes that the infinitesimal conformal holonomy and the ambient holonomy of a conformal manifold agree up to the order of the ambient metric's definition, linking two key geometric notions.

Contribution

It proves the equivalence of conformal and ambient holonomy up to the ambient metric's defining order, clarifying their relationship in conformal geometry.

Findings

Infinitesimal conformal and ambient holonomies agree up to the ambient metric's order.

Provides a precise relation between tractor and ambient holonomy.

Enhances understanding of conformal invariants in geometric analysis.

Abstract

This paper studies the relation between two notions of holonomy on a conformal manifold. The first is the conformal holonomy, defined to be the holonomy of the normal tractor connection. The second is the holonomy of the Fefferman-Graham ambient metric of the conformal manifold. It is shown that the infinitesimal conformal holonomy and the infinitesimal ambient holonomy always agree up to the order that the ambient metric is defined.