Differential geometric connection of finite order on composite manifold, infinitesimal polygonal loops and generalized Fermi-Walker transport
type: Item
instance of: scholarly article
title: Differential geometric connection of finite order on composite manifold, infinitesimal polygonal loops and generalized Fermi-Walker transport
zbMATH Open document ID: 0763.53020
DOI: 10.1016/0034-4877(91)90044-N
author: D. Yu. Glazkov
published in: Reports on Mathematical Physics
publication date: 1993-01-16T00:00:00Z
review text: A composite manifold \(L_ m(B_ n)\) (similar to a fiber bundle) is considered. The base space \(B_ n\) has coordinates \(y^ k\), \(k,\ell=1,\dots,n\), and the local space \(L_ m\) has coordinates \(\eta^ a\), \(a,b=1,\dots,m\); \(s\) runs over \(1,\dots,r\). A differential geometric connection of the \(r\)-th order is given by \(d\eta^ a/dt+\Gamma^ a\left(y^ k,d^ sy^ \ell/dt^ s,\eta^ b\right)=0\), where \(\Gamma^ a\) are the coefficients of the connection. (The nonlinear connection is that special case, when \(s=1\) and the \(\Gamma^ a\) are linear in \(dy^ \ell/dt\).) The holonomy group transformations belonging to the infinitesimal 4-gonal loops in \(L_ m(B_ n)\), calculated with an accuracy up to terms of the second order of smallness are already known in the literature and they are determined by the curvature vector (tensor) of the connection. Now the coordinate 4-gonal loops are replaced by polygonal loops, and the holonomy group transformations are calculated in this case, where beside the curvature vector other additional terms appear. In the second part of the paper \(L_ m(B_ n)\) is replaced by a vector bundle, and other simplifications are also investigated. Finally, as an illustration, the generalized Fermi-Walker transport is considered on a Riemannian tangent bundle.
Mathematics Subject Classification ID: 53B15
zbMATH DE Number: 89988
zbMATH Keywords: \(r\)-th order connection
zbMATH Keywords: holonomy group transformations
zbMATH Keywords: curvature vector
Wikidata QID: Q115364215
reviewed by: Lajos Tamássy
MaRDI profile type: Publication
cites work: Q5801830
cites work: Q3260127
cites work: Q5801822
cites work: Global Geometry of Polygons. I: The Theorem of Fabricius-Bjerre
cites work: Q5726225
cites work: Q4725259
cites work: Q4141810
cites work: Q4692418
cites work: Q3263435
full work available at URL: 0034-4877(91)90044-n
OpenAlex ID: W2110653695
description: scientific article; zbMATH DE number 89988