= Relative Moser theorem
{title2=$\kappa^*\omega_1=\omega_0,\quad\kappa|_S=\operatorname{id}_S$}
Two closed <symplectic forms> agreeing as ambient bilinear forms along a compact embedded submanifold are equivalent near it by a <diffeomorphism> fixing it pointwise. Shrink until their affine interpolation is <nondegenerate>. A relative homotopy primitive $\eta$ for their difference vanishes as an ambient covector on the submanifold. Solving $\iota_{Z_t}\omega_t=-\eta$ gives a <vector field> vanishing there; its local flow fixes the submanifold and makes $\omega_t$ constant under pullback.
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