Holonomy groups and parallel spinors on totally reducible pseudo-Riemannian manifolds
type: Item
instance of: scholarly article
title: Holonomy groups and parallel spinors on totally reducible pseudo-Riemannian manifolds
zbMATH Open document ID: 1057.53039
author: Aziz Ikemakhen
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-08-20T00:00:00Z
review text: The note deals with the following problem: What are the possible holonomy groups of simply connected pseudo-Riemannian spin manifolds \((M^{(p,q)},g)\) which admit parallel spinors? \textit{M.Y. Wang} [Ann. Global Anal. Geom. 7, No. 1, 59--68 (1989; Zbl 0688.53007)], \textit{H. Baum} and \textit{I. Kath} [Ann. Global Anal. Geom. 17, 1--17 (1999; Zbl 0928.53027)] solved the problem when \((M^{(p,q)},g)\) is irreducible and non-locally symmetric. \textit{Th. Leistner} [Holonomy and parallel spinors in Lorentzian geometry, Ph.D. Thesis, Humboldt University of Berlin, 2003] solved the problem in the Lorentzian case. The author [C. R., Math., Acad. Sci. Paris, Ser. 337, 179--184 (2003; Zbl 1052.53042)] also characterized, by their holonomy groups, the simply connected spin pseudo-Riemannian manifolds which admit parallel pure spinors. In this note, the author gives a complete answer to the above problem when \((M^{(p,q)},g)\) is totally reducible, i.e. every invariant subspace by the holonomy representation admits an invariant complementary.
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 2094047
zbMATH Keywords: pseudo-Riemannian spin manifold
zbMATH Keywords: parallel spinor
zbMATH Keywords: holonomy group
zbMATH Keywords: Clifford multiplication
zbMATH Keywords: spinor representation.
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
full work available at URL: j.crma.2004.04.029
OpenAlex ID: W2046078206
cites work: A Theorem on Holonomy
cites work: Q4001351
cites work: Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds
cites work: Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes
cites work: Parallel pure spinors and holonomy
cites work: Q3024753
cites work: Parallel spinors and parallel forms
cites work: On the de Rham decomposition theorem
DOI: 10.1016/J.CRMA.2004.04.029
description: scientific article; zbMATH DE number 2094047