Moser's trick

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, which exists for all $t \in [0, 1]$ because $M$ is compact. By Cartan's formula for the Lie derivative, $$\frac{d}{dt} \phi_t^* \omega_t = \phi_t^* \Big( \mathcal{L}_{X_t} \omega_t + \frac{d}{dt}\omega_t \Big) = \phi_t^* \big( d\iota_{X_t}\omega_t + d\sigma_t \big) = 0.$$ 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.

Going deeper

For a student

The same argument proves Darboux's theorem, the Weinstein neighbourhood theorem for Lagrangian submanifolds, 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.