Subvariety, involutive

(in symplectic geometry)

Let $V$ be a vector space of dimension $2n$ and $\omega$ a non-degenerate alternating $2$-form on it. Given a subspace $W$ of $V$, one defines (as usual)

$$W ^ \perp = \{ {x \in V } : {\omega ( x, w)= 0 \textrm{ for all } w \in W } \} .$$

One says that $W$ is an isotropic subspace if $W \subset W ^ \perp$, that it is an involutive subspace (or co-isotropic subspace) if $W \supset W ^ \perp$, and that it is a Lagrangian subspace if $W = W ^ \perp$. Note that for $W$ to be involutive it is necessary that $\mathop{\rm dim} ( W) \geq n$.

Now, let $V$ be a subvariety (possibly with singularities; or, more generally, an analytic subset) of a symplectic manifold $X$. Let $\mathop{\rm Reg} ( V)$ be the set of points of $V$ which have a neighbourhood in $V$ that is free of singular points. Then $V$ is an involutive subvariety of $X$ if for all $p \in \mathop{\rm Reg} ( V)$ the subspace $V _ {p}$ of $X _ {p}$ is involutive. The notions of an isotropic subvariety and a Lagrangian subvariety are defined analogously. If $\mathop{\rm Reg} ( V)$ is dense in $V$, then $V$ is involutive if and only if for every two $C ^ {1}$-functions $f$, $g$ on $X$ which vanish on $V$ the Poisson bracket $\{ f, g \}$ (defined by the symplectic $2$-form on $X$) also vanishes on $V$.

References

[a1]	P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) MR0882548 Zbl 0643.53002