Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 118 / 3 / a

A connection on a vector bundle is a real-linear map (complex-linear for a complex bundle)It extends to the covariant exterior derivative by $D(\eta \otimes s)=d\eta \otimes s+(-1{)}^k\eta \wedge Ds$ when $\eta$ has degree $k$. Its square is linear over smooth functions: in $D^2(fs)$ the two $df\wedge Ds$ terms have opposite signs and $d^{2}f=0$. The curvature form of a connection is therefore the endomorphism-valued two-formIn a local frame with $D=d+\theta$, it is $\Theta =d\theta +\theta \wedge \theta$.

Define the tensor product connection on decomposable sections byThe Leibniz rule for $D_1,D_2$ makes this well-defined over smooth functions and makes it a connection on a vector bundle. Applying its covariant exterior derivative again givesThe mixed terms cancel by the graded sign. Thus curvature adds on the two tensor factors:Equivalently, the mixed terms in $({\theta }_1\otimes I+I\otimes {\theta }_2)\wedge ({\theta }_1\otimes I+I\otimes {\theta }_2)$ cancel because operators on the different factors commute. This also proves the curvature of a tensor product connection formula in local frames.