Gauge equivalence of principal connections ($A\sim g^{-1}Ag+g^{-1}dg$)

A gauge transformation is an automorphism of a principal bundle covering the identity on the base and commuting with its right group action. Pullback relates gauge-equivalent principal connections. Locally it acts by the displayed formula; the gauge curvature transforms by conjugation. Gauge functions on overlapping charts satisfy the bundle's transition compatibility. Infinitesimally $g=e^{\epsilon }$ gives $\delta A=D_A\epsilon$.