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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?

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1 Answer 1

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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 $h:N\to M$.

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  • $\begingroup$ Does $i^\ast\omega_0=i^\ast\omega_1=0$ not imply that $\omega_0|_p=\omega_1|_p=0$ for all $p\in X$? . $\endgroup$ Commented Feb 1 at 7:00

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