Holonomy and Parallel Spinors in Lorentzian Geometry

Thomas Leistner

ISBN 978-3-8325-0472-4
190 pages, year of publication: 2004
price: 40.50 €
The group of parallel displacements, the so-called holonomy group, is an important tool in order to study geometric structures on a smooth manifold. It describes parallel objects in tensor bundles as well as in other geometric vector bundles, such as the spinor bundle.

The question: Which groups may occur as holonomy group of a semi-Riemannian manifold and which of these admit parallel spinors? is a classical one in differential geometry. It is essentially solved for Riemannian manifolds, because here the holonomy group acts completely reducible. This dissertation is an aproach to answer this question for Lorentzian manifolds. Lorentzian manifolds are relevant for modern physics and their holonomy representation is not necessarily completely reducible. There are manifolds with indecomposable, but non-irreducible holonomy representation. These are interesting for the existence of parallel spinors. This is explained in the mainly introductory first chapter. Here also a decomposition theorem for the existence of parallel spinors is proven. Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation are described in detail in the second chapter. Their holonomy group is contained in the parabolic group and its essential part is the screen holonomy.

A construction method is given for manifolds, for which the screen holonomy is a Riemannian holonomy group. In the third chapter the existence of parallel spinors and its consequences for the holonomy group and the screen holonomy are studied. In the last chapter a partial classification of indecomposable, non-irreducible Lorentzian holonomy groups is given by proving that in most of the cases the screen holonomy has to be a Riemannian holonomy. The consequences for the existence of parallel spinors are drawn. This classification result uses methods of representation theory of real, semisimple Lie algebras. The main facts of this theory are briefly explained in the appendix.

  • Lorentzian manifolds
  • holonomy groups
  • parallel spinors


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