Moser's trick

Intuition [Beginner]

Moser's trick is a method for proving that two symplectic forms on the same manifold are the same up to a diffeomorphism. The setup: you have a manifold $M$ and a smooth path of symplectic forms ${\omega }_t$, $t\in [0,1]$, all in the same de Rham cohomology class. The conclusion: there is an isotopy ${\psi }_t$ of $M$ (a smooth path of diffeomorphisms with ${\psi }_0=\mathrm{id}$) that pulls ${\omega }_t$ back to ${\omega }_0$. At time $t=1$ this gives a single diffeomorphism intertwining ${\omega }_0$ and ${\omega }_1$.

The technique is purely calculational: pick a primitive of the time-derivative ${\dot{\omega }}_t$, use non-degeneracy of ${\omega }_t$ to convert that primitive into a vector field $X_t$, integrate $X_t$ to get ${\psi }_t$. The proof that ${\psi }_t^{\ast }{\omega }_t$ is constant in $t$ is one application of Cartan's formula plus the cohomology assumption.

Moser introduced the method in 1965 for volume forms; Weinstein extended it to symplectic forms and to neighbourhoods of Lagrangian submanifolds. Almost every standard local-form result in symplectic geometry — Darboux, Weinstein neighbourhood, equivariant Darboux, the smooth structure of regular symplectic reduction — is one Moser argument away.

Visual [Beginner]

A manifold $M$ with two symplectic forms drawn schematically as two patterned tilings, connected by a path of intermediate forms. An arrow labelled ${\psi }_1$ connects them, indicating the diffeomorphism the trick produces.

The key picture is the flow of the time-dependent vector field along the path of forms.

Worked example [Beginner]

Take $M={\mathbb{R}}^2$ with two symplectic forms: the standard ${\omega }_0=dx\wedge dy$ and the perturbed ${\omega }_1=(1+e^{-x^2-y^2})dx\wedge dy$. Both are exact on ${\mathbb{R}}^2$ since de Rham cohomology of ${\mathbb{R}}^2$ is zero in positive degree, so the cohomology hypothesis is automatic.

The path ${\omega }_t=(1-t){\omega }_0+t{\omega }_1=(1+t{e}^{-x^2-y^2})dx\wedge dy$ stays positive-definite and closed for every $t$. Its time-derivative ${\dot{\omega }}_t=e^{-x^2-y^2}dx\wedge dy$ has a primitive $\alpha$ that goes to zero rapidly at infinity, so the Moser vector field $X_t$ is compactly-essentially-supported and the flow exists for all $t$.

The output: a smooth diffeomorphism ${\psi }_1$ of ${\mathbb{R}}^2$ that pulls ${\omega }_1$ back to ${\omega }_0$. The diffeomorphism stretches and shrinks the plane to even out the local symplectic density, but you never need to compute it explicitly — the Cartan-formula calculation guarantees its existence.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ be a smooth manifold and ${\omega }_t$, $t\in [0,1]$, a smooth one-parameter family of symplectic forms (each closed, non-degenerate; smooth dependence on $t$). Suppose ${\omega }_t-{\omega }_0$ is exact for all $t$ — equivalently, ${\dot{\omega }}_t=d{\alpha }_t$ for some smooth one-parameter family of one-forms ${\alpha }_t$.

The Moser construction is the time-dependent vector field $X_t$ on $M$ defined by

$${\iota }_{X_t}{\omega }_t=-{\alpha }_t,$$

uniquely solvable for $X_t$ at each point and each time $t$ because ${\omega }_t$ is non-degenerate. The corresponding Moser flow is the isotopy ${\psi }_t:M\to M$ obtained by integrating $X_t$ from the identity at $t=0$. (When $M$ is non-compact the flow may not be defined for all $t$; the standard hypothesis is that $X_t$ is compactly supported, or that one works on a relatively compact subset.)

Key theorem with proof [Intermediate+]

Theorem (Moser stability). Suppose ${\omega }_t$ is a smooth path of symplectic forms on $M$ with ${\omega }_t-{\omega }_0$ exact for all $t$. Let $X_t$ be the Moser vector field and ${\psi }_t$ its flow (assumed defined for $t\in [0,1]$). Then $\psi_t^ \omega_t = \omega_0$forevery$t$.*

Proof. Differentiate ${\psi }_t^{\ast }{\omega }_t$ in $t$:

$$\frac{d}{dt}({\psi }_t^{\ast }{\omega }_t)={\psi }_t^{\ast }({\mathcal{L}}_{X_t}{\omega }_t+{\dot{\omega }}_t).$$

