Poisson manifold

This article describes an additional structure on a differential manifold
View other additional structures

This article or section of article is sourced from:Wikipedia

Definition

A Poisson manifold is a differential manifold equipped with an additional structure called Poisson structure described below:

A Poisson structure is a $\mathbb{R}$-bilinear map $\{,\}:C^{\mathrm{\infty }}(M)\times C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$, turning $C^{\mathrm{\infty }}(M)$ into a Poisson algebra. More explicitly, the following identities are satisfied:

• $\{f,g\}=-\{g,f\}$
• $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f.g\}\}=0$
• $\{fg,h\}=f\{g,h\}+g\{f,h\}$

Relation with other structures

Stronger structures

• Symplectic manifold