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Moser's trick

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In differential geometry, a branch of mathematics, the Moser's trick (or Moser's argument) is a method to relate two differential forms and on a smooth manifold by a diffeomorphism such that , 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 such that .

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

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Let be a family of differential forms on a compact manifold . If the ODE admits a solution , then there exists a family of diffeomorphisms of such that and . In particular, there is a diffeomorphism such that .

Proof

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The trick consists in viewing as the flows of a time-dependent vector field, i.e. of a smooth family of vector fields on . Using the definition of flow, i.e. for every , one obtains from the chain rule that By hypothesis, one can always find such that , hence their flows satisfies . In particular, as is compact, this flows exists at .

Application to volume forms

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Let be two volume forms on a compact -dimensional manifold . Then there exists a diffeomorphism of such that if and only if .[1]

Proof

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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 . Since , the de Rham cohomology class vanishes, as a consequence of Poincaré duality and the de Rham theorem. Then for some , hence . By Moser's trick, it is enough to solve the following ODE, where we used the Cartan's magic formula, and the fact that is a top-degree form:However, since is a volume form, i.e. , given one can always find such that .

Application to symplectic structures

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In the context of symplectic geometry, the Moser's trick is often presented in the following form.[3][4]

Let be a family of symplectic forms on such that , for . Then there exists a family of diffeomorphisms of such that and .

Proof

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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 is closed. However, since is non-degenerate, i.e. , given one can always find such that .

Corollary

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Given two symplectic structures and on such that for some point , there are two neighbourhoods and of and a diffeomorphism such that and .[3][4]

This follows by noticing that, by Poincaré lemma, the difference is locally for some ; then, shrinking further the neighbourhoods, the result above applied to the family of symplectic structures yields the diffeomorphism .

Darboux theorem for symplectic structures

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The Darboux's theorem for symplectic structures states that any point in a given symplectic manifold admits a local coordinate chart such thatWhile 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 , one can always find local coordinates such that . Then it is enough to apply the corollary of Moser's trick discussed above to and , and consider the new coordinates .[3][4]

Application: Moser stability theorem

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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 a family of symplectic form on which are cohomologous, i.e. the deRham cohomology class does not depend on . Then there exists a family of diffeomorphisms of such that and .

Proof

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It is enough to check that ; then the proof follows from the previous application of Moser's trick to symplectic structures. By the cohomologous hypothesis, is an exact form, so that also its derivative is exact for every . The actual proof that this can be done in a smooth way, i.e. that for a smooth family of functions , 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

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  1. ^ a b c d 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. ^ a b 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. ^ a b c d e f 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. ^ a b c d e 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.