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On the holonomic equivalence of two curves

Tamer Tlas

Department of Mathematics, American University of Beirut, Beirut, Lebanon

https://doi.org/10.1142/S0129167X16500555Cited by:1 (Source: Crossref)

Abstract

Given a principal $G$ $G$ -bundle $P \to M$ $P \to M$ and two $C^{1}$ $C^{1}$ curves in $M$ $M$ 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 $P$ $P$ . The main result in this paper is that if $G$ $G$ 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, $C^{1}$ $C^{1}$ 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.

Keywords:

AMSC: 53C29, 57R35

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