Holonomy groups of Lorentzian manifolds: classification, examples, and applications

GALAEV, Anton and Thomas LEISTNER. Holonomy groups of Lorentzian manifolds: classification, examples, and applications. In LEISTNER, Thomas. Recent developments in pseudo-Riemannian geometry. Vienna, Austria: European Mathematical Society, 2008, p. 53-97. ESI Lectures in Mathematics and Physics. ISBN 978-3-03719-051-7. Available from: https://dx.doi.org/10.4171/051.

Basic information

• Original name: Holonomy groups of Lorentzian manifolds: classification, examples, and applications
• Authors: GALAEV, Anton (643 Russian Federation, guarantor, belonging to the institution) and Thomas LEISTNER (276 Germany).
• Edition: Vienna, Austria, Recent developments in pseudo-Riemannian geometry, p. 53-97, 45 pp. ESI Lectures in Mathematics and Physics, 2008.
• Publisher: European Mathematical Society

Other information

• Original language: English
• Type of outcome: Chapter(s) of a specialized book
• Field of Study: 10101 Pure mathematics
• Country of publisher: Austria
• Confidentiality degree: is not subject to a state or trade secret
• Publication form: printed version "print"
• WWW: URL
• RIV identification code: RIV/00216224:14310/08:00050991
• Organization unit: Faculty of Science
• ISBN: 978-3-03719-051-7
• Doi: http://dx.doi.org/10.4171/051
• UT WoS: 000268749200002
• Keywords (in Czech): Holonomy groups, Lorentzian manifolds, parallel spinors, pp-waves
• Keywords in English: Holonomy groups; Lorentzian manifolds; parallel spinors; pp-waves
• Tags: Pb, rivok
• Tags: International impact, Reviewed

Abstract

We review recent developments in the theory of Lorentzian holonomy groups focussing on the classification results. We present the list of indecomposable, nonirreducible Lorentzian holonomy groups, explain the idea of its proof, and describe a method of constructing metrics which realise all the possible groups. This method is then used to construct many examples of metrics. Finally, we give some applications for the existence of parallel spinors and a short outlook on other signatures. As a new result we obtain the holonomy classification for indecomposable, non-irreducible Lorentzian Einstein spaces.