Poissonmanifold

A Poisson manifold is a smooth manifold M equipped with a Poisson bracket {,} on the space C∞(M) of smooth functions, which makes C∞(M) into a Lie algebra and satisfies the Leibniz rule {fg,h} = f{g,h} + g{f,h}. Equivalently, a Poisson structure is given by a bivector field π ∈ Γ(∧^2 TM) with vanishing Schouten–Nijenhuis bracket [π,π] = 0, such that {f,g} = π(df,dg) for all f,g ∈ C∞(M). The map π#: T*M → TM, α ↦ π(α,·), encodes the Hamiltonian dynamics.

For every smooth function f, the Hamiltonian vector field X_f = π#(df) generates a flow that preserves the Poisson bracket. The image of π#, Im π# ⊂ TM defines a characteristic distribution whose integral manifolds are the symplectic leaves: each leaf carries a nondegenerate two-form inherited from π, making it a symplectic manifold. The rank of π can vary across M; if the rank is constant, the Poisson structure is regular, whereas variable rank leads to singular Poisson manifolds with leaves of different dimensions.

Examples include symplectic manifolds, where π is nondegenerate and π^−1 is a symplectic form; the dual of a Lie algebra g* with the Lie–Poisson bracket; product Poisson structures, and the zero Poisson structure. Casimir functions are smooth functions that Poisson-commute with all others and are constant on each leaf.

Maps between Poisson manifolds that preserve the bracket are called Poisson maps: φ:M→N is Poisson if {f∘φ,g∘φ} = {f,g}∘φ for all f,g ∈ C∞(N). Local structure is described by Weinstein’s splitting theorem, a Poisson analogue of Darboux, giving coordinates in which π takes a standard block form with a symplectic part and a transverse Poisson part.

Poisson manifolds admit rich geometric and algebraic structures, including Poisson cohomology, deformations, and integration to symplectic groupoids, informing aspects of quantization and Morita theory.