Symplectic leaf - Imported Knowlpedia Content

Definition

Let $(M,\pi)$ be a Poisson manifold, and define its characteristic subspace at $p$ by

\mathcal D_p=\pi^\sharp_p(T_p^*M)\subseteq T_pM.

A symplectic leaf through $p$ is the maximal connected immersed submanifold $L\subseteq M$ containing $p$ and satisfying $T_qL=\mathcal D_q$ for every $q\in L$ . Its symplectic form is determined by

\omega_L(\pi^\sharp\alpha,\pi^\sharp\beta)=\pi(\alpha,\beta).

This is well-defined, closed, and nondegenerate. The Poisson integrability theorem guarantees that every point lies on a unique such leaf, even when the rank of $\pi$ varies.

Why the leaf form is canonical

If $\pi^\sharp\alpha=0$ , then $\pi(\alpha,\beta)=0$ for every $\beta$ , so the displayed formula is independent of the covectors representing tangent vectors. It is nondegenerate because the tangent space is exactly the image of $\pi^\sharp$ . Closedness and integrability ultimately follow from the Jacobi identity for the Poisson bracket. These facts are developed in Vaisman, “The Symplectic Foliation of a Poisson Manifold”.

Structure and consequences

The leaf dimension equals the rank of $\pi$ along the leaf and is therefore even and locally constant on that leaf. Different leaves may have different dimensions, so the decomposition is generally a singular foliation rather than a regular fiber bundle.

Hamiltonian vector fields are tangent to every leaf. Conversely, their values span the characteristic distribution, so a leaf is the region reachable by piecewise Hamiltonian flows. The inclusion $L\hookrightarrow M$ is a Poisson immersion when $L$ is equipped with the bracket induced by $\omega_L$ .

Examples and non-examples

For $\mathbb R^3$ with $\pi=\partial_x\wedge\partial_y$ , the leaves are the planes $z=c$ , each with its standard symplectic form. If $\pi=0$ , every leaf is a single point. The coadjoint orbits in the dual of a finite-dimensional Lie algebra are the leaves of its linear Poisson structure.

A proper open disk in one plane $z=c$ is an integral symplectic submanifold, but it is not a leaf because it is not maximal.

Conventions and scope

References

Izu Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118, Birkhäuser, 1994. Publisher record. Relevant: “The Symplectic Foliation of a Poisson Manifold,” pp. 19–30.
Alan Weinstein, “The Local Structure of Poisson Manifolds,” Journal of Differential Geometry 18, no. 3 (1983), 523–557. DOI record. Relevant: §1, Poisson manifolds and mappings, and §2, the splitting theorem.