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On Jacobi Manifolds and Jacobi Bundles

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Symplectic Geometry, Groupoids, and Integrable Systems

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 20))

Abstract

We introduce the notion of a Jacobi bundle, which generalizes that of a Jacobi manifold. The construction of a Jacobi bundle over a conformal Jacobi manifold has, as particular cases, the constructions made by A. Weinstein [21] of a Le Brun-Poisson manifold over a contact manifold, and that of a Heisenberg-Poisson manifold over a symplectic (or Poisson) manifold. We show that the total space of a Jacobi bundle has a natural homogeneous Poisson structure, and that with each section of that bundle is associated a Hamiltonian vector field, defined on the total space of the bundle, tangent to the zero section, which projects onto the base manifold.

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Author information

Authors and Affiliations

  1. Université Pierre et Marie Curie, Mathématiques, 4, place Jussieu, 75252, Paris Cedex 05, France

    Charles-Michel Marle

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  1. Charles-Michel Marle
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Editor information

Editors and Affiliations

  1. Département de Mathématiques, Université Lyon I, 43 Boulevard du Novembre 1918, 69622, Villeurbanne Cedex, France

    Pierre Dazord

  2. Department of Mathematics, University of California, 94720, Berkeley, CA, USA

    Alan Weinstein

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© 1991 Springer-Verlag New York, Inc.

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Marle, CM. (1991). On Jacobi Manifolds and Jacobi Bundles. In: Dazord, P., Weinstein, A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol 20. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9719-9_16

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  • DOI: https://doi.org/10.1007/978-1-4613-9719-9_16

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9721-2

  • Online ISBN: 978-1-4613-9719-9

  • eBook Packages: Springer Book Archive

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