Classifications of bundle connection pairs by parallel translation and lassos
type: Item
instance of: scholarly article
title: Classifications of bundle connection pairs by parallel translation and lassos
zbMATH Open document ID: 0676.53033
DOI: 10.1016/0022-1236(89)90035-9
author: Bruce K. Driver
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let E be a vector bundle over a smooth connected manifold M with typical fiber space V (V is a complex or real finite-dimensional vector space) and with structure group G (G is a closed Lie subgroup of \(G\ell (V))\). Pairs (E,V) and \((E',V')\) of connections on E are equivalent if there is a vector bundle isomorphism K: \(E\to E'\) with \(KV=V'K\) such that K respects the structure group G. The set of equivalence classes [E,V] will be denoted by \({\mathcal E}={\mathcal E}(M,V,G).\) Let \(\Omega_ m\) be the space of \(H^ 1\)-loops in M starting at \(m\in M\) and \({\mathcal M}^ a \)subset of \(C^{\infty}(\Omega_ m,G)\) where every \(P\in {\mathcal M}\subset C^{\infty}(\Omega_ m,G)\) is multiplicative \((P(\sigma \tau)=P(\sigma)\cdot P(\tau)\) for \(\sigma,\tau \in \Omega_ m\), \(\sigma\) \(\tau\) is the join of the loops \(\sigma\) and \(\tau)\), parametrization invariant \((P(\sigma ')=P(\sigma)\), if \(\sigma '\) is a reparametrization of \(\sigma)\) and strongly differentiable. Two functions \(P,P'\in {\mathcal M}\) are equivalent if for all \(\sigma \in \Omega_ m\), \(P'(\sigma)=g\cdot P(\sigma)\cdot g^{-1}\) for any \(g\in G\). The set of all equivalence classes [P] will be denoted by \(\bar {\mathcal M}=\bar {\mathcal M}(M,m,V,G).\) For a connection \(\nabla\) on E the parallel transfer operator on \(\Omega_ m\) is denoted by \(P^{\nabla}\) and the element of \(\bar {\mathcal M}\) defined by \([K\otimes P^{\nabla}(.)\otimes K^{-1}]\) by \([P^{\nabla}]\) where K: \(E_ m\to V\) is a linear isomorphism such that for every admissible local trivialization \(\psi\) of E, \(K\otimes \psi_ m^{-1}\in G\). One of the main results of this remarkable memoir is theorem 5.1 which states that the correspondence [E,\(\nabla]\in {\mathcal E}\to [P^{\nabla}]\in \bar {\mathcal M}\) is one to one. A similar statement was announced by \textit{S. Kobayashi} [C. R. Acad. Sci., Paris 238, 318-319 (1954; Zbl 0055.166)]. Another main result states that if M is simply connected then t…
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 55R25
zbMATH DE Number: 4107606
zbMATH Keywords: vector bundle
zbMATH Keywords: connections
zbMATH Keywords: loops
zbMATH Keywords: parallel transfer
zbMATH Keywords: integrated lassos
zbMATH Keywords: bundle connection pairs
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90035-9
OpenAlex ID: W2024447621
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description: scientific article; zbMATH DE number 4107606