In differential geometry, a branch of mathematics, the Moser's trick (or Moser's argument) is a method to relate two differential forms
, provided that one can find a family of vector fields satisfying a certain ODE.
More generally, the argument holds for a family
and produce an entire isotopy
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.
be a family of differential forms on a compact manifold
as the flows of a time-dependent vector field, i.e. of a smooth family
be two volume forms on a compact
[1]One implication holds by the invariance of the integral by diffeomorphisms:
For the converse, we apply Moser's trick to the family of volume forms
, the de Rham cohomology class
vanishes, as a consequence of Poincaré duality and the de Rham theorem.
By Moser's trick, it is enough to solve the following ODE, where we used the Cartan's magic formula, and the fact that
In the context of symplectic geometry, the Moser's trick is often presented in the following form.
be a family of symplectic forms on
.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
[3][4]This follows by noticing that, by Poincaré lemma, the difference
; then, shrinking further the neighbourhoods, the result above applied to the family
of symplectic structures yields the diffeomorphism
The Darboux's theorem for symplectic structures states that any point
admits a local coordinate chart
While the original proof by Darboux required a more general statement for 1-forms,[5] Moser's trick provides a straightforward proof.
, one can always find local coordinates
Then it is enough to apply the corollary of Moser's trick discussed above to
Moser himself provided an application of his argument for the stability of symplectic structures,[1] which is known now as Moser stability theorem.
a family of symplectic form on
; then the proof follows from the previous application of Moser's trick to symplectic structures.
One option is to prove it by induction, using Mayer-Vietoris sequences;[3] another is to choose a Riemannian metric and employ Hodge theory.