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Symmetry Reduction in Symplectic and Poisson Geometry

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Abstract

We present a quick review of several reduction techniques for symplectic and Poisson manifolds using local and global symmetries compatible with these structures. Reduction based on the standard momentum map (symplectic or Marsden–Weinstein reduction) and on generalized distributions (the optimal momentum map and optimal reduction) is emphasized. Reduction of Poisson brackets is also discussed and it is shown how it defines induced Poisson structures on cosymplectic and coisotropic submanifolds.

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Authors and Affiliations

  1. Centre National de la Recherche Scientifique, Département de Mathématiques de Besançon, Université de Franche-Comté. UFR des Sciences et Techniques, 16, route de Gray, F-25030, Besançon cedex, France

    Juan-Pablo Ortega

  2. Centre Bernoulli, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Judor S. Ratiu

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  1. Juan-Pablo Ortega
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  2. Judor S. Ratiu
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Correspondence to Juan-Pablo Ortega.

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Ortega, JP., Ratiu, J.S. Symmetry Reduction in Symplectic and Poisson Geometry. Lett Math Phys 69, 11–61 (2004). https://doi.org/10.1007/s11005-004-0898-x

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