The proof is trivial in the Abelian case by the Stokes' theorem.How to prove it in the non-Abelian case?



Prove that the holonomies along any two homotopic paths are the same if the curvature of the connection vanishes
A possible answer would be to invoke *Ambrose-Singer Holonomy theorem*:


**Theorem (Ambrose-Singer)** Let $M$ be a (smooth) connected manifold, $E\to M$ a vector bundle over $M$, and $\nabla$ a connection on $E$. Then, for each $x\in M$, $\mathfrak{hol}\_x(\nabla)$ is a Lie subalgebra of $\text{End}(E\_x)$ which, as a vector space, is spanned by all elements of $\text{End}(E\_x)$ of 
the form $P\_{\gamma}^{-1}[ (F\_{\nabla})\_y\cdot{(v\wedge w)}] P\_{\gamma}$, where $\gamma:[0,1]\to M$ is a piece-wise smooth path with end points $\gamma(0) = x$ and $\gamma(1) = y$, $P\_{\gamma}: E\_x \to E\_y$ is the associated parallel translation map, and $v,w\in T\_yM$.


In particular, if the connection $\nabla$ is *flat* (i.e. if $F\_{\nabla}=0$), then $\mathfrak{hol}\_x(\nabla)=0$, and therefore the restricted holonomy group $\text{Hol}\_x^0(\nabla)$ is trivial, for each $x\in M$. Recalling
$$
\text{Hol}\_x^0(\nabla)=\{P\_{\gamma}: \gamma\text{ is a null-homotopic loop based at }x\},
$$ the conclusion you need follows. Indeed, given two homotopic paths $\gamma\_1$ and $\gamma\_2$ (I'm assuming this homotopy preserves the base-points), say with $\gamma\_1(0)=\gamma\_2(0)=x$, then you can consider the null-homotopic concatenation $\gamma\_2^{-1}\cdot{\gamma\_1}$, which is a (piece-wise smooth) loop based at $x$. By the above, we have that $1=P\_{\gamma\_2^{-1}\cdot{\gamma\_1}} = P\_{\gamma\_2}^{-1}\circ P\_{\gamma\_1}$, i.e. $P\_{\gamma\_1}=P\_{\gamma\_2}$, as we wanted.


A principal bundle version of this theorem (together with a proof) can be found e.g. in Kobayashi&Nomizu's book [Foundations of differential geometry, vol. I](http://www.ams.org/journals/bull/1964-70-02/S0002-9904-1964-11094-6/), Theorem 8.1, p. 89. The statement I give here was based on Theorem 2.4.3. (a) of Joyce's book [Riemannian Holonomy Groups and Calibrated Geometry](http://link.springer.com/chapter/10.1007/978-3-642-19004-9_1).


