Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 55 / 3 / a / Solution

Choose local sections related by $s^{\prime }=su$, where $u:U\to G$. Keeping the same point of the principal bundle requires $s\gamma =s^{\prime }{\gamma }^{\prime }$, so ${\gamma }^{\prime }=u^{-1}\gamma$. For reconstruction of a principal connection from local gauge potentials, the local principal connection form transforms asUsing $d(u^{-1})=-u^{-1}(du)u^{-1}$, computeandThe inhomogeneous terms cancel. Thus $\boxed{{\omega }^{\prime }=\omega }$: both local expressions define the same one-form on the total space. This is compatibility across trivializations, whereas $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$ describes the separate right-action equivariance of a principal connection. In particular a change of trivialization is not an assertion that the local gauge potential of a principal connection $A$ itself is unchanged.