Characteristic distribution of a coisotropic submanifold

Definition

Let $C$ be a coisotropic submanifold of a symplectic manifold $(M,\omega)$ , and write $\omega_C=\iota^*\omega$ for the restricted two-form. The characteristic distribution on $C$ is the vector subbundle $\mathcal K\subseteq TC$ whose fiber at $p\in C$ is

\mathcal K_p=(T_pC)^\omega =\ker\!\left(\omega_C^\flat:T_pC\to T_p^*C\right).

Coisotropy ensures $(T_pC)^\omega\subseteq T_pC$ , while symplectic linear algebra gives $\operatorname{rank}\mathcal K=\operatorname{codim}_M C$ . Thus these null spaces have constant rank and form a smooth distribution. It is also called the null distribution of the restricted form.

Involutivity

The characteristic distribution is involutive. If $X$ and $Y$ are vector fields tangent to $\mathcal K$ , then $\iota_X\omega_C=\iota_Y\omega_C=0$ . Because $d\omega_C=0$ , Cartan's formula gives

\iota_{[X,Y]}\omega_C =\mathcal L_X(\iota_Y\omega_C)-\iota_Y(\mathcal L_X\omega_C)=0.

Hence $[X,Y]$ is again tangent to $\mathcal K$ . The Frobenius theorem therefore integrates $\mathcal K$ to the characteristic foliation of $C$ .

Examples

For a coisotropic hypersurface, $\mathcal K$ is a line field. If the hypersurface is a regular energy level $H^{-1}(c)$ , its characteristic line is spanned by the Hamiltonian vector field $X_H$ wherever $dH\ne0$ . At the opposite extreme, if $C$ is Lagrangian, then $\mathcal K=TC$ , so each connected component is a characteristic leaf.

Role in reduction

The restricted form $\omega_C$ is horizontal along $\mathcal K$ and invariant under vector fields tangent to $\mathcal K$ . Consequently it is the candidate pullback of a two-form on the leaf space $C/\mathcal K$ . If that quotient is a smooth manifold and the projection is a submersion, the descended form is symplectic. The distribution alone does not guarantee that the leaf space is Hausdorff or even a manifold.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Springer DOI record. Relevant: characteristic foliations and symplectic reduction.
Paulette Libermann and Charles-Michel Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications 35, D. Reidel, 1987. Springer DOI record. Relevant: Chapter III, coisotropic submanifolds and characteristic distributions.