The Symplectic Foliation of a Poisson Manifold

Abstract

Let (Mⁿ, w) be a Poisson manifold. According to the definition, as given at the beginning of Chapter 1, we expect the set ${{X}_{w}}(M)\mathop{ = }\limits^{{def}} \left. {\{ {{X}_{f}}\mid f \in {{C}^{\infty }}(M)} \right\}$ of the Hamiltonian vector fields of M to play an important role. In order to discover it, let us define

$${{S}_{x}}_{0}(M) = \{ v \in {{T}_{x}}_{0}M/\exists f \in {{C}^{\infty }}(M),{{X}_{f}}({{x}_{0}}) = v\} ({{x}_{0}} \in M) $$

(2.1)

. Notice that, since X_f = #(df), where # is the homomorphism (1.3), we have S_x0(M) = im#x₀. In order to speak of the set S(M) of all these planes, we shall adopt the following terminology, which is slightly different from that of the textbooks on differentiable manifolds.