Constant Poisson Structures, Regular and Symplectic Manifolds

  • Camille Laurent-Gengoux
  • Anne Pichereau
  • Pol Vanhaecke
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 347)


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.


Poisson Bracket Symplectic Manifold Poisson Structure Symplectic Structure Ahler Manifold 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Camille Laurent-Gengoux
    • 1
  • Anne Pichereau
    • 2
  • Pol Vanhaecke
    • 3
  1. 1.CNRS UMR 7122, Laboratoire de MathématiquesUniversité de LorraineMetzFrance
  2. 2.CNRS UMR 5208, Institut Camille JordanUniversité Jean MonnetSaint-EtienneFrance
  3. 3.CNRS UMR 7348, Lab. Mathématiques et ApplicationsUniversité de PoitiersFuturoscope ChasseneuilFrance

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