Poisson Manifold - The Symplectic Leaves of A Poisson Structure

The Symplectic Leaves of A Poisson Structure

A Poisson manifold (M,{⋅,⋅}_M) can be split into a collection of symplectic leaves. This splitting arises from the foliation of disjoint regions of M where the Poisson bivector field has constant rank. Each leaf of the foliation is thus an even-dimensional sub-manifold of M that is itself a symplectic manifold. Distinct symplectic leaves may have different dimensions. Two points lie in the same leaf if and only if they are joined by a piecewise-smooth curve where each piece is the integral curve of a Hamiltonian vector field. The relation "piecewise-connected by integral curves of Hamiltonian fields" is an equivalence relation on M, and the equivalence classes of this equivalence relation are the symplectic leaves.