The Symplectic Slice Theorem

  • Juan-Pablo Ortega
  • Tudor S. Ratiu
Part of the Progress in Mathematics book series (PM, volume 222)


Most of the good technical behavior of proper Lie group actions is a direct consequence of the existence of slices and tubes; they provide a privileged system of semiglobal coordinates in which the group action takes on a particularly simple form. Proper symplectic Lie group actions turn out to behave similarly: the tubular chart can be constructed in such a way that the expression of the symplectic form is very natural and, moreover, if there is a momentum map associated to this canonical action, this construction provides a normal form for it. The statement and proof of this Symplectic Slice Theorem is the main goal of this chapter.


Symplectic Form Symplectic Manifold Horizontal Lift Horizontal Curve Lagrangian Subspace 


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Copyright information

© Juan Pablo Ortega and Tudor S. Ratiu 2004

Authors and Affiliations

  • Juan-Pablo Ortega
    • 1
  • Tudor S. Ratiu
    • 2
  1. 1.CNRS-Laboratoire de Mathématiques de BensançonUniversité de Franche-Comté, UFR des Sciences et TechniquesBensançon CedexFrance
  2. 2.Départment de MathématiquesLausanneSwitzerland

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