Corollary: Contact modifications relative to the sheet

Let $N:\mathcal{D}\to \mathcal{D}$ be any map and ${\mathsf{R}}^N(F_c)=\hat{\varphi }(d_c[h]+Nc)$.

1. ${\mathsf{R}}^N$ satisfies (eq:sheet) if and only if ${\Delta }_{S}Nc=0$ for all $c$, if and only if $\hat{\varphi }(Nc)=0$ for all $c$; in that case ${\mathsf{R}}^N=R_1(V_{1,g}[h];\cdot )$ as maps into ${\mathfrak{A}}_0$.

2. Example of a sheet-compatible modification: $Nc=P_m(\chi c)$ with $\chi \in C^{\infty }(M)$; it is invisible in ${\mathfrak{A}}_0$.

3. Example of a sheet-incompatible modification: $Nc=wc$ with $w\in C^{\infty }(M)$ real and $w\not\equiv 0$; in particular $w=\lambda g{b}_1[h]$ with $\lambda \ne 0$ whenever $g{b}_1[h]\not\equiv 0$.

Proof. (i) ${\mathrm{ev}}_u{\mathsf{R}}^N(F_c)-{\mathrm{ev}}_u{R}_1(F_c)=⟨Nc,u⟩$; apply Lemma lem:detect to $Nc$ for each $c$, and Lemma lem:graded. (ii) $P_m(\chi c)\in P_m\mathcal{D}$. (iii) Choose $x_0$ with $w(x_0)\ne 0$. The Cauchy problem with smooth compactly supported data $u{|}_{t={x_0}^0}=\psi$, ${\partial }_{t}u{|}_{t={x_0}^0}=0$, $\psi ({\mathbf{x}}_0)\ne 0$, has a unique smooth solution, spatially compact by finite propagation BGP (Thm. 3.2.11); hence $u\in {\Delta }_S\mathcal{D}$ BGP (Thm. 3.4.7), and $u(x_0)\ne 0$. On a small ball around $x_0$ the real function $\mathrm{Re}(w\bar{u})$ or $\mathrm{Im}(w\bar{u})$ has a fixed sign; choose $c\ge 0$, $c\not\equiv 0$, supported in that ball with the corresponding phase so that $⟨wc,u⟩\ne 0$. Thus ${\Delta }_S(wc)\ne 0$ and (i) fails.