In a comment on my last post, plm suggests that my condition about the rules of turning energy functions into flows be itself time invariant is the only way to justify requiring that symplectic forms be closed.
While I agree that this is a good way of thinking about what closeness is supposed to mean, and maybe the best, I would dispute that it’s the only one. It’s a very reasonable condition from the pure math side as a kind of “flatness.”
I think it’s a fairly commonly accepted principle there is a triad of the most basic “kinds of structures” on a smooth manifold (of course, there are lots of variants of these, so there there is no claim of completeness here; I don’t want to be attacked by angry contact or Finsler geometers): Riemannian, complex, and symplectic. In each case, this structure consists of a structure on the tangent spaces (a symmetric bilinear form, a complex vector space structure or an antisymmetric bilinear form) which varies smoothly; of course, since both complex and symplectic have extra conditions, I really should say almost complex and almost symplectic.
A vector space which carries a non-degenerate antisymmetric bilinear form thought of as a manifold is a almost symplectic manifold in the obvious way: identify the tangent spaces with the vector space itself and think of the anti-symmetric bilinear form as a 2-form. Call this a “constant” structure. You do the same thing in the Riemannian and complex cases.
Definition. Call a Riemannian/almost complex/almost symplectic manifold flat if it is locally isomorphic to a constant structure on a vector space.
The remarkable theorem about flat structures is that there is a tensor exactly obstructing the flatness of one of these structures. Let the “curvature tensor” of such a structure be the usual Riemann curvature tensor in the Riemannian case, the Nijenhuis tensor in the almost complex case and the exterior derivative of the associated 2-form in the almost symplectic case.
Theorem. A Riemannian/almost complex/almost symplectic manifold is “flat” if and only if its “curvature” tensor vanishes.
In the Riemannian case, I’m not even sure who ascribe this theorem to; it’s so ingrained in people’s consciousness that I’m not sure there’s a name for it. In the complex case, this is the Newlander–Nirenberg theorem, and in symplectic geometry, this is called the Darboux theorem. Of course, the terminology is chosen to make this sound “obvious” but it’s not. One way is; it’s easy to see that these quantities all vanish on a “constant” structure, but the other way is quite difficult, as one has to use some differential equations to cook up the right coordinates.
Flat Riemannian manifolds are called just that; flat almost complex and almost symplectic manifolds are called “complex” and “symplectic” with no further modifiers. I would say this is for historical reasons; the importance of non-flat Riemmannian manifolds was so important, so early in the theory (even before general relativity) that there was no reason to think of the flat case as the basic one and the non-flat as some weird variation on it, whereas I think it took a lot longer for people to recognize that almost complex manifolds had interesting uses and I honestly know of no reason to think about almost symplectic ones, but someone can correct me.
Thus, I think the Darboux theorem is a pretty good justifier of the closed definition, but the ultimate justification is that it seems to be the right level of generality; it covers the examples that interest us and any weakening of it (to, say, exact) throws out lots of examples we like. It may be that 50 years from now, people will think almost symplectic manifolds are great, and we were silly not to have studied them all this time. We’ll just have to see.