On the construction of some complete metrics with exceptional holonomy
type: Item
instance of: scholarly article
title: On the construction of some complete metrics with exceptional holonomy
zbMATH Open document ID: 0681.53021
DOI: 10.1215/S0012-7094-89-05839-0
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Until the first author's recent paper [Ann. Math., II. Ser. 126, 525-576 (1987; Zbl 0637.53042)], the two Lie groups \(G_ 2\) and Spin(7) were the only two members of \textit{M. Berger}'s list of holonomy groups of irreducible Riemannian manifolds [Bull. Soc. Math. Fr. 83, 279-330 (1955; Zbl 0068.36002)] whose existence was in doubt. The present paper improves upon the above by providing explicit construction of three distinct complete Riemannian metrics whose holonomy group is \(G_ 2\), one complete metric whose holonomy group is Spin(7), and various other incomplete metrics with exceptional holonomy. The construction techniques may be of interest to those studying Ricci flat and Einstein spaces.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 4115451
zbMATH Keywords: vector bundle
zbMATH Keywords: Einstein space
zbMATH Keywords: spin bundle
zbMATH Keywords: irreducible Riemannian manifolds
zbMATH Keywords: holonomy group
zbMATH Keywords: exceptional holonomy
Wikidata QID: Q56387073
author: Robert L. Bryant
author: Simon Salamon
reviewed by: James Hebda
MaRDI profile type: Publication
cites work: Q5563881
cites work: Self-duality in four-dimensional Riemannian geometry
cites work: Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes
cites work: Q3333683
cites work: Q3756941
cites work: Q5345512
cites work: Metrics with exceptional holonomy
cites work: Construction and Properties of Some 6-Dimensional Almost Complex Manifolds
cites work: Métriques kählériennes et fibrés holomorphes
cites work: Q3763283
cites work: A classification of Riemannian manifolds with structure group Spin (7)
cites work: Weak holonomy groups
cites work: Kählerian Twistor Spaces
cites work: Calibrated geometries
cites work: Alcune osservazioni riguardanti i gruppi di Lie \(G_ 2\) e Spin(7), candidati a gruppi di olonomia
cites work: Einstein metrics on quaternionic line bundles
full work available at URL: s0012-7094-89-05839-0
OpenAlex ID: W1560094496
description: scientific article; zbMATH DE number 4115451