Berger’s Holonomy Theorem, and a First Incursion into Lie Algebroid Holonomy

Abstract

The holonomy group of a connection is very intimately related to the curvature of the connection and to the existence and quantity of parallel sections, thus controlling an important part of the geometry. The Lie groups that can arise as the holonomy group of the Levi-Civita connection of a Riemannian manifold were classified by Berger, resulting in a list of seven possible groups. These holonomies give rise to special geometries, like Kähler, Calabi–Yau or hyperkähler geometries. It was not until fifty years later that Olmos offered a geometric proof of Berger’s theorem, as an alternative to Berger’s more algebraic proof. The first part of this work is dedicated to presenting Olmos’s proof, orderly developing the requisites needed to understand it. In the second part we introduce the generalization of holonomy to the Lie algebroid setting. Lie algebroids are, in a sense, a generalization of the tangent bundle and, as such, it makes sense to consider Lie algebroid connections and Lie algebroid holonomy. This new concept presents some remarkable new features. The first one is the failure of the Ambrose–Singer theorem: the holonomy algebra is not only determined by curvature, but also by the isotropy of the algebroid. We give a new proof of this Lie algebroid Ambrose–Singer theorem, and provide some original examples of flat Lie algebroid connections with non-discrete holonomy. Secondly, the notion of Lie algebroid holonomy is a leafwise notion, so the holonomy can jump from leaf to leaf. When considering general Lie algebroid connections on vector bundles, this behavior can be quite wild: it can jump either up or down when changing to smaller leaves. We provide as well original examples of such behaviors.