2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69859The motivation for this paper stems \cite{CR} from the need to construct explicit isomorphisms of (possibly nontrivial) principal $G$-bundles on the space of loops or, more generally, of paths in some manifold $M$, over which I consider a fixed principal bundle $P$; the aforementioned bundles are then pull-backs of $P$ w.r.t. evaluation maps at different points. The explicit construction of these isomorphisms between pulled-back bundles relies on the notion of {\em parallel transport}. I introduce and discuss extensively at this point the notion of {\em generalized gauge transformation between (a priori) distinct principal $G$-bundles over the same base $M$}; one can see immediately that the parallel transport can be viewed as a generalized gauge transformation for two special kind of bundles on the space of loops or paths; at this point, it is possible to generalize the previous arguments for more general pulled-back bundles. Finally, I discuss how flatness of the reference connection, w.r.t. which I consider holonomy and parallel transport, is related to horizontality of the associated generalized gauge transformation.56 pagesDifferential Geometry53C05; 53C29; 81Q70text