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
Famous quotes containing the word manifold:
“She ran down the stair
A twelve-year-old darling
And laughing and calling
She tossed her bright hair;”
—John Streeter Manifold (b. 1915)