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arXiv · 1311.6611

On the Holonomic Equivalence of Two Curves

Tamer Tlas
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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.

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Tamer Tlas. 2013-11-26. On the Holonomic Equivalence of Two Curves. https://doi.org/10.1142/s0129167x16500555

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