# A Canonical Form Theorem for $n$-forms?

I have been working with what I call measured-manifolds'', i.e., an $n$-dimensional smooth manifold $M$ together with a non-vanishing $n$-form $\omega$ on $M$. At a certain point I realized that a lot of things would be greatly simplified if I knew that at each point $p$ of $M$ I could find a "canonical coordinate system" $(x_1,\ldots,x_n)$, meaning that in these coordinates $\omega = dx_1\wedge\ldots,\wedge\ dx_n$. I couldn't recall ever seeing such a result, and I spent about a day looking in the obvious textbooks, searching with Google, and asking friends if they had heard of such an existence theorem. Then suddenly I realized that it was obvious !
If $(y_1,\ldots,y_n)$ is any coordinate system centered at $p$ and $\omega = \rho(y_1,\ldots , y_n) dy_1\wedge \ldots\wedge dy_n$ then define
$x_1(y_1,\ldots , y_n) := \int_0^{y_1} \rho(t,y_2,\ldots,y_n) dt$ and let $x_i := y_i$ for $i = 2,3,\ldots,n$. Then $dx_1 = \rho(y) dy_1 + \sum_{j=2}^n a_j(y) dy_j$ so $dx_1 \wedge(dy_2 \wedge \ldots \wedge dy_n) = \omega$ and $(x_1,\ldots,x_n)$ are indeed canonical coordinates. Now I am not so dumb as to imagine I am the first person to observe this triviality, and I would even be willing to bet that E. Cartan knew it way back when, and that a few zillion others have noticed it and used it since. So my question is, where can I find the best reference for the existence of canonical coordinates for $n$-forms.

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This would be the «Darboux's theorem» for $n$-forms :-) – Mariano Suárez-Alvarez May 30 '13 at 5:37
@Mariano. Right, that is the way I think of it, and it was Darboux Theorem that made me think it might be true. – Dick Palais May 30 '13 at 5:51
Maybe I'm missing someting, but it seems to me that this is the content of the section "No local structure" in this wikipedia article: en.wikipedia.org/wiki/Volume_form The reference given is Kobayashi's book "Transformation groups in differential geometry" – Francesco Polizzi May 30 '13 at 9:22
Moser's trick is a fancier, but less elementary, proof. I think Abraham and Marsden show how to use Moser's trick both for symplectic forms and also for volume forms. – Ben McKay May 30 '13 at 12:12
@Francesco YES ! I don't think you are missing anything---that is essentially the same proof as mine. I'll have to get Kobayashi's book from the library and have a look. Many thanks for this clue. – Dick Palais May 30 '13 at 14:08

## 1 Answer

This probably will not actually answer your question about earliest references, but it's too long for a comment.

You are right that this 'normal form' was known to É. Cartan, and I'll be that it was known to Lie as well earlier than that. However, I am not sure that you'll be able to find it explicitly stated as a result in Cartan's works. More likely, he would have regarded it as obvious, and for the reason that you gave: The proof is one line.

The reason I have for claiming this is probably overkill, but here goes: Cartan classified the 'infinite transitive primitive Lie groups' in a series of early papers starting around 1904. ('Infinite' means 'not finite dimensional, 'transitive' has its usual meaning, 'primitive' means 'leaves no nontrivial foliation invariant', and 'Lie group' means 'a transformation (pseudo-)group defined as the solutions of some system of PDE'.) His list (all of which were known to Lie) goes like:

1. All transformations (the PDE system is empty)
2. Transformations that preserve a volume form up to a constant multiple (Jacobian determinant is constant)
3. Transformations that preserve a volume form (Jacobian determinant is 1)
4. Transformations that preserve a symplectic form on a domain, up to a constant multiple. (Only happens in even dimensions. THe PDE system is the obvious one.)
5. Transformations that preserve a symplectic form on the domain. (Only happens in even dimensions. THe PDE system is the obvious one.)
6. Transformations that preserve a contact structure. (Only happens in odd dimensions. The PDE system is the obvious one.)

Cartan implicitly assumes that there is (locally) only one kind of volume form when stating this result for Items 2 and 3, just as he implicitly assumes Darboux' Theorem in the statement of Items 4 and 5. From this, I conclude that he knew then that there was, up to diffeomorphism, only one kind of volume form (i.e., form of top degree). However, I don't remember him actually stating this 'normal form' for volume forms explicitly (and I'm traveling right now and don't have access to the works of Cartan, so I can't check); it's more likely that he just assumed that 'everybody knows this'.

Remark: By the way, if you look at the above list, you probably will object that it doesn't have some other obvious items, such as the biholomorphic transformations in a given (even) dimension and the transformations that preserve a volume form up to a sign (i.e., the transformations for which the absolute value of the Jacobian determinant is 1 , etc. Cartan's result needs some interpretation. While he didn't say this explicitly, he generally assumed in those papers that it didn't matter whether one was working with real or complex variables and he didn't bother to distinguish groups that had the same Lie algebra of 'infinitesimal transformations'. When Singer and Sternberg revisited this work in the 1960s, they did kind of clean this up and extend the list to distinguish real and complex variables, so that the list gets a bit longer, but only by the obvious additions that this entails.

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Sophus Lie, in volume 5 of his collected works, gives a list of all Lie algebra actions on the plane, up to local isomorphism, and case III A 2 is the action of the divergence free vector fields. So at least Lie was aware that all area forms on surfaces are locally isomorphic. – Ben McKay May 30 '13 at 12:11
@Robert, you are trying to make us believe that you ever leave home without the complete works of Cartan?! :-) – Mariano Suárez-Alvarez May 30 '13 at 14:13
No problem, Mariano: NUMDAM has the two relevant papers of Cartan available on the internet, so we are always in the presence of the master: numdam.org/item?id=ASENS_1904_3_21__153_0, numdam.org/item?id=ASENS_1905_3_22__219_0 – Ben McKay May 30 '13 at 17:25
@Ben & Mariano: Indeed, I have downloaded several of Cartan's papers from NUMDAM for easy reference, but not all of Cartan's papers are there and not all of his books. I haven't had time to check through the papers Ben is mentioning, but I am pretty sure that I remember as much as I wrote above. What I am not sure about is whether Cartan might have mentioned the normal form for closed decomposable $n$-forms in his "Leçons sur les invariants intégraux" (ca. 1922, I think), which would have been a natural place for him to do this, and I was expecting to check that before attempting an answer. – Robert Bryant May 30 '13 at 18:55