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Constant Poisson Structures, Regular and Symplectic Manifolds

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Poisson Structures

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 347))

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Abstract

Symplectic manifolds appear as phase spaces of many mechanical systems and are as such, historically, the first examples of Poisson manifolds. They can be characterized as Poisson manifolds whose rank is constant and equal to the dimension of the manifold. Since the rank of their Poisson structure is constant, symplectic manifolds are regular Poisson manifolds. Locally their Poisson structure looks like the standard Poisson structure, i.e., locally they are constant Poisson structures. Important examples of symplectic manifolds which are discussed include Kähler manifolds, cotangent bundles and quotients of symplectic vector spaces.

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Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P. (2013). Constant Poisson Structures, Regular and Symplectic Manifolds. In: Poisson Structures. Grundlehren der mathematischen Wissenschaften, vol 347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31090-4_6

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