Moser's trick

Let ${\displaystyle \{\omega _{t}\}_{t\in [0,1]}\subset \Omega ^{k}(M)}$  be a family of differential forms on a compact manifold ${\displaystyle M}$ . If the ODE ${\displaystyle {\frac {d}{dt}}\omega _{t}+{\mathcal {L}}_{X_{t}}\omega _{t}=0}$  admits a solution ${\displaystyle \{X_{t}\}_{t\in [0,1]}\subset {\mathfrak {X}}(M)}$ , 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 particular, there is a diffeomorphism ${\displaystyle \psi :=\psi _{1}}$  such that ${\displaystyle \psi ^{*}\omega _{1}=\omega _{0}}$ .

The trick consists in viewing ${\displaystyle \{\psi _{t}\}_{t\in [0,1]}}$  as the flows of a time-dependent vector field, i.e. of a smooth family ${\displaystyle \{X_{t}\}_{t\in [0,1]}}$  of vector fields on ${\displaystyle M}$ . Using the definition of flow, i.e. ${\displaystyle {\frac {d}{dt}}\psi _{t}=X_{t}\circ \psi _{t}}$  for every ${\displaystyle t\in [0,1]}$ , one obtains from the chain rule that ${\displaystyle {\frac {d}{dt}}(\psi _{t}^{*}\omega _{t})=\psi _{t}^{*}{\Big (}{\frac {d}{dt}}\omega _{t}+{\mathcal {L}}_{X_{t}}\omega _{t}{\Big )}.}$  By hypothesis, one can always find ${\displaystyle X_{t}}$  such that ${\displaystyle {\frac {d}{dt}}\omega _{t}+{\mathcal {L}}_{X_{t}}\omega _{t}=0}$ , hence their flows ${\displaystyle \psi _{t}}$  satisfies ${\displaystyle \psi _{t}^{*}\omega _{t}=\mathrm {const} =\psi _{0}^{*}\omega _{0}=\omega _{0}}$ . In particular, as ${\displaystyle M}$  is compact, this flows exists at ${\displaystyle t=1}$ .

Let ${\displaystyle \alpha _{0},\alpha _{1}}$  be two volume forms on a compact ${\displaystyle n}$ -dimensional manifold ${\displaystyle M}$ . Then there exists a diffeomorphism ${\displaystyle \psi }$  of ${\displaystyle M}$  such that ${\displaystyle \psi ^{*}\alpha _{1}=\alpha _{0}}$  if and only if ${\displaystyle \int _{M}\alpha _{0}=\int _{M}\alpha _{1}}$ .^[1]

One implication holds by the invariance of the integral by diffeomorphisms: ${\displaystyle \int _{M}\alpha _{0}=\int _{M}\psi ^{*}\alpha _{1}=\int _{\psi (M)}\alpha _{1}=\int _{M}\alpha _{1}}$ .

For the converse, we apply Moser's trick to the family of volume forms ${\displaystyle \alpha _{t}:=(1-t)\alpha _{0}+t\alpha _{1}}$ . Since ${\displaystyle \int _{M}(\alpha _{1}-\alpha _{0})=0}$ , the de Rham cohomology class ${\displaystyle [\alpha _{0}-\alpha _{1}]\in H_{dR}^{n}(M)}$  vanishes, as a consequence of Poincaré duality and the de Rham theorem. Then ${\displaystyle \alpha _{1}-\alpha _{0}=d\beta }$  for some ${\displaystyle \beta \in \Omega ^{n-1}(M)}$ , hence ${\displaystyle \alpha _{t}=\alpha _{0}+td\beta }$ . By Moser's trick, it is enough to solve the following ODE, where we used the Cartan's magic formula, and the fact that ${\displaystyle \alpha _{t}}$  is a top-degree form:$${\displaystyle 0={\frac {d}{dt}}\alpha _{t}+{\mathcal {L}}_{X_{t}}\alpha _{t}=d\beta +d(\iota _{X_{t}}\alpha _{t})+\iota _{X_{t}}({\cancel {d\alpha _{t}}})=d(\beta +\iota _{X_{t}}\alpha _{t}).}$$ However, since ${\displaystyle \alpha _{t}}$  is a volume form, i.e. ${\textstyle TM\xrightarrow {\cong } \wedge ^{n-1}T^{*}M,\quad X_{t}\mapsto \iota _{X_{t}}\alpha _{t}}$ , given ${\displaystyle \beta }$  one can always find ${\displaystyle X_{t}}$  such that ${\displaystyle \beta +\iota _{X_{t}}\alpha _{t}=0}$ .

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]