On compact Riemannian manifolds with noncompact holonomy groups
type: Item
instance of: scholarly article
title: On compact Riemannian manifolds with noncompact holonomy groups
zbMATH Open document ID: 0979.53057
author: Burkhard Wilking
published in: Journal of Differential Geometry
publication date: 2002-02-17T00:00:00Z
review text: The notion of holonomy groups of Riemannian manifolds first appeared in the work of E.~Cartan, who used them as a tool in his classification of symmetric spaces. Over the years, several mathematicians (such as Ambrose and Singer, Berger and Simons, to name only a few) contributed to their study. By the middle of the eighties, most problems concerning holonomy groups were solved. (We refer to Chapter~10 in [\textit{A. Besse}, ``Einstein manifolds'', Ergeb. Math. Grenzgeb. 3. Folge 10, Springer, Berlin (1987, Zbl 0613.53001)] for the state of affairs in~1987, and essentially still in~2001.) One major question remained open: is the holonomy group of a compact Riemannian manifold compact? The present article gives the first fully satisfying answer to this question. The author presents the first explicit examples of compact Riemannian manifolds with noncompact holonomy group and then proceeds to examine the structure of such manifolds. To do this, he defines the holonomy representation of the fundamental group of a Riemannian manifold and he shows that a compact Riemannian manifold has compact holonomy group if and only if the image of the fundamental group under this representation is finite. Moreover, he describes all possible holonomy representations on the algebraic level, deducing from this the existence of many compact Riemannian spaces with rather exotic holonomy representations. As a consequence of his detailed study of holonomy representations, he derives the main structure theorem: a finite cover of a compact \(n\)-dimensional Riemannian manifold with noncompact holonomy group is the total space of a bundle over some compact \(b\)-dimensional space, \(b\leq n-4\), whose fibers are flat tori.
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1547116
zbMATH Keywords: non-compact holonomy group
zbMATH Keywords: holonomy representation
zbMATH Keywords: fundamental group
Wikidata QID: Q115193793
reviewed by: Eric Boeckx
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214425277
OpenAlex ID: W1503742440
DOI: 10.4310/JDG/1214425277
description: scientific article; zbMATH DE number 1547116