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cianfa72

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- Application of Darboux theorem to symplectic manifold

Hi,

I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem.

We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form is closed (##d\omega=0##) then from Poincaré lemma there exist locally a 1-form ##\theta## such that ##\omega=d\theta##.

The point I'm missing is why ##\theta \wedge (d\theta)^m## is identically the null form.

Thank you.

p.s. ##(d\theta)^m## should be ##d\theta \wedge d\theta \wedge d\theta \wedge ...## ##m## times

I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem.

We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form is closed (##d\omega=0##) then from Poincaré lemma there exist locally a 1-form ##\theta## such that ##\omega=d\theta##.

The point I'm missing is why ##\theta \wedge (d\theta)^m## is identically the null form.

Thank you.

p.s. ##(d\theta)^m## should be ##d\theta \wedge d\theta \wedge d\theta \wedge ...## ##m## times

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