Moser's trick explained

In differential geometry, a branch of mathematics, the Moser's trick (or Moser's argument) is a method to relate two differential forms

\alpha _{0} and \alpha _{1} on a smooth manifold by a diffeomorphism \psi \in Diff (M) such that \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

\{ \alpha _{t} \} _{t} and produce an entire isotopy \psi _{t} such that \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

\{ \omega _{t} \} _{t} \subset \Omega ^{k} (M) be a family of differential forms on a compact manifold M . If the ODE (d)/(dt) \omega _{t} + l{L} _{X _{t}} \omega _{t} = 0 admits a solution \{ X _{t} \} _{t} \subset ak{X}(M) , then there exists a family \{ \psi _{t} \} _{t} of diffeomorphisms of M such that \psi ^{*\omega} _{t} = \omega _{0} and \psi _{0} = id _{M} .

In particular, there is a diffeomorphism

\psi : = \psi _{1} such that \psi ^{*\omega} _{1} = \omega _{0} .

Proof

The trick consists in viewing

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

Application to volume forms

Let

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

Proof

One implication holds by the invariance of the integral by diffeomorphisms:

\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

\alpha _{t} : = ( 1- t) \alpha _{0} + t \alpha _{1} . Since \int _{M} (\alpha _{1} - \alpha _{0)} = 0 , the de Rham cohomology class [\alpha _{0} - \alpha _{1]} \in H ^{n} _{dR} (M) vanishes, as a consequence of Poincaré duality and the de Rham theorem. Then \alpha _{1} - \alpha _{0} = d\beta for some \beta \in \Omega ^{n-1} (M) , hence \alpha _{t} = \alpha _{0} + t d\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 \alpha _{t} is a top-degree form:$$0\; =\; \backslash frac\; \backslash alpha\_t\; +\; \backslash mathcal\_\; \backslash alpha\_t\; =\; d\backslash beta\; +\; d\; (\backslash iota\_\; \backslash alpha\_t)\; +\; \backslash iota\_\; (\backslash cancel)\; =\; d\; (\backslash beta\; +\; \backslash iota\_\; \backslash alpha\_t).$$However, since \alpha _{t} is a volume form, i.e. $TM\; \backslash xrightarrow\; \backslash wedge^\; T^*M,\; \backslash quad\; X\_t\; \backslash mapsto\; \backslash iota\_\; \backslash alpha\_t$, given \beta one can always find X _{t} such that \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.

Let

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

Proof

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

$$0\; =\; \backslash frac\; \backslash omega\_t\; +\; \backslash mathcal\_\backslash omega\_t\; =\; d\; \backslash sigma\_t\; +\; \backslash iota\_\; (\backslash cancel)\; +\; d\; (\backslash iota\_\; \backslash omega\_t)\; =\; d\; (\backslash sigma\_t\; +\; \backslash iota\_\; \backslash omega\_t),$$where we used the hypothesis, the Cartan's magic formula, and the fact that

\omega _{t} is closed. However, since \omega _{t} is non-degenerate, i.e. $TM\; \backslash xrightarrow\; T^*M,\; \backslash quad\; X\_t\; \backslash mapsto\; \backslash iota\_\; \backslash omega\_t$, given \sigma _{t} one can always find X _{t} such that \sigma _{t} + \iota _{X _{t}} \omega _{t} = 0 .

Corollary

Given two symplectic structures

\omega _{0} and \omega _{1} on M such that (\omega _{0)} _{x} = (\omega _{1)} _{x} for some point x \in M , there are two neighbourhoods U _{0} and U _{1} of x and a diffeomorphism \phi: U _{0} \to U _{1} such that \phi(x) = x and \phi ^{*\omega} _{1} = \omega _{0} .

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

Darboux theorem for symplectic structures

The Darboux's theorem for symplectic structures states that any point

x in a given symplectic manifold (M,\omega) admits a local coordinate chart (U, x ^{1,\ldots,x} ^{n,y} ^{1,\ldots,y} ^{n)} such that$$\backslash omega|\_U\; =\; \backslash sum\_^n\; dx^i\; \backslash wedge\; dy^i.$$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 (T _{x} M,\omega _{x)} , one can always find local coordinates (\tilde{U}, \tilde{x} ^{1,\ldots,\tilde{x}} ^{n,\tilde{y}} ^{1,\ldots,\tilde{y}} ^{n)} such that \omega _{x} = \sum ^{n} _{i = i} (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 \omega _{0} = \omega | _{\tilde{U} } and \omega _{1} = \sum ^{n} _{i = i} d\tilde{x} ^{i} \wedge d\tilde{y} ^{i} , and consider the new coordinates x ^{i} = \tilde{x} ^{i} \circ \phi, y ^{i} = \tilde{y} ^{i} \circ \phi .

Application: Moser stability theorem

Moser himself provided an application of his argument for the stability of symplectic structures, which is known now as Moser stability theorem.

Let

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

Proof

It is enough to check that $\backslash frac\; \backslash omega\_t\; =\; d\; \backslash sigma\_t$; then the proof follows from the previous application of Moser's trick to symplectic structures. By the cohomologous hypothesis,

\omega _{t} - \omega _{0} is an exact form, so that also its derivative $\backslash frac\; (\backslash omega\_t\; -\; \backslash omega\_0)\; =\; \backslash frac\; \backslash omega\_t$ is exact for every t . The actual proof that this can be done in a smooth way, i.e. that $\backslash frac\; \backslash omega\_t\; =\; d\; \backslash sigma\_t$ for a smooth family of functions \sigma _{t} , requires some algebraic topology. One option is to prove it by induction, using Mayer-Vietoris sequences; another is to choose a Riemannian metric and employ Hodge theory.

References

1. Moser . Jürgen . Jürgen Moser . 1965 . On the volume elements on a manifold . . en . 120 . 2 . 286–294 . 10.1090/S0002-9947-1965-0182927-5 . 0002-9947 . free.
2. Weinstein . Alan . Alan Weinstein . 1971-06-01 . Symplectic manifolds and their lagrangian submanifolds . . en . 6 . 3 . 329–346 . 10.1016/0001-8708(71)90020-X . 0001-8708 . free.
3. Book: McDuff, Dusa . Introduction to Symplectic Topology . Salamon . Dietmar . 2017-06-22 . . 978-0-19-879489-9 . 1 . en . 10.1093/oso/9780198794899.001.0001 . Dusa McDuff . Dietmar Salamon.
4. Book: Cannas Silva, Ana . Lectures on Symplectic Geometry . . 2008 . 978-3-540-42195-5 . en . 10.1007/978-3-540-45330-7 . Ana Cannas da Silva.
5. Book: Sternberg, Shlomo . Lectures on Differential Geometry . . 1964 . 9780828403160 . 140–141 . Shlomo Sternberg.