Transitive conformal holonomy groups

TL;DR

This paper classifies all connected conformal holonomy groups acting transitively on the Möbius sphere for conformal manifolds of signature (p,q), revealing their structure and geometric implications.

Contribution

It provides a complete description of transitive conformal holonomy groups, including irreducible and decomposable cases, and links them to special Einstein product metrics.

Findings

Classified all transitive conformal holonomy groups on the Möbius sphere.

Identified conditions under which the holonomy acts irreducibly or decomposably.

Showed that certain holonomy groups imply the existence of special Einstein product metrics.

Abstract

For $(M,[g])$ a conformal manifold of signature $(p,q)$ and dimension at least three, the conformal holonomy group $\mathrm{Hol}(M,[g]) \subset O(p+1,q+1)$ is an invariant induced by the canonical Cartan geometry of $(M,[g])$. We give a description of all possible connected conformal holonomy groups which act transitively on the M\"obius sphere $S^{p,q}$, the homogeneous model space for conformal structures of signature $(p,q)$. The main part of this description is a list of all such groups which also act irreducibly on $\R^{p+1,q+1}$. For the rest, we show that they must be compact and act decomposably on $\R^{p+1,q+1}$, in particular, by known facts about conformal holonomy the conformal class $[g]$ must contain a metric which is locally isometric to a so-called special Einstein product.