Characteristic foliation - Imported Knowlpedia Content

Definition

Let $C$ be a coisotropic submanifold of a symplectic manifold $(M,\omega)$ . Its characteristic foliation is the regular foliation whose tangent distribution is the characteristic distribution

\mathcal K_p=(T_pC)^\omega=\ker\!\left((\iota^*\omega)_p:T_pC\longrightarrow T_p^*C\right).

The closedness of $\iota^*\omega$ makes $\mathcal K$ involutive, so the Frobenius theorem supplies a unique maximal connected integral manifold through every point of $C$ . These integral manifolds are the characteristic leaves. The leaf space $C/\mathcal K$ is only a set in general; it need not be Hausdorff or carry a smooth-manifold structure.

Why the distribution integrates

For vector fields $X,Y$ tangent to $\mathcal K$ , one has $\iota_X\iota^*\omega=\iota_Y\iota^*\omega=0$ . Cartan's formula and $d(\iota^*\omega)=0$ then imply

\iota_{[X,Y]}\iota^*\omega=0.

Thus $\mathcal K$ is closed under Lie brackets. Its rank is constant because coisotropy gives $\dim\mathcal K_p=\operatorname{codim}_M C$ . Frobenius therefore applies as a regular-foliation theorem, not merely as a statement about a possibly singular distribution Cannas da Silva, §1.4.

Examples and geometry of the leaves

If $C$ is a coisotropic hypersurface, its characteristic foliation is one-dimensional. For a regular energy surface $C=H^{-1}(c)$ , its leaves are the unparameterized trajectories of the Hamiltonian vector field $X_H$ restricted to $C$ . If $C$ is Lagrangian, then $\mathcal K=TC$ , so the leaves are the connected components of $C$ .

By contrast, a positive-dimensional symplectic submanifold has zero null distribution. Its point leaves form a trivial foliation, but the submanifold is not coisotropic unless it is open in $M$ .

Leaf space and reduction

The restricted form $\iota^*\omega$ annihilates directions tangent to the leaves and is invariant along them. If the leaf space is a smooth manifold and the quotient map is a surjective submersion, these facts allow the form to descend. The resulting symplectic manifold is the coisotropic reduction of $C$ .

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.4, coisotropic submanifolds, characteristic distributions, and reduction.
Paulette Libermann and Charles-Michel Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications 35, D. Reidel, 1987. DOI record. Relevant: Chapter III, characteristic systems of presymplectic forms.