Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 15 / 3 / c / Solution

Use the covariant exterior derivative on the endomorphism bundle and remember that $F$ has degree two. The Bianchi identity is $d_{A}F=dF+A\wedge F-F\wedge A=0$. The local formula from part (b) givesThe two lines cancel, establishingBecause this is an endomorphism-valued equation and the operators transform covariantly under a change of local frame, this local calculation proves the global identity.

For a line bundle over $k=\mathbb{R}$ or $\mathbb{C}$, every fibre endomorphism is multiplication by a scalar. Thus $\mathrm{End}E$ is canonically the trivial scalar bundle, even when $E$ itself is nontrivial. In a local frame $A$ is a scalar one-form, so $A\wedge A=0$ and $F=dA$ locally. Also $A\wedge F=F\wedge A$, since their degrees are one and two. The Bianchi identity therefore reduces to the ordinary closedness conditionThis does not assert that the local scalar one-forms $A$ combine into a global one-form.

For two connections on a vector bundle $\nabla ,{\nabla }^{\prime }$, the difference is $C^{\infty }(M)$-linear in the section: the two $df\otimes s$ terms cancel. It is consequently a global endomorphism-valued one-form $a$. In rank one, $a$ is a global ordinary scalar one-form. Locally $A^{\prime }=A+a$, and the scalar cross terms and $a\wedge a$ vanish, givingThis is a global equality, so the two closed differential forms differ by an exact differential form. It proves the connection-independent curvature class of a line bundle: