High Quality Content by WIKIPEDIA articles! In mathematics, a Poisson manifold is a differential manifold M such that the algebra C?(M) of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra. Every symplectic manifold is a Poisson manifold but not vice versa. For a symplectic manifold, ? is nothing other than the pairing between tangent and cotangent bundle induced by the symplectic form ?, which exists because it is nondegenerate. The difference between a symplectic manifold and a Poisson manifold is that the symplectic form must be nowhere singular, whereas the Poisson bivector does not need to be of full rank everywhere. When the Poisson bivector is zero everywhere, the manifold is said to possess the trivial Poisson structure.

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