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An introduction to symplectic geometry

  • Augustin Banyaga
Chapter
Part of the Progress in Mathematics book series (PM, volume 117)

Abstract

A symplectic form on a vector space V is a skew-symmetric bilinear form α: V × V → R such that \( \tilde \alpha :V \to V^ \star \tilde \alpha (x)(y) = \alpha (x,y) \) is an isomorphism. Here V* denotes the dual of V.

Keywords

Vector Bundle Symplectic Form Symplectic Manifold Symplectic Structure Lagrangian Submanifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [1]
    R. Abraham, J.E. Marsden, Foundations of Mechanics, Benjamin, NY, 1967.Google Scholar
  2. [2]
    V.I. Arnold, A characteristic class entering in quantization conditions, Funct. Anal. Appl. 1 (1967), 1–13.CrossRefGoogle Scholar
  3. [3]
    V.I. Arnold, Mathematical methods of classical mechanics classique, Graduate Texts in Mathematics, Springer, 1978.Google Scholar
  4. [4]
    V.I. Arnold, The Sturm theorems and symplectic geometry, Funct. Anal. Appl. 19 (1985), 251–259.CrossRefGoogle Scholar
  5. [5]
    V.I. Arnold, A.B. Givental, Dynamical systems IV, Encyclopaedia of Math. Sci., Springer, 1990.Google Scholar
  6. [6]
    M. Audin, Classes caractéristiