Natural principal connections on the principal gauge prolongation of a principal bundle
type: Item
instance of: scholarly article
title: Natural principal connections on the principal gauge prolongation of a principal bundle
zbMATH Open document ID: 1195.53040
DOI: 10.1016/S0034-4877(10)00002-9
author: Josef Janyška
author: Jan Vondra
published in: Reports on Mathematical Physics
publication date: 2010-04-07T00:00:00Z
review text: For a principal bundle \(P\) over a manifold \(M\) with structure group \(G\) and an integer \(r\), one can form the \(r\)-th principal prolongation \(W^rP\), which essentially consists of \(r\)-jets of local principal bundle charts. This is a principal bundle with structure group the \(r\)-th prolongation \(W^r_mG\), where \(m\) is the dimension of \(M\). The aim of this article is to determine all ways to naturally associate a principal connection on \(W^rP\) to a principal connection on \(P\) and an affine connection on \(M\). Using the Utiyama theorem, the authors arrive at a description of such connections for general \(r\) in terms of a number of tensor fields depending naturally on the two given connections. They also give a description of such tensor fields in terms of Ad-invariant elements in \(\mathfrak g\) and Ad-equivariant linear maps \(\mathfrak g\to\mathfrak g\), where \(\mathfrak g\) denotes the Lie algebra of the structure group \(G\). For \(r=1\) and \(r=2\), the classification is carried out completely. Finally, the Ad-invariant elements and Ad-equivariant linear maps needed in the description are determined in the case \(\mathfrak g=\mathfrak{gl}(n,\mathbb R)\).
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53A55
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 58A20
Mathematics Subject Classification ID: 58A32
zbMATH DE Number: 5689970
zbMATH Keywords: gauge natural bundle
zbMATH Keywords: principal bundle
zbMATH Keywords: principal connection
zbMATH Keywords: natural operator
reviewed by: Andreas Cap
MaRDI profile type: Publication
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full work available at URL: s0034-4877(10)00002-9
OpenAlex ID: W2085664306
description: scientific article; zbMATH DE number 5689970