Torsion-free \(G_{3}\)-connections that are not analytic
type: Item
instance of: scholarly article
title: Torsion-free \(G_{3}\)-connections that are not analytic
zbMATH Open document ID: 1077.53023
author: Quo-Shin Chi
published in: Differential Geometry and its Applications
publication date: 2004-11-19T00:00:00Z
review text: In 1955 \textit{M. Berger} [Bull. Soc. Math. Fr. 83, 279--330 (1955; Zbl 0068.36002)] classified the possible irreducible Riemannian holonomies. Berger also, partially classified the possible non-Riemannian holonomies of torsion free connections. However, his classification omits a finite number of possibilities, which are referred to as exotic holonomies. \textit{R. Bryant} [Proc. Symp. Pure Math. 53, 33--88 (1991; Zbl 0758.53017)] gave examples of torsion free connections on four-dimensional manifolds whose holonomy is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of \(SL(2,\mathbb{R})\) (\(GL(2,\mathbb{R})\), respectively) and these connections are called \(H_{3}\)-connections (\(G_{3}\)-connections, respectively). A complete classification of homogeneous \(G_{3}\)-connections was given by \textit{L. J. Schwachhöfer} [Geom. Dedicata 62, 193--208 (1996; Zbl 0860.53015)]. The moduli space of the \(H_{3}\) family is analytic and finite-dimensional whereas the moduli space of the generic (nondegenerate) analytic \(G_{3}\)-connections is parametrized by four functions in three variables. Up to now, all the known torsion-free \(G_{3}\)-connections have been analytic. In this paper, the author give examples of torsion-free \(G_{3}\)-connections that are not analytic. Actually, he provides an explicit PDE-solving approach to construct a family of smooth torsion-free \(G_{3}\)-connections that are not equivalent to any analytic ones. More exactly, it is proven: There are smooth torsion-free \(G_{3}\)-connections that are not equivalent to any analytic ones, although each of these connections is locally homogeneous (in fact, is left-invariant on a local solvable group) away from a hypersurface of the base manifold.
reviewed by: Ilie Burdujan
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 2116887
zbMATH Keywords: torsion-free linear connection
zbMATH Keywords: holonomy
MaRDI profile type: Publication
full work available at URL: j.difgeo.2004.05.001
OpenAlex ID: W2030952716
Wikidata QID: Q115358035
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DOI: 10.1016/J.DIFGEO.2004.05.001
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description: scientific article; zbMATH DE number 2116887