Cartan's formula gives ${\mathcal{L}}_{X_t}{\omega }_t=d{\iota }_{X_t}{\omega }_t+{\iota }_{X_t}d{\omega }_t$. Closedness $d{\omega }_t=0$ kills the second term. The first term, by the defining equation of $X_t$, equals $d(-{\alpha }_t)=-d{\alpha }_t=-{\dot{\omega }}_t$. Hence

$$\frac{d}{dt}({\psi }_t^{\ast }{\omega }_t)={\psi }_t^{\ast }(-{\dot{\omega }}_t+{\dot{\omega }}_t)=0,$$

so ${\psi }_t^{\ast }{\omega }_t$ is constant in $t$ and equals its $t=0$ value ${\psi }_0^{\ast }{\omega }_0={\omega }_0$. $\square$

Bridge. The Moser computation here builds toward 05.05.02 (Weinstein Lagrangian neighbourhood theorem), where the same path-method appears again in a tubular neighbourhood: the symplectic form $\omega$ pulled back from $M$ and the canonical form $-d\lambda$ on $T^{\ast }L$ are connected by a linear path of forms vanishing-on-$L$ on a neighbourhood, and Moser's trick produces the symplectomorphism. The construction is exactly the smooth-structure argument behind regular symplectic reduction 05.04.02: different transversal slices give Moser-isotopic reduced symplectic forms. Putting these together, the foundational reason every standard local-form theorem in symplectic geometry holds is exactly that the Moser vector field $X_t$ defined by ${\iota }_{X_t}{\omega }_t=-{\alpha }_t$ identifies primitives with diffeomorphisms — the cohomology hypothesis is dual to the diffeomorphism conclusion.

Exercises [Intermediate+]

Advanced results [Master]

The Moser argument is the prototype of a deformation method in geometry: a small set of cohomological hypotheses combined with non-degeneracy yields a diffeomorphism intertwining the data. The same structure recurs in many guises throughout symplectic and contact geometry.

Standard applications.

• Darboux's theorem. Every symplectic manifold is locally $({\mathbb{R}}^{2n},\sum d{q}^i\wedge d{p}_i)$. Moser produces the chart change as above.
• Weinstein neighbourhood theorem. A neighbourhood of a closed Lagrangian $L$ in $(M,\omega )$ is symplectomorphic to a neighbourhood of the zero section in $T^{\ast }L$ with its canonical form. The path connects $\omega$ pulled back to a tubular neighbourhood with $-d\lambda$; both vanish on $L$, so the relative Poincaré lemma gives a primitive vanishing on $L$, hence the Moser vector field vanishes on $L$ and the flow fixes $L$ pointwise.
• Equivariant Darboux. When a compact group $G$ acts on $M$ preserving $\omega$, choose a $G$-invariant primitive ${\alpha }_t$ (by averaging) and the resulting Moser flow is $G$-equivariant.
• Symplectic stability of fibre bundles (Thurston-style symplectic forms on fibre bundles): two Thurston-compatible forms in the same cohomology class are connected by a Moser isotopy.
• Smooth structure on regular symplectic reduction (Marsden-Weinstein): different transversal slices of a moment-map level set give Moser-isotopic reduced forms.
• Generic Hamiltonian non-degeneracy (Floer-theoretic transversality): paths of Hamiltonians in the right cohomology class are Moser-isotopic up to the transversality fix.

Failure modes.

• No cohomology hypothesis. If ${\omega }_1-{\omega }_0$ is not exact, ${\dot{\omega }}_t$ has no global primitive and $X_t$ cannot be constructed globally. Two volume forms with different total integrals on a closed manifold are not Moser-equivalent.
• Non-degeneracy fails along the path. If some ${\omega }_t$ for $t\in (0,1)$ is degenerate, the inversion ${\omega }_t^{-1}$ is undefined and the Moser field has a singularity. Pre-emptive convexity-style arguments (path of forms staying in the symplectic locus) handle this.
• Compactness of support / global flow. On non-compact $M$ the flow may not exist for $t\in [0,1]$; one solves on relatively compact subsets or chooses ${\alpha }_t$ with controlled support.

Synthesis. Moser's trick generalises the classical Poincaré-lemma-plus-flow strategy from one-forms to symplectic forms: the cohomology hypothesis is exactly the existence of the primitive, and the non-degeneracy hypothesis is exactly what converts the primitive into a vector field. Read in the opposite direction, the Moser construction is dual to the period map: the cohomology class $[{\omega }_t]\in H^2(M;\mathbb{R})$ is the obstruction to Moser-equivalence, and putting these together one sees that the moduli space of symplectic forms on a fixed closed manifold up to symplectomorphism is locally the same as the affine space of cohomology classes — exactly the statement that "symplectic geometry has no local invariants beyond dimension." The central insight is that the bridge between the analytic equation ${\iota }_{X_t}{\omega }_t=-{\alpha }_t$ and the geometric conclusion ${\psi }_1^{\ast }{\omega }_1={\omega }_0$ is the foundational reason every standard local-form theorem in symplectic geometry follows from a single calculational template — and that template identifies primitives with diffeomorphisms.

