Moser's trick

In the context of symplectic geometry, the Moser's trick is often presented in the following form.^[3][4]

Let ${\displaystyle \{\omega _{t}\}_{t\in [0,1]}\subset \Omega ^{2}(M)}$ be a family of symplectic forms on ${\displaystyle M}$ such that ${\displaystyle {\frac {d}{dt}}\omega _{t}=d\sigma _{t}}$, for ${\displaystyle \{\sigma _{t}\}_{t\in [0,1]}\subset \Omega ^{1}(M)}$. Then there exists a family ${\displaystyle \{\psi _{t}\}_{t\in [0,1]}}$ of diffeomorphisms of ${\displaystyle M}$ such that ${\displaystyle \psi _{t}^{*}\omega _{t}=\omega _{0}}$ and ${\displaystyle \psi _{0}=\mathrm {id} _{M}}$.

In order to apply Moser's trick, we need to solve the following ODE

$${\displaystyle 0={\frac {d}{dt}}\omega _{t}+{\mathcal {L}}_{X_{t}}\omega _{t}=d\sigma _{t}+\iota _{X_{t}}({\cancel {d\omega _{t}}})+d(\iota _{X_{t}}\omega _{t})=d(\sigma _{t}+\iota _{X_{t}}\omega _{t}),}$$where we used the hypothesis, the Cartan's magic formula, and the fact that ${\displaystyle \omega _{t}}$ is closed. However, since ${\displaystyle \omega _{t}}$ is non-degenerate, i.e. ${\textstyle TM\xrightarrow {\cong } T^{*}M,\quad X_{t}\mapsto \iota _{X_{t}}\omega _{t}}$, given ${\displaystyle \sigma _{t}}$ one can always find ${\displaystyle X_{t}}$ such that ${\displaystyle \sigma _{t}+\iota _{X_{t}}\omega _{t}=0}$.

Given two symplectic structures ${\displaystyle \omega _{0}}$ and ${\displaystyle \omega _{1}}$ on ${\displaystyle M}$ such that ${\displaystyle (\omega _{0})_{x}=(\omega _{1})_{x}}$ for some point ${\displaystyle x\in M}$, there are two neighbourhoods ${\displaystyle U_{0}}$ and ${\displaystyle U_{1}}$ of ${\displaystyle x}$ and a diffeomorphism ${\displaystyle \phi :U_{0}\to U_{1}}$ such that ${\displaystyle \phi (x)=x}$ and ${\displaystyle \phi ^{*}\omega _{1}=\omega _{0}}$.^[3][4]

This follows by noticing that, by Poincaré lemma, the difference ${\displaystyle \omega _{1}-\omega _{0}}$ is locally ${\displaystyle d\sigma }$ for some ${\displaystyle \sigma \in \Omega ^{1}(M)}$; then, shrinking further the neighbourhoods, the result above applied to the family ${\displaystyle \omega _{t}:=(1-t)\omega _{0}+t\omega _{1}}$ of symplectic structures yields the diffeomorphism ${\displaystyle \phi$ :=\psi _{1}} .

The Darboux's theorem for symplectic structures states that any point ${\displaystyle x}$ in a given symplectic manifold ${\displaystyle (M,\omega )}$ admits a local coordinate chart ${\displaystyle (U,x^{1},\ldots ,x^{n},y^{1},\ldots ,y^{n})}$ such that$${\displaystyle \omega |_{U}=\sum _{i=1}^{n}dx^{i}\wedge dy^{i}.}$$While the original proof by Darboux required a more general statement for 1-forms,^[5] Moser's trick provides a straightforward proof. Indeed, choosing any symplectic basis of the symplectic vector space ${\displaystyle (T_{x}M,\omega _{x})}$, one can always find local coordinates ${\displaystyle ({\tilde {U}},{\tilde {x}}^{1},\ldots ,{\tilde {x}}^{n},{\tilde {y}}^{1},\ldots ,{\tilde {y}}^{n})}$ such that ${\displaystyle \omega _{x}=\sum _{i=i}^{n}(d{\tilde {x}}^{i}\wedge d{\tilde {y}}^{i})|_{x}}$. Then it is enough to apply the corollary of Moser's trick discussed above to ${\displaystyle \omega _{0}=\omega |_{\tilde {U}}}$ and ${\displaystyle \omega _{1}=\sum _{i=i}^{n}d{\tilde {x}}^{i}\wedge d{\tilde {y}}^{i}}$, and consider the new coordinates ${\displaystyle x^{i}={\tilde {x}}^{i}\circ \phi ,y^{i}={\tilde {y}}^{i}\circ \phi }$.^[3][4]

Moser himself provided an application of his argument for the stability of symplectic structures,^[1] which is known now as Moser stability theorem.^[3][4]

Let ${\displaystyle \{\omega _{t}\}_{t\in [0,1]}\subset \Omega ^{2}(M)}$ a family of symplectic form on ${\displaystyle M}$ which are cohomologous, i.e. the deRham cohomology class ${\displaystyle [\omega _{t}]\in H_{dR}^{2}(M)}$ does not depend on ${\displaystyle t}$. Then there exists a family ${\displaystyle \psi _{t}}$ of diffeomorphisms of ${\displaystyle M}$ such that ${\displaystyle \psi ^{*}\omega _{t}=\omega _{0}}$ and ${\displaystyle \psi _{0}=\mathrm {id} _{M}}$.

It is enough to check that ${\textstyle {\frac {d}{dt}}\omega _{t}=d\sigma _{t}}$; then the proof follows from the previous application of Moser's trick to symplectic structures. By the cohomologous hypothesis, ${\displaystyle \omega _{t}-\omega _{0}}$ is an exact form, so that also its derivative ${\textstyle {\frac {d}{dt}}(\omega _{t}-\omega _{0})={\frac {d}{dt}}\omega _{t}}$ is exact for every ${\displaystyle t}$. The actual proof that this can be done in a smooth way, i.e. that ${\textstyle {\frac {d}{dt}}\omega _{t}=d\sigma _{t}}$ for a smooth family of functions ${\displaystyle \sigma _{t}}$, requires some algebraic topology. One option is to prove it by induction, using Mayer-Vietoris sequences;^[3] another is to choose a Riemannian metric and employ Hodge theory.^[1]