Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 22 / 2 / a / Solution

Use a row of local frame sections $e=(e_1,\dots ,e_m)$ and columns of coefficients: $D(eu)=e(du+Au)$. If $e^{\prime }=eg$ for a smooth invertible matrix-valued function $g$, thenHence the change of frame of a vector-bundle connection isThis convention specifies both the placement of the frame matrix and the sign of the inhomogeneous term. The vector-bundle curvature transforms by ${\Theta }^{\prime }=g^{-1}\Theta g$, either by applying $D^2$ to the new frame or by substituting in the formula of the next subpart.