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Characteristic distribution of a coisotropic submanifold is involutive

Statement

If $C$ is a coisotropic submanifold of $(M, ω)$ , then

$K = \ker (ω ∣_{T C}) = (T C)^{ω}$

is a smooth constant-rank distribution on $C$ , and it is involutive. It is called the characteristic distribution.

Facts & Assumptions

Given: A coisotropic submanifold $C \subseteq (M, ω)$ .

[F1]

Coisotropic means $(T_{p} C)^{ω} \subseteq T_{p} C$ at every $p$ . Isotropic, coisotropic, symplectic, and Lagrangian submanifolds.

[F2]

Cartan's formula relates Lie derivative, contraction, and exterior differentiation. Cartan's magic formula.

Proof

technique · direct

1.1

By [F1], the kernel of $ω ∣_{T_{p} C}$ is exactly $(T_{p} C)^{ω}$ . If $\dim M = 2 n$ and $\dim C = k$ , symplectic linear algebra gives its dimension $2 n - k$ , independent of $p$ . It is the kernel of a smooth constant-rank bundle map $T C \to T^{*} C$ , hence is a smooth subbundle.

F1givenalgebra

2.1

Let $X, Y$ be local sections of $K$ and $Z$ a tangent vector field on $C$ . In the formula for $d (ω ∣_{C}) (X, Y, Z)$ , every differentiated pairing vanishes identically and all bracket terms except $- ω ([X, Y], Z)$ contain $X$ or $Y$ as an argument. Since $d ω = 0$ , it follows that $ω ([X, Y], Z) = 0$ for every $Z$ . Thus $[X, Y]$ is a section of $K$ , proving involutivity.

F2step 1.1given∎

Depends on

Used by

Local normal form near a coisotropic submanifold Theorem

Dependency tree · two levels

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Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)