# Normal coordinates for isotropic submanifolds

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.)

• Your last statement looks like Theorem 2.2 in Sjamaar-Lerman "Stratified symplectic spaces and reduction". – SHP Jan 10 '19 at 8:29
• 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. – KSackel Jan 10 '19 at 13:04