Construction of connection inducing maps between principal bundles. I
type: Item
instance of: scholarly article
title: Construction of connection inducing maps between principal bundles. I
zbMATH Open document ID: 0639.53035
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: Let \(X\to V\) and \(Y\to W\) be two \(C^{\infty}\)-smooth principal bundles with the same structure group G and with \(C^{\infty}\)-connection \(\Gamma\) on X and \(\Delta\) on Y, respectively. The problem which the author deals with, is to find a \(C^{\infty}\)-map \(f: V\to K\) such that the induced bundle \(f^*(Y)\) over V with the induced connection \(f^*(\Delta)\) is isomorphic to (X,\(\Gamma)\). She gives some general criteria for the existence of connection inducing maps, and she states the following main result. Let \(\Gamma\) be an arbitrary connection on a trivial 0(p)-bundle over a stably parallelizable n-dimensional manifold V. If \(q\geq p(n+3)/2\), then there exists a connection inducing map \(f: V\to Gr_ p({\mathbb{R}}^ q).\) From a technical point of view, the construction of a connection inducing map amounts to solving a system of partial differential equations. If \(q\geq p(n+1)\), then the PDE for connection inducing maps \(f: V\to Gr_ p({\mathbb{R}}^ q)\) can be reduced to an algebraic system. Moreover, a suitable regularity condition on f makes the linearized PDE algebraically solvable. Then, Nash's implicit function theorem can be applied to the local study and the theory of topological sheaves gives global results.
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 4042753
zbMATH Keywords: principal bundles
zbMATH Keywords: connection
zbMATH Keywords: connection inducing maps
zbMATH Keywords: Nash's implicit function theorem
zbMATH Keywords: topological sheaves
author: Giuseppina D'Ambra
reviewed by: Anna Maria Pastore
Wikidata QID: Q115184746
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441176
OpenAlex ID: W2164925024
DOI: 10.4310/JDG/1214441176
description: scientific article; zbMATH DE number 4042753