Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 3 / a

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- Solution a

Solution

In a local frame of

E

, a connection on a vector bundle has the form

d^{A} = d + A \land,

where

A

is a matrix of one-forms. Under a frame change

g

, its matrix transforms as

A^{'} = g^{- 1} A g + g^{- 1} d g .

The covariant exterior derivative on an

E

-valued

r

-form

σ

is

d^{A} σ = d σ + A \land σ .

The curvature form of a connection is

F = d A + A \land A .

Using the graded Leibniz rule,

(d^{A})^{2} σ = d (A \land σ) + A \land d σ + A \land A \land σ = (d A + A \land A) \land σ = F \land σ .

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