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 x ∈ M and y ∈ N are to be held constant. In other words, if
- ,
then both
and
are implied.
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