# Difference between relative Moser and Weinstein's Lagrangian neighbourhood theorem

The statement of relative Moser:

Let M be a manifold, X a compact submanifold, i : X → M the inclusion map, and $$ω_0$$ and $$ω_1$$ symplectic forms on M such that $$i^\ast ω_0 = i^\ast ω_1$$. Then there exist neighborhoods $$U_0$$ and $$U_1$$ of X in M and a diffeomorphism $$ϕ : U_0 → U_1$$ such that $$\phi|_X=id_X$$ and $$\phi^\ast\omega_1=\omega_0$$.

The Statement of WLT:

Let M be a 2n-dimensional manifold, X a compact n-dimensional submanifold, i : X → M the inclusion map, and $$ω_0$$ and $$ω_1$$ symplectic forms on M such that $$i^\ast ω_0 = i^\ast ω_1 = 0$$, i.e., X is a lagrangian submanifold of both (M, $$ω_0$$) and (M, $$ω_1$$). Then there exist neighborhoods $$U_0$$ and $$U_1$$ of X in M and a diffeomorphism $$ϕ : U_0 → U_1$$ such that $$\phi|_X=id_X$$ and $$\phi^\ast\omega_1=\omega_0$$.

Is the WLT not just a special case of relative Moser's Theorem? That said, anywhere I look up or read the proof of WLT, the Whitney extension theorem is used in addition to relative Moser's. I do not understand why the WLT does not directly follow from relative Moser. Could someone clarify this?

WLT's hypotheses are not quite framed correctly for a direct application of relative Moser. For relative Moser to be applied, we need that $$\omega_0|_p = \omega_1|_p$$ for all $$p\in X$$. For WLT, we rather assume that $$X$$ is Lagrangian with respect to $$\omega_0$$ and $$\omega_1$$, so that $$i^*\omega_0 = i^*\omega_1 = 0$$.
Let $$p\in M$$. Assuming the above, symplectic linear algebra tells us that there is a linear isomorphism $$L_p: T_pM \to T_pM$$ such that $$L_p|_{T_pX} = Id_{T_pX}$$ and $$L_p^*\omega_1|_p = \omega_0|p$$. Whitney extension theorem says that an embedding $$h:N\to M$$ exists where $$N$$ is a neighbourhood of $$X$$ such that $$h|_X = Id_X$$ and $$dh_p = L_p$$. One can check then that $$(h^*\omega_1)p = \omega_0|p.$$ So now we're in a place to apply relative Moser: we have $$i:X\hookrightarrow N$$ an embedded submanifold, with two symplectic forms $$h^*\omega_1$$ and $$h^*\omega_0$$ on $$N$$, such that for all $$p\in X$$, $$(h^*\omega_1)p = \omega_0|p.$$
WLT then follows by composing the resulting diffeomorphism from Moser with the embedding $$h:N\to M$$.
• Does $i^\ast\omega_0=i^\ast\omega_1=0$ not imply that $\omega_0|_p=\omega_1|_p=0$ for all $p\in X$? . Commented Feb 1 at 7:00