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.

This week David Kazhdan gave a series of talks about BV integration (and other things). One of the basic constructions assigns to any odd symplectic manifold a line bundle of “semidensities”, and a differential on it. Actually, the line bundle makes sense for odd almost-symplectic manifolds. The differential comes from realizing that there is a spectral sequence starting with a certain bicomplex, and this bicomplex exists only when the almost-symplectic structure is in fact symplectic.

On a different direction, is there an easy way to see all three “curvature tensors” as arising from the same formula/construction?

That’s a really nice insight. Thanks alot. I was wrong.

With this added perspective and reviewing the literature a little I would comment:

Darboux’s theorem is mentioned very little in Arnold’s classic book, and most uses he makes of closedness are related to invariance of Hamilton’s equations, and integrals of various forms, under canonical transformations(=symplectomorphisms). Most of his objectives in chapters 9 and 10, after introducing “symplectic” revolve around transforming equations of mechanical systems to simpler forms and solving them, in particular he defines integrable systems and constructs action-angle variables for them.

Conceptually Arnold uses Darboux’s theorem by giving much weight to local coordinate calculations, so this leans toward seeing Darboux’s theorem as justifying closedness. A related main argument for Darboux’s theorem is that it is equivalent to existence of a symplectic atlas, i.e. where coordinate changes preserve the standard symplectic structure on local charts. Then we may further observe that all symplectic vector spaces have a symplectic basis (or work with general symplectic vector spaces as charts all along).

But on the other hand we do not require flat atlases in riemannian geometry, so there must be an extra that makes “symplectic” include flatness. Then we may view Darboux’s theorem as the mathematical part/ingredient of the justification of closedness, and flow-invariance as the sociological or physical part.

Regarding this sociological part, it is hammered in wikipedia and other articles that symplectic manifolds have no local invariants, because of Darboux’s theorem. This, as Ben indicated, tells us that people do not see almost symplectic manifolds as interesting.

It is interesting that Darboux proved his theorem in a setting (superficially) very different from global symplectic geometry. And I guess there is a unified presentation of those flatness results (probably from Cartan), as Theo mentions.

Finally I could remark that symplectic was introduced by Weyl, in “The Classical Groups”, as I used to be proud of having figured on my own, but now is easily found in wikipedia. And the first (interesting?) symplectic manifold was defined by Lagrange (1808), as Weinstein tells us in his historical article “Symplectic geometry”.

Then I guess it was Arnold who popularized the terms “symplectic geometry” and “symplectic manifold”.

Re(finally) I find it quite interesting that this kind of discussion is recorded nowadays, ideas that would have been thought and lost repeatedly are longer-lasting now. The beauty of the web, but perhaps a little sad that great past mathematicians have not been so accessible. Though there are still many modes of thinking nowadays which shy of being web-publicized (but there is certainly already a lack of privacy).

That was very interesting, thank you.

Isn’t this “flatness” theorem in the Riemannian case due to Riemann himself? At least that’s the impression I got from reading Spivak’s account of Riemann’s lecture in volume 2 of his bible.

Do you mean “forms” instead of “manifolds” in the title?

“In each case, a vector space carrying the structure induced on the tangent spaces itself as a canonical one of these structures thought of a real manifold; call this a “constant” structure.”

I found this sentence very difficult to parse (maybe due to a typo, or maybe due to the fact that I am not a native English speaker).

There was a typo, but I was also trying too hard to talk about all cases at once. In each case we are talking about a linear algebraic structure on the tangent space; if a real vector space has that structure, thought of as a real manifold its tangent space do as well. Anyways, I’ve rewritten that bit, as it is key.

@Theo

Are notes from Kazhdan’s talk available?

In the Riemannian case — the 2d case does go back to Gauss’

Egregium, of course; when I finally took a course on these things, Niky Kamran used the phrase “Frobenius Integration”, but he’d already outlined the frames/Maurer-Cartan forms by then, so I really don’t understand the historical chronology; but I think that the Riemann Tensor is effectively the only obstruction to local flatness is properly due to Weyl.