Inherited statements

ial_1\), $h=x^1{\delta }^2\otimes {\delta }^2$, ${\varphi }_0=t{x}^1$, $⟨d_1[{\mathcal{L}}_{\xi }h],{\varphi }_0⟩=2\int f{\partial }_1{\partial }_t({{\mathsf{E}}_0}^{\mathrm{R}}{)}^2\chi \ne 0$ for suitable $f,g,\delta$.

• Lemma 5.1 and Proposition 5.2 of X-E: for a diffeomorphism $F$ equal to the identity outside a compact set, a natural Lagrangian density integrates to the same value before and after pull-back, and its Euler–Lagrange tensor density is natural, $E[F^{\ast }\text{fields}]=F^{\ast }E[\text{fields}]$.