When does Darboux’s theorem on symplectic manifolds work globally

Question

I'm a Physics PhD student working on Hamiltonian systems in the context of General Relativity. Recently, I was working on adding a perturbation to a dynamical system with known solutions.

Without going to much into detail, we are working on a 2n-dimensional manifold where the new symplectic 2-form after the perturbation takes the form

\begin{equation}
\Omega_{AB}=\Omega^0_{AB}+\epsilon \Omega^1_{AB}
\end{equation}

where $\epsilon$ is supposed to be a small number and $\Omega^0=dp_\alpha \wedge dz^\alpha$ is the symplectic 2-form of the unperturbed system (I'm using the canonical coordiantes $(z^\alpha,p_\alpha)$ with $\alpha=1,2\dots n$. Since $\Omega^1$ has to be closed and nondegenerate like $\Omega$ then there's a set of local coordinates $(\bar{z}^\alpha,\bar{p})_\alpha$ where it takes the form

\begin{equation}
\Omega^1 = d\bar{p}_\alpha \wedge d\bar{z}^\alpha
\end{equation}

Which means that we can use the diffeomorphism connecting the two sets of variables to write the perturbation as the pullback of $\Omega^0$ like

\begin{equation}
\Omega^1 = \mathcal{L}_X \Omega^0
\end{equation}
where $X$ is the vector field that generates the diffeomorphism connecting the bared and unbared coordinates.

Now, this whole thing works due to Darboux's theorem which guarantees that there are coordinates where $\Omega^1$ takes the canonical form, at least locally. The question is: Are there conditions for this to work globally? I think that there are probably some topological conditions on the manifold but I wouldn't know how to approach the question. Any directions will be welcomed.

Edit: Based on the comments I want to clarify what the goal is. The general question is what are the conditions for Darboux Theorem to hold globally. That is, what conditions need to be satisfied for the perturbation to be $\Omega^1 =d\bar{p}_\alpha\wedge d\bar{z}^\alpha$ globally. In particular, I want to know what conditions have to be satisfied for the expression $\Omega^1 = \mathcal{L}_X \Omega^0$ to work globally. But I think both concerns are the same question.

Answer 1

I am still unsure what are you after; here are some relevant results:

1. Suppose that $(M,\omega_0)$ is a compact symplectic manifold. Consider a smooth perturbation of $\omega_0$, i.e. a smooth family of symplectic forms $\omega_t$, $t\in [0,T]$. One question to ask is if there is a smooth family of diffeomorphisms $f_t: M\to M, t\in [0,T]$, such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$. There is an obvious topological obstruction to the existence of such a family, namely, the cohomology classes $[\omega_t]\in H^2(M, {\mathbb R})$ have to be constant (i.e. the same as the one given by $\omega_0$). In other words, for each $t$ there should be a 1-form $\alpha_t\in \Omega^1(M)$ such that $\omega_t- \omega_0= d\alpha_t$. Now, the relevant theorem is known as Moser's Stability Theorem:

Theorem 1. Assume that in the above setting $[\omega_t]=[\omega_0]$ for all $t$. Then indeed, there is a smooth family of diffeomorphisms $f_t: M\to M, t\in [0,T]$, such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$.

2. Moser's theorem generalizes to noncompact manifolds, for instance:

Theorem 2. Suppose that $(M,\omega_t)$ is a symplectic manifold and $\omega_t$ as above is:

a. A compactly supported deformation of $\omega_0$ in the sense that:

There is a compact $K\subset M$ such that $\omega_0=\omega_t$ outside of $K$ for all $t\in [0,T]$, and the compactly supported cohomology class of $(\omega_t-\omega_0), t\in [0,T]$, is zero.

Then there exists a smooth family of diffeomorphisms $f_t: M\to M, t\in [0,T]$, such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$ and, furthermore, $f_t=id, t\in [0,T]$, outside of a compact subset $C\subset M$.

b. In the case when $\omega_0$ is the standard symplectic form on $M={\mathbb R}^{2n}$ one can do a bit better and find a family of diffeomorphisms $f_t: M\to M, t\in [0,T]$ such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$, provided that the difference $\omega_t(x)-\omega_0(x)$ merely decays sufficiently fast (in a suitable sense) as $x\to \infty$.

One can think of Theorem 2 as a version of the Global Darboux Theorem on ${\mathbb R}^{2n}$ for "small perturbations" of the standard symplectic form.

3. One can also ask if the Global Darboux Theorem holds for arbitrary symplectic manifolds $(M,\omega)$. One obvious obstruction, of course is that $M=M^{2n}$ is supposed to be diffeomorphic to a domain in ${\mathbb R}^{2n}$. With this restriction, Global Darboux again holds for planar surfaces ($n=1$), due to Greene and Shiohama, generalizing Moser's proof. However, Global Darboux fails in dimensions $\ge 4$ even if $M={\mathbb R}^{2n}$, $n\ge 2$. This was first observed by Gromov (who left a proof as an exercise as he tends to). Explicit examples were found later, for instance, in works by Bates, Peschke and Casals:

Theorem 3. For every $n\ge 2$ there exists a symplectic form $\omega$ on ${\mathbb R}^{2n}$ such that there is no smooth embedding $f: {\mathbb R}^{2n}\to {\mathbb R}^{2n}$ satisfying
$$ f^*(\omega_0)= \omega, $$ where $\omega_0$ is the standard symplectic form on ${\mathbb R}^{2n}$.

References:

1. Larry Bates, George Peschke, A remarkable symplectic structure, J. Differ. Geom. 32, No. 2, 533-538 (1990). ZBL0714.53028.

2. Roger Casal, Exotic symplectic structures, ZBL07152607.

3. Robert Greene, Katsuhiro Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Am. Math. Soc. 255, 403-414 (1979). ZBL0418.58002.

4. Jürgen Moser, On the volume elements on a manifold, Trans. Am. Math. Soc. 120, 286-294 (1965). ZBL0141.19407.

5. Xiudi Tang, "Symplectic Stability and New Symplectic Invariants of Integrable Systems", Ph.D. thesis, 2018.

See also this lecture by Weimin Chen for a self-contained treatment of Moser's theorem.