Isometry groups of Riemannian fibrations
type: Item
instance of: scholarly article
title: Isometry groups of Riemannian fibrations
zbMATH Open document ID: 0674.53045
DOI: 10.1007/BF00572442
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: Let M and F be smooth finite dimensional Riemannian manifolds with isometry groups I(M) and I(F) respectively. Let G be a finite dimensional Lie group, and let P be a principal G-bundle over M with a fixed connection \(\omega\) in P. Suppose G acts on F through isometries by means of a Lie group homomorphism \(\gamma\) : \(G\to I(F)\). Then there is a canonically defined Riemannian metric on the associated fiber bundle \(E=P\times_ GF\). Let \(I_ M(E)\) denote the group of all fiber preserving isometries of E, and let \(\beta\) : \(I_ M(E)\to I(M)\) be canonical Lie group homomorphism. The main theorem of this paper identifies the kernel of \(\beta\) with a certain subgroup of the gauge group of the principal I(F)-bundle \(Q=P\times_ GI(F)\). Indeed, ker \(\beta\) is isomorphic to the isotropy group of the connection in Q associated to \(\omega\) under the action of the gauge group of Q on the set of connections on Q. This in turn is isomorphic to the centralizer in I(F) of the image of the holonomy group of \(\omega\) under \(\gamma\). This result is applied to describe \(I_ M(TM)\) where TM is the tangent bundle of M, and to characterize the Kähler condition for complex manifolds.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 4103913
zbMATH Keywords: Riemannian fibrations
zbMATH Keywords: isometry groups
zbMATH Keywords: principal G-bundle
zbMATH Keywords: connection
zbMATH Keywords: isotropy group
zbMATH Keywords: gauge group
zbMATH Keywords: tangent bundle
zbMATH Keywords: Kähler condition
author: Hans R. Fischer
author: Robert J. jun. Fisher
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00572442
OpenAlex ID: W1993245407
description: scientific article; zbMATH DE number 4103913