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

Solution

Use a row of local frame sections

e = (e_{1}, \dots, e_{m})

and columns of coefficients:

D (e u) = e (d u + A u)

. If

e^{'} = e g

g

, then

D (e^{'}) = D (e g) = e (A g + d g) = e^{'} (g^{- 1} A g + g^{- 1} d g) .

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

This convention specifies both the placement of the frame matrix and the sign of the inhomogeneous term. The vector-bundle curvature transforms by

Θ^{'} = g^{- 1} Θ g

, either by applying

D^{2}

to the new frame or by substituting in the formula of the next subpart.

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