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

For a section represented by a column $u$, expand the covariant exterior derivative using the graded Leibniz rule:Therefore the Cartan curvature matrix equation isThe product includes endomorphism composition; it need not vanish for a matrix-valued one-form. For a scalar frame change $g$, $A^{\prime }=A+g^{-1}dg$ and $d(g^{-1}dg)=0$, confirming that the line-bundle two-forms $dA$ patch globally.