Let $p:E\longrightarrow B$ be a smooth surjective submersion and $\sigma, \sigma^\prime: p^*(TB)\longrightarrow TE$ be two complete connections. Given a path $\gamma:I\longrightarrow B$ we can consider the holonomomies:
$$\textrm{Hol}^\sigma_{\gamma}, \textrm{Hol}^{\sigma^\prime}_{\gamma}: E_{\gamma(0)}\longrightarrow E_{\gamma(1)},$$ which are diffeomorphisms between the fibers $E_{\gamma(j)}:=p^{-1}(\gamma(j))$. Is there a way to compare those maps? It has to do with [this question][1]. 

Thanks.


  [1]: http://math.stackexchange.com/questions/1878930/path-connecting-textrmhol-sigma-gammae-and-textrmhol-sigma-prime