In differential geometry, a branch of mathematics, 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 \mathrm{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 \in [0,1]} 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 \in [0,1]} \subset \Omega^k (M) be a family of differential forms on a compact manifold M. If the ODE \frac{d}{dt} \omega_t + \mathcal{L}_{X_t} \omega_t = 0 admits a solution \{ X_t \}_{t \in [0,1]} \subset \mathfrak{X}(M), then there exists a family \{ \psi_t \}_{t \in [0,1]} of diffeomorphisms of M such that \psi_t^*\omega_t = \omega_0 and \psi_0 = \mathrm{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 \in [0,1]} as the flows of a time-dependent vector field, i.e. of a smooth family \{ X_t \}_{t \in [0,1]} of vector fields on M. Using the definition of flow, i.e. \frac{d}{dt} \psi_t = X_t \circ \psi_t for every t \in [0,1], one obtains from the chain rule that \frac{d}{dt} (\psi_t^* \omega_t) = \psi_t^* \Big( \frac{d}{dt} \omega_t + \mathcal{L}_{X_t}\omega_t \Big). By hypothesis, one can always find X_t such that \frac{d}{dt} \omega_t + \mathcal{L}_{X_t} \omega_t = 0, hence their flows \psi_t satisfies \psi_t^* \omega_t = \mathrm{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.[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 = \frac{d}{dt} \alpha_t + \mathcal{L}_{X_t} \alpha_t = d\beta + d (\iota_{X_t} \alpha_t) + \iota_{X_t} (\cancel{d \alpha_t}) = d (\beta + \iota_{X_t} \alpha_t).

However, since \alpha_t is a volume form, i.e. TM \xrightarrow{\cong} \wedge^{n-1} T^*M, \quad X_t \mapsto \iota_{X_t} \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.[3][4]

Let \{ \omega_t \}_{t \in [0,1]} \subset \Omega^2 (M) be a family of symplectic forms on M such that \frac{d}{dt} \omega_t = d \sigma_t, for \{ \sigma_t \}_{t \in [0,1]} \subset \Omega^1 (M). Then there exists a family \{ \psi_t \}_{t \in [0,1]} of diffeomorphisms of M such that \psi_t^*\omega_t = \omega_0 and \psi_0 = \mathrm{id}_M.

Proof

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

0 = \frac{d}{dt} \omega_t + \mathcal{L}_{X_t}\omega_t = d \sigma_t + \iota_{X_t} (\cancel{d\omega_t}) + d (\iota_{X_t} \omega_t) = d (\sigma_t + \iota_{X_t} \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 \xrightarrow{\cong} T^*M, \quad X_t \mapsto \iota_{X_t} \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.[3][4]
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

\omega|_U = \sum_{i=1}^n dx^i \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_{i=i}^n (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_{i=i}^n 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.[3][4]

Application: Moser stability theorem

Moser himself provided an application of his argument for the stability of symplectic structures,[1] which is known now as Moser stability theorem.[3][4]

Let \{ \omega_t \}_{t \in [0,1]} \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 = \mathrm{id}_M.

Proof

It is enough to check that \frac{d}{dt} \omega_t = d \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 \frac{d}{dt} (\omega_t - \omega_0) = \frac{d}{dt} \omega_t is exact for every t. The actual proof that this can be done in a smooth way, i.e. that \frac{d}{dt} \omega_t = d \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;[3] another is to choose a Riemannian metric and employ Hodge theory.[1]

References

  1. ^ Moser, Jürgen (1965). "On the volume elements on a manifold". Transactions of the American Mathematical Society. 120 (2): 286–294. doi:10.1090/S0002-9947-1965-0182927-5. ISSN 0002-9947
  2. ^ Weinstein, Alan (1971-06-01). "Symplectic manifolds and their lagrangian submanifolds". Advances in Mathematics. 6 (3): 329–346. doi:10.1016/0001-8708(71)90020-X. ISSN 0001-8708
  3. ^ McDuff, Dusa & Salamon, Dietmar (2017-06-22). Introduction to Symplectic Topology. Vol. 1. Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879489-9.
  4. ^ Cannas Silva, Ana (2008). Lectures on Symplectic Geometry. Springer. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.
  5. ^ Sternberg, Shlomo (1964). Lectures on Differential Geometry. Prentice Hall. pp. 140–141. ISBN 9780828403160.