Let $(M,\omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p\in N$, can we always find coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ in a neighbourhood $U$ of $p$ such that both $\omega=\sum dx_i\wedge dy_i$ (Darboux coordinates) and $$N\cap U=\{(x_1,\ldots,x_n,y_1,\ldots,y_n):x_{k+1}=\cdots=x_n=y_1=\cdots=y_n=0\}$$ where $k=\dim N$?


The answer is yes, and in fact, you can let $x_1,\ldots,x_k$ be any chosen coordinate patch on $N$.

Here's one way to go about it. The first point is that the symplectic normal bundle $TN^{\omega}/TN \rightarrow N$ is locally trivial (as a symplectic vector bundle). Feeding a particular trivialization into the standard tubular neighborhood theorem shows that you can at least find a diffeomorphism from a neighborhood of $p$ to this model so that the restriction of $\omega$ to $TM|_N$ matches the model. This is enough data to feed into a Moser-type argument.

I'll leave the first part of that argument as a sketch, but let me expand on this last step. In these coordinates, we now have two symplectic forms $\omega_0$ (the one coming from the actual symplectic form on $M$) and $\omega_1$ (the model symplectic form), which match on $TM|_N$. Note that the homotopy of forms $\omega_t := (1-t)\omega_0 + t\omega_1$ remains symplectic in some small enough neighborhood $Open(p)$ by the matching of $\omega_0$ and $\omega_1$ along $TM|_N$. Our goal now is to find a sequence of diffeomorphisms $f_t \colon Open(p) \rightarrow Open(p)$ which are the identity along $N$ and such that $f_t^*(\omega_t) = \omega_0$, since then $f_1$ yields a symplectomorphism desired.

The Moser trick is to take $f_0$ to be the identity, and suppose that $f_t$ is built out of the flow of some vector field $X_t$. Then, taking a derivative, we find

$$0 = \frac{d}{dt}\omega_0 = \frac{d}{dt}f_t^*\omega_t = f_t^*(\dot{\omega_t} + \mathcal{L}_{X_t}\omega_t) = f_t^*(\omega_1-\omega_0 + \mathcal{L}_{X_t}\omega_t).$$

In other words, it suffices that $X_t$ satisfies $$\mathcal{L}_{X_t}\omega_t = \omega_0-\omega_1.$$ Since $\omega_t$ is closed, this is just $$di_{X_t}\omega_t = \omega_0-\omega_1.$$ Since $\omega_0 = \omega_1$ along $N$ and $\omega_0 - \omega_1$ is closed, we can choose a $1$-form $\beta$ such that $\beta$ is $0$ on its restriction to $TM|_N$ and $d\beta = \omega_0-\omega_1$ (this is a general Poincare lemma). Non-degeneracy of $\omega_t$ now allows us to define $X_t$ uniquely by $i_{X_t}\omega_t = \beta$. In particular, since $\beta|_{TM|_N} = 0$, we therefore have $X_t|_N = 0$, and so $f_t$ is always the identity along $N$.

A further careful analysis shows that the derivative of $f_1$ along $TM|_N$ is the identity (this is because $d\beta = \omega_0 - \omega_1$ is $0$ on $TM|_N$). This gives a little bit more, in case you wanted it.

You might also notice, by the way, that nowhere did I actually use the explicit coordinates of the model, except in the first sentence where I suggested that you could choose any coordinate patch on $x_1,\ldots,x_k$. That's because this argument works in tons more generality. All you need in symplectic geometry to see that two submanifolds $N_1$ and $N_2$ of symplectic manifolds $M_1$ and $M_2$ have symplectomorphic neighborhoods is a diffeomorphism $\phi \colon N_1 \rightarrow N_2$ and a bundle map $\Phi \colon TM_1|_{N_1} \rightarrow TM_2|_{N_2}$ lying over $\phi$ such that $\Phi^*\omega_2 = \omega_1$. Then there is a symplectomorphism of neighborhoods realizing $\Phi$. I do not know of a source where this very general statement is written, though it is literally the same proof that I have just written. It has been known to experts for a long time. If anyone knows where this is written, I'd love to know! (Regardless, and as a little bit of self-promotion, I plan to include this, along with a contact-geometric version, and a bit more, in my thesis, so at least there will be one source.)

  • $\begingroup$ Your last statement looks like Theorem 2.2 in Sjamaar-Lerman "Stratified symplectic spaces and reduction". $\endgroup$ – SHP Jan 10 '19 at 8:29
  • $\begingroup$ Yes, the Sjamaar-Lerman result is a special case. (I didn't mention existence in the above general statement, but this is just the condition $\omega|_N$ is closed, since any such form has a closed extension to a neighborhood of $N$.) However, the last statement really is more general, since it also covers non-constant rank embeddings - for example a generically symplectic embedding with a coisotropic locus. $\endgroup$ – KSackel Jan 10 '19 at 13:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.