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On Jacobi Manifolds and Jacobi Bundles

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Part of the Mathematical Sciences Research Institute Publications book series (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.

Keywords

  • Vector Field
  • Symplectic Manifold
  • Poisson Structure
  • Zero Section
  • Contact Manifold

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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

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