Parallel Transports in Webs
Abstract
For connected reductive linear algebraic structure groups it is proven that every web is holonomically isolated. The possible tuples of parallel transports in a web form a Lie subgroup of the corresponding power of the structure group. This Lie subgroup is explicitly calculated and turns out to be independent of the chosen local trivializations. Moreover, explicit necessary and sufficient criteria for the holonomical independence of webs are derived. The results above can even be sharpened: Given an arbitrary neighbourhood of the base points of a web, then this neighbourhood contains some segments of the web whose parameter intervals coincide, but do not include 0 (that corresponds to the base points of the web), and whose parallel transports already form the same Lie subgroup as those of the full web do.
Keywords
Cite
@article{arxiv.math-ph/0304001,
title = {Parallel Transports in Webs},
author = {Christian Fleischhack},
journal= {arXiv preprint arXiv:math-ph/0304001},
year = {2007}
}
Comments
22 pages, 1 figure, LaTeX. -- v2: Main results (application to reductive groups and webs) unchanged. Def. 4.1, two refs. and technical assumption in Thm. 4.1 added; Acknowl., Cor. 4.2, Thm. 4.9, proofs of Thm. 4.1 and Prop. 4.11 accordingly modified