Aspects of conformal holonomy

Abstract

This is an expository article, which gives an overview about aspects of the theory of conformal holonomy. In particular, we announce a complete geometric description of compact Riemannian conformal manifolds with decomposable conformal holonomy representation. Furthermore, we discuss the relation to almost Einstein structures and generalised Fefferman constructions. Generically, the latter conformal geometries have irreducible conformal holonomy. Reduced conformal holonomy is related to the existence of solutions of certain overdetermined conformally covariant PDE systems. We explain this relation in a unified approach using BGG-sequences.