Normal coordinates for isotropic submanifolds

Question

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 $k=\dim N$?

Answer 1

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.)