Remarks on generally covariant Lagrangians for electromagnetism and {$G^k_n$}-invariance

The authors derive the conditions for a first-order Lagrangian in classical field theory to be generally covariant by writing out the infinitesimal generators of the action of the diffeomorphism group on the first jet bundle. They solve explicitly the case of a scalar field and of a field of 1-forms on four-dimensional space-time with a fixed background metric $g_{\mu\nu}$. For a scalar field $\Phi$ the expected result is that $\scr L=\sqrt gL(\Phi, g^{\mu\nu}\partial\Phi_\mu\partial\Phi_\nu)$, and for a 1-form field $A=A_\mu\,dx^\mu$ the Lagrangian becomes a function of the four scalars that can be formed out of $A_\mu$ and $F_{\mu\nu}=2\partial_{[\nu}A_{\mu]}$.