Poisson Manifold - The Product of Poisson Manifolds

The Product of Poisson Manifolds

Given two Poisson manifolds (M,{⋅,⋅}M) and (N,{⋅,⋅}N), a Poisson bracket may be defined on the product manifold M × N. Letting f1 and f2 be two smooth functions defined on M × N, one can define a new Poisson bracket {⋅,⋅}M × N in terms of {⋅,⋅}M and {⋅,⋅}N as follows:

,

where xM and yN are to be held constant. In other words, if

,

then both

and

are implied.

Read more about this topic:  Poisson Manifold

Famous quotes containing the word product:

    Culture is a sham if it is only a sort of Gothic front put on an iron building—like Tower Bridge—or a classical front put on a steel frame—like the Daily Telegraph building in Fleet Street. Culture, if it is to be a real thing and a holy thing, must be the product of what we actually do for a living—not something added, like sugar on a pill.
    Eric Gill (1882–1940)