Poisson manifold

In geometry, a Poisson structure on a smooth manifold M {displaystyle M} is a Lie bracket { ⋅ , ⋅ } {displaystyle {cdot ,cdot }} (called a Poisson bracket in this special case) on the algebra C ∞ ( M ) {displaystyle {C^{infty }}(M)} of smooth functions on M {displaystyle M} , subject to the Leibniz rule In geometry, a Poisson structure on a smooth manifold M {displaystyle M} is a Lie bracket { ⋅ , ⋅ } {displaystyle {cdot ,cdot }} (called a Poisson bracket in this special case) on the algebra C ∞ ( M ) {displaystyle {C^{infty }}(M)} of smooth functions on M {displaystyle M} , subject to the Leibniz rule Said in another manner, it is a Lie algebra structure on the vector space of smooth functions on M {displaystyle M} such that X f = df { f , ⋅ } : C ∞ ( M ) → C ∞ ( M ) {displaystyle X_{f}{stackrel { ext{df}}{=}}{f,cdot }:{C^{infty }}(M) o {C^{infty }}(M)} is a vector field for each smooth function f {displaystyle f} , which we call the Hamiltonian vector field associated to f {displaystyle f} . These vector fields span a completely integrable singular foliation, each of whose maximal integral sub-manifolds inherits a symplectic structure. One may thus informally view a Poisson structure on a smooth manifold as a smooth partition of the ambient manifold into even-dimensional symplectic leaves, which are not necessarily of the same dimension. Poisson structures are one instance of Jacobi structures introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. Let M {displaystyle M} be a smooth manifold. Let C ∞ ( M ) {displaystyle {C^{infty }}(M)} denote the real algebra of smooth real-valued functions on M {displaystyle M} , where multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on M {displaystyle M} is an R {displaystyle mathbb {R} } -bilinear map