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 $k=\dim N$?



Normal coordinates for isotropic submanifolds
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.)


