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