In mathematics, a Poisson manifold is a smooth manifold M equipped with a bilinear map {⋅,⋅}M (called a Poisson bracket) on the algebra C∞(M) of smooth functions on M such that (C∞(M),{⋅,⋅}M) is a Poisson algebra. One usually denotes a Poisson manifold by the ordered pair (M,{⋅,⋅}M). Since their introduction by André Lichnerowicz in 1977, the subjects of Poisson geometry and the cohomology of Poisson manifolds have developed into a wide field of research, which includes modern-day non-commutative geometry.
It is a fact that every symplectic manifold is a Poisson manifold but not vice-versa. This will be explained in Section 2.
Read more about Poisson Manifold: Definition, The Poisson Bivector, Poisson Maps, The Product of Poisson Manifolds, The Symplectic Leaves of A Poisson Structure, Example (Lie-Poisson Manifold), Complex Structure
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“They had met, and included in their meeting the thrust of the manifold grass stems, the cry of the peewit, the wheel of the stars.”
—D.H. (David Herbert)