Why should a symplectic form be closed?

Question

Thanks for reading my question. I'm wonder why a symplectic form should be closed. I found many different answers in the internet, but it sounds like a technical requirement (if we omit this requisit, we obtain almost symplectic structures, insteresting as well). Why do yo think? I just want to have a fresh perspective. Thank you in advance.

Answer 1

There are many reasons why we might want a symplectic form to be closed by definition. Here are a few:

• A closed $2$-form represents a cohomology class in $H^2(M; \Bbb R)$.

• When $\omega$ is closed, we get a one-to-one correspondence between $1$-parameter groups of symplectomorphisms and symplectic vector fields. We also get Hamiltonian diffeomorphisms from this, which turn out to be interesting for symplectic geometers.

• The local rigidity theorems for symplectic manifolds, such as Darboux's theorem, Moser's stability theorem, Weinstein's tubular neighborhood theorem, and so on rely on the closedness of the symplectic form. I think some such results hold more generally, for example Gotay's coisotropic neighborhood theorem holds for presymplectic manifolds if I recall correctly. In any case, these results are what give symplectic geometry its global nature.

There's other reasons based on physical considerations as well. In short, closedness of a symplectic form is a rather mild condition that gives rise to a lot of nice structure we wouldn't otherwise have.