Relative Poincaré primitive near a submanifold

Spend ${\mathrm{A}\mathrm{C}}_{\omega }$ exactly through [F1] to identify a neighbourhood of $S$ with a neighbourhood of the zero section in its normal bundle. Shrink it to be invariant under fibrewise dilation and let $H_t(s,v)=(s,tv)$. This deformation retracts the tube to the zero section and is independent of the parameter.

Orient the homotopy from $H_0$ to $H_1=\mathrm{id}$ and put ${\beta }_p=K_H{\alpha }_p$. Because $d{\alpha }_p=0$ and $H_0^{\ast }{\alpha }_p=0$, [F2] gives ${\alpha }_p=d{\beta }_p$. The integral defining $K_H$ is jointly smooth in the supplied parameters. In local bundle coordinates, the coefficients of ${\alpha }_p(s,v)$ are $O(\mathrm{\mid }v\mathrm{\mid })$ and contraction with the radial homotopy velocity contributes another factor $v$; hence ${\beta }_p(s,v)=O(\mathrm{\mid }v{\mathrm{\mid }}^2)$. Tangential derivatives vanish as well because ${\beta }_p(s,0)=0$ identically in $s$. Thus its first jet vanishes along $S$.