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:
“There is then creative reading as well as creative writing. When the mind is braced by labor and invention, the page of whatever book we read becomes luminous with manifold allusion. Every sentence is doubly significant, and the sense of our author is as broad as the world.”
—Ralph Waldo Emerson (1803–1882)