On the Holonomic Equivalence of Two Curves

Abstract

Given a principal $G$-bundle $P\to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The main result in this paper is that if $G$ is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, $C^1$ homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.

20 pages, 2 figures

References in corpus (1)

• Uniqueness for the signature of a path of bounded variation and the reduced path group

Cited by in corpus (3)

• On the Ashtekar-Lewandowski Measure as a Restriction of the Product One
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• Thin homotopy and the holonomy approach to gauge theories