Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 4 / d

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- Solution d

Solution

Let

D_{1}

and

D_{2}

each be a connection on a vector bundle. Their difference is tensorial, so

D_{1} = D_{2} + a

with

a \in Ω^{1} (End E)

. Extend

D_{2}

to the endomorphism bundle connection. Expanding

(D_{2} + a)^{2}

with the supplied graded Leibniz rule gives the curvature difference formula

F_{D_{1}} = F_{D_{2}} + D_{2} (a) + a \land a .

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