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:
“Thy love is such I can no way repay,
The heavens reward thee manifold I pray.
Then while we live, in love lets so persever,
That when we live no more, we may live ever.”
—Anne Bradstreet (c. 1612–1672)