Moser's trick

In differential geometry, a branch of mathematics, the Moser's trick (or Moser's argument) is a method to relate two differential forms ${\displaystyle \alpha _{0}}$ and ${\displaystyle \alpha _{1}}$ on a smooth manifold by a diffeomorphism ${\displaystyle \psi \in \mathrm {Diff} (M)}$ such that ${\displaystyle \psi ^{*}\alpha _{1}=\alpha _{0}}$, provided that one can find a family of vector fields satisfying a certain ODE.

More generally, the argument holds for a family ${\displaystyle \{\alpha _{t}\}_{t\in [0,1]}}$ and produce an entire isotopy ${\displaystyle \psi _{t}}$ such that ${\displaystyle \psi _{t}^{*}\alpha _{t}=\alpha _{0}}$.

It was originally given by Jürgen Moser in 1965 to check when two volume forms are equivalent,^[1] but its main applications are in symplectic geometry. It is the standard argument for the modern proof of Darboux's theorem, as well as for the proof of Darboux-Weinstein theorem^[2] and other normal form results.^[2][3][4]

General statement

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 ${\displaystyle \psi ^{*}\omega _{1}=\omega _{0}}$.

Proof

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 ${\displaystyle t=1}$.

Application to volume forms

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]

Proof

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:

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 ${\displaystyle \beta +\iota _{X_{t}}\alpha _{t}=0}$.

Application to symplectic structures

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 ${\displaystyle \psi _{0}=\mathrm {id} _{M}}$.

Proof

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

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 ${\displaystyle \sigma _{t}+\iota _{X_{t}}\omega _{t}=0}$.

Corollary

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 ${\displaystyle \phi :=\psi _{1}}$.

Darboux theorem for symplectic structures

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

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]

Application: Moser stability theorem

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 ${\displaystyle \psi _{0}=\mathrm {id} _{M}}$.

Proof

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]

References