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Reading about the geometrical theory of systems of first order pdes, I have met a result from symplectic geometry, that is easy to prove, but I am unable to give a reference for it. So my question is:

Where could I find a reference to the following easy result?

Let $W$ be a coisotropic embedded submanifold of a symplectic manifold $(M,\omega)$. Let $T_W M$ be the restriction to $W$ of the tangent bundle of $M$. In the symplectic vector bundle $(T_W M,\omega|_{T_W M})$, let the symplectic complement of $TW$ be denoted by $(TW)^\perp$.

Then $(TW)^\perp$, being a vector subbundle of $TW$ because of the coisotropy of $W$, is completely integrable.

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I don't know an explicit reference to this exact statement, but doesn't this follow directly from Darboux' theorem? After all, $(TW)^\perp\subset TW$ is the null space of the pullback of the closed $2$-form $\omega$ to $W$, and Darboux' theorem gives that this null space is integrable. There are many places you can see this statement, but, for example, if you look at what I called Darboux' Reduction Theorem in my Park City Lectures, you'll see this exact statement. – Robert Bryant Jul 2 '11 at 13:46
Dear Robert Bryant, thanks very much for your attention. If I had wait a moment before to post the question, perhaps I could have remembered of the Darboux' reduction theorem in your lectures or of the equivalent Propositions 5.1.2-3 in Foundations of Mechanics by A&M, but I have preferred to get an occasion to communicate with others on the subject I try to learn. Thank you once again. – Giuseppe Tortorella Jul 2 '11 at 14:54
up vote 1 down vote accepted

This is an exercise in McDuff and Salamon's Introduction to Symplectic Topology. In the second edition, it's exercise 3.29. I don't know if an exercise is a great reference, but I've seen them used before.

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Dear Jordan Watts, thank you for the reference. Sure it works well and, by the way, hints a proof as the one that initially I have thought. – Giuseppe Tortorella Jul 2 '11 at 14:13

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