Full proof set [Master]

Lemma (Moser identity). For a smooth path of differential forms ${\omega }_t$ and a smooth path of vector fields $X_t$ with flow ${\psi }_t$,

$$\frac{d}{dt}({\psi }_t^{\ast }{\omega }_t)={\psi }_t^{\ast }({\mathcal{L}}_{X_t}{\omega }_t+{\dot{\omega }}_t).$$

Proof. Direct from the chain rule: $\frac{d}{dt}{\psi }_t^{\ast }{\omega }_t{|}_{t=t_0}={\psi }_{t_0}^{\ast }{\dot{\omega }}_{t_0}+\frac{d}{ds}{|}_{s=0}{\psi }_{t_0+s}^{\ast }{\omega }_{t_0}$. The second piece is ${\psi }_{t_0}^{\ast }{\mathcal{L}}_{X_{t_0}}{\omega }_{t_0}$ by definition of Lie derivative along a time-dependent flow. $\square$

Lemma (relative Poincaré). Let $L\subset M$ be a closed embedded submanifold and let $U$ be a tubular neighbourhood of $L$. If $\eta$ is a closed $k$-form on $U$ vanishing along $L$, then $\eta =d\beta$ for some $(k-1)$-form $\beta$ on $U$ vanishing along $L$.

Proof sketch. Tubular neighbourhood structure gives $U\cong$ disk subbundle of $\nu (L)$. Use the standard fibrewise Poincaré-lemma cone construction: $\beta (x)={\int }_0^1{\iota }_{{\partial }_r}({\rho }_s^{\ast }\eta )(x)ds$, where ${\rho }_s$ is fibrewise scaling by $s$. Vanishing along $L$ follows from the integrand vanishing at $s=0$. $\square$

Theorem (Darboux, via Moser). Around any point of a symplectic manifold, there exist coordinates $(q^1,p_1,\dots ,q^n,p_n)$ in which $\omega =\sum d{q}^i\wedge d{p}_i$.

Proof. See Exercise 2 above for the path argument. The relative Poincaré lemma is what gives the primitive $\alpha$ vanishing at $p$, and the Moser flow integrates to the chart change. $\square$

Connections [Master]

• Symplectic manifold 05.01.02. Moser's trick is the standard tool for proving local and semi-local rigidity of symplectic structures — Darboux, Weinstein, equivariant Darboux all reduce to the same calculation.

• Darboux's theorem 05.01.04. The first major application; the modern proof of Darboux is exactly Moser's argument applied to the linear path between the standard form and the given form on a chart.

• Weinstein Lagrangian neighbourhood 05.05.02. The signature application beyond Darboux; Moser's trick produces the symplectomorphism between the tubular neighbourhood of $L$ in $(M,\omega )$ and the neighbourhood of the zero section in $T^{\ast }L$.

• Symplectic reduction 05.04.02. The smooth structure of the regular reduced manifold is established by a Moser argument comparing different transversal slices.

• Hamiltonian vector field 05.02.01. Two Moser primitives differ by a closed one-form; on simply-connected $M$ this difference is exact and induces a Hamiltonian-isotopy correction between the two Moser flows.

Historical & philosophical context [Master]

Jürgen Moser introduced the path method in 1965 On the volume elements on a manifold (Trans. AMS 120) ^{[Moser 1965]} for the volume-form setting: he proved that on a closed orientable manifold, any two cohomologous volume forms are connected by a diffeomorphism. The motivation was a question in dynamical systems about the genericity of measure-preserving maps. The construction of the Moser vector field and the Cartan-formula calculation are due to Moser; the only ingredients are exterior calculus and the existence of flows.

Alan Weinstein's 1971 paper Symplectic manifolds and their Lagrangian submanifolds (Adv. Math. 6) ^{[Weinstein 1971]} extended Moser's argument to the symplectic setting and to neighbourhoods of Lagrangian submanifolds. Weinstein's framing — "Lagrangian submanifolds are the right submanifolds in symplectic geometry" — turned the path method from a technical tool into the conceptual foundation of an entire programme.

Cannas da Silva's Lectures on Symplectic Geometry (Springer LNM 1764, 2001/2008) ^{[Cannas da Silva]} gives the standard pedagogical exposition of the trick along with its main applications. McDuff-Salamon Introduction to Symplectic Topology ^{[McDuff-Salamon]} develops the same material with more emphasis on global rigidity and the role of Moser stability in pseudoholomorphic-curve theory.

The path method has since been extended far beyond its original setting: equivariant Moser arguments in the presence of compact group actions, parametric Moser arguments for families of symplectic structures, Moser-style proofs in contact geometry (Gray's theorem), and infinite-dimensional analogues used in Floer theory. Each generalisation uses the same template — primitive of the time-derivative, vector field via non-degeneracy, Cartan-formula closure of the calculation.

Bibliography [Master]

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