I am trying to prove that for a vector bundle with a flat connection, holonomies between points $p$ and $q$ are the same for smoothly homotopic paths. If we talk about a local trivialisation, and all the curves are within it, the problem can be solved with straightforward computation using path ordered exponents. I cannot understand what to do if curves travel all around the manifold and do not stay within one trivialisation patch.
So far, I have the idea that the image of homotopy $\gamma_s:[0,1]^2 \rightarrow M$ is compact as an image of compact space. Therefore, the image is coverable with a finite number of coordinate trivialisations. Looking for different pictures of how it possibly can happen, I think that some induction on the number of charts can help here. I hope the following images clarify what I mean.
Coloured regions here are different trivialisation charts. Red lines are curves that help to reduce a problem with the given number of charts to the subproblem with a smaller amount of patches. It seems like it is always possible to come up with some red line. However, it is unclear how to define it in the general case.
Coloured regions here are different trivialisation charts. Red lines are curves that help to reduce a problem with the given number of charts to the subproblem with a smaller amount of patches. It seems like it is always possible to come up with some red line. However, it is unclear how to define it in the general case.
Another idea is to consider the question in terms of contractible loops. I thought it could be possible to represent holonomy around a big loop as a sum over smaller ones and use that a holonomy around a small loop for a vanishing curvature is $O(\epsilon^3)$, where $\epsilon$ is the size of the loop. It seems not to be the case.