On homogeneous connections with exotic holonomy
type: Item
instance of: scholarly article
title: On homogeneous connections with exotic holonomy
zbMATH Open document ID: 0860.53015
DOI: 10.1007/BF00147811
published in: Geometriae Dedicata
publication date: 1997-04-21T00:00:00Z
review text: \textit{M. Berger} [Bull. Soc. Math. Fr. 83, 279-330 (1955; Zbl 0068.36002)] classified the irreducible Riemannian holonomies and partially the non-Riemannian holonomies of torsion free connections. \textit{R. L. Bryant} [Proc. Symp. Pure Math. 53, 33-88 (1991; Zbl 0758.53017)] found some torsion free connections on 4-manifolds whose holonomy is the irreducible 4-dimensional representation of \(\text{SL}(2,\mathbb{R})\) (called \(H_3\)-connections) and of \(\text{GL}(2,\mathbb{R})\) (called \(G\)-connections). The \(H_3\)-connections and \(G_3\)-connections are not contained in Berger's list (their holonomy is exotic). The author gives a complete classification of homogeneous \(G_3\)-connections. Such a connection has a symmetry group of dimension at most 4, and every generic homogeneous \(G_3\)-connection is locally equivalent to a left-invariant connection on a 4-dimensional Lie group. The classification of these connections is realised by the orbit space of some polynomials which can be completely classified.
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C29
zbMATH DE Number: 929899
zbMATH Keywords: holonomy
zbMATH Keywords: affine connections
zbMATH Keywords: \(G\)-structures
zbMATH Keywords: classification of homogeneous \(G_ 3\)-connections
author: Lorenz Schwachhöfer
reviewed by: I. D. Albu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 929899