A Poisson manifold is a particular type of differentiable manifold equipped with a Poisson bracket, which is a bilinear operation that satisfies certain algebraic properties.
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A Poisson manifold is a smooth manifold with a bilinear bracket on smooth functions that is antisymmetric, obeys the Jacobi identity, and is a derivation in each argument. Its Poisson bivector may have nonconstant rank, unlike the inverse of a symplectic form.