---
tag: 82YGJ392SJ
kind: statement
role: theorem
name: Moser's trick
msc: [53D05, 53D35]
uses: [CAN490EPRT, 0ZV14XRNF0, PJWK0PKN6Q, 06738GGWWK, 0KZZ70ZHXP, H8ACPTMX08]
about: [CAN490EPRT]
sources:
  - url: https://en.wikipedia.org/wiki/Moser%27s_trick
same_as:
  wikidata: Q3333609
  wikipedia: Moser's trick
---

## Statement

Let $M$ be a closed (compact, without boundary) smooth manifold.
Let $(\omega_t)_{t \in [0, 1]}$ be a family of [[symplectic forms|CAN490EPRT]] on $M$ that depends smoothly on $t$.
Suppose that the [[de Rham class|0ZV14XRNF0]] $[\omega_t] \in H^2_{dR}(M)$ does not depend on $t$.
Then there is a smooth family of [[diffeomorphisms|0KZZ70ZHXP]] $\phi_t \colon M \to M$, $t \in [0, 1]$, with $\phi_0 = \id$ and $\phi_t^* \omega_t = \omega_0$ for all $t$.

## Remarks

Since the class is constant, $\tfrac{d}{dt}\omega_t$ is exact, and one can choose $1$-forms $\sigma_t$ depending smoothly on $t$ with $\tfrac{d}{dt}\omega_t = d\sigma_t$.
Nondegeneracy gives a unique time-dependent vector field $X_t$ with $\iota_{X_t} \omega_t = -\sigma_t$.
Let $\phi_t$ be its [[flow|06738GGWWK]], which exists for all $t \in [0, 1]$ because $M$ is compact.
By Cartan's formula for the [[Lie derivative|PJWK0PKN6Q]],
$$\frac{d}{dt} \phi_t^* \omega_t = \phi_t^* \Big( \mathcal{L}_{X_t} \omega_t + \frac{d}{dt}\omega_t \Big) = \phi_t^* \big( d\iota_{X_t}\omega_t + d\sigma_t \big) = 0.$$
The trick is to look for the diffeomorphisms as the flow of a vector field, which turns a nonlinear problem into a linear equation for $X_t$.

Example of use: two area forms on a closed connected oriented surface, both positive for the orientation and with the same total area, are related by a diffeomorphism.
Apply the theorem to $\omega_t = (1 - t)\omega_0 + t\omega_1$, which stays nondegenerate because both forms are positive for the orientation, so at each point one is a positive multiple of the other.

Compactness is needed for the flow to exist up to time $1$; on open manifolds one adds conditions, as in local versions near a point or a submanifold.

## Explanations

### L1 human

If one moves a symplectic form continuously without changing its cohomology class, nothing really changes.
The theorem gives a family of diffeomorphisms that carries the moving form back to the starting one.
So on a closed manifold, symplectic forms in a fixed class that can be joined by a path are all the same up to diffeomorphism.

### L2 human

The same argument proves Darboux's theorem, the Weinstein neighbourhood theorem for [[Lagrangian submanifolds|H8ACPTMX08]], and Moser's theorem on volume forms.
In each case one joins two structures by a path that agrees on a set, solves $\iota_{X_t}\omega_t = -\sigma_t$, and integrates.
The method shows that symplectic structures have no local invariants, in contrast with Riemannian metrics, which have curvature.