Lemma: Admitted transformations

1. $F_{\kappa }$ is the identity outside a compact subset of $U$; in particular on a neighbourhood of $\{t\le \epsilon \}$ and of $\{t\ge T\}$. It maps $U$ onto $U$ and $M_T$ onto $M_T$.

2. $F_{\kappa }^{\ast }g=g$ and $F_{\kappa }^{\ast }{\sigma }_{\epsilon }={\sigma }_{\epsilon }$.

3. For every smooth tensor field or tensor density $\mathsf{T}$,

   $$F_{\kappa }^{\ast }\mathsf{T}=\mathsf{T}+\kappa {\mathcal{L}}_{\xi }\mathsf{T}+{\kappa }^2({\mathcal{L}}_{\zeta }+\frac{1}{2}{{\mathcal{L}}_{\xi }}^2)\mathsf{T}+O({\kappa }^3),$$

   with the Lie derivative of the appropriate type and the remainder $O({\kappa }^3)$ in the smooth topology on compact sets.

4. The formal operators $\exp (\kappa {\mathcal{L}}_{\xi }+{\kappa }^2{\mathcal{L}}_{\zeta })(\mathrm{mod}{\kappa }^3)$, $\xi ,\zeta \in {\mathfrak{g}}_U$, form a group ${\mathcal{G}}_U$ under composition.

Proof. (a) The flow of a vector field supported in a compact $K_0\subset U$ is the identity outside $K_0$ and preserves every set containing $K_0$ whose complement it fixes. (b) On $U$ both $g$ and ${\sigma }_{\epsilon }$ equal one by (eq:U); outside $U$, $F_{\kappa }=\mathrm{id}$. (c) Let $\Psi (s)=({\mathrm{Fl}}_{V_{\kappa }}^s{)}^{\ast }\mathsf{T}$. Then ${\partial }_s\Psi =({\mathrm{Fl}}_{V_{\kappa }}^s{)}^{\ast }{\mathcal{L}}_{V_{\kappa }}\mathsf{T}$, so

because ${{\mathcal{L}}_{{\mathcal{V}}_{\kappa }}}^3=O({\kappa }^3)$. Insert ${\mathcal{L}}_{V_{\kappa }}=\kappa {\mathcal{L}}_{\xi }+{\kappa }^2{\mathcal{L}}_{\zeta }$. (d) By the Baker–Campbell–Hausdorff formula modulo ${\kappa }^3$,

and $[\xi ,{\xi }^{\prime }]\in {\mathfrak{g}}_U$. Inverses are obtained with $(\xi ,\zeta )\mapsto (-\xi ,-\zeta )$.