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2 results

  • 3 - Symplectic Geometry

  • M. J. D. Hamilton, Universität Stuttgart, D. Kotschick, Ludwig-Maximilians-Universität München
  • Book:
    Künneth Geometry
    Published online:
    07 December 2023
    Print publication:
    21 December 2023, pp 30-41

  • Summary

    This chapter is a crash course on symplectic geometry. We first introduce symplectic manifolds and their Lagrangian submanifolds and give some examples. Then we discuss the Moser method [Mos-65], and use it to prove results about the existence of local normal forms and of standard tubular neighbourhoods. This is of course just the beginning of the vast area of symplectic geometry. A more extensive account is contained, for example, in the book of McDuff-Salamon [McS-95]; see especially Chapter 3 of that book. Other useful references for this material include the writings of Weinstein [Wei-71, Wei-77].

  • The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces

  • David Martínez Torres
  • Journal:
    Canadian Mathematical Bulletin / Volume 58 / Issue 3 / 01 September 2015
    Published online by Cambridge University Press:
    20 November 2018, pp. 575-579
    Print publication:
    01 September 2015
    • Article
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  • A surface $\sum$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration, $G\left( \pi \right)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\text{ }\!\!\pi\!\!\text{ }$ is not an area form on the 2-sphere, then $G\left( \pi \right)$ is diffeomorphic to the cotangent bundle $T*\sum$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.