A <diffeomorphism> $\phi:M\to M$ is a smooth bijection with smooth inverse. It acts covariantly on a differential form by the <pullback of a differential form>:
$$
\omega\longmapsto\phi^*\omega.
$$
For vector fields $X_0,\ldots,X_p$,
$$
(\phi^*\omega)_x(X_1,\ldots,X_p)
=\omega_{\phi(x)}(d\phi_xX_1,\ldots,d\phi_xX_p).
$$
The coordinate definition of the <exterior derivative>, or its characterization as the unique natural graded derivation extending the differential of functions, gives
$$
d(\phi^*\omega)=\phi^*(d\omega).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 4 / a / Solution

Transform both fields geometrically, $g\mapsto\phi^*g$ and $A\mapsto\phi^*A$. Naturality gives
$$
R_{\phi^*g}=\phi^*R_g,\quad
\operatorname{vol}_{\phi^*g}=\phi^*\operatorname{vol}_g,\quad
d(\phi^*A)=\phi^*F,
$$
and
$$
*_{\phi^*g}(\phi^*F)=\phi^*(*_gF).
$$
Each Lagrangian four-form therefore transforms by pullback. Integration of a top-degree form is unchanged under an orientation-preserving <diffeomorphism>, so both the Einstein-Hilbert and Maxwell terms, and hence the full action, are invariant.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 4 / c / Solution

{wiki}

A diffeomorphism is a smooth bijection between smooth manifolds whose inverse is also smooth.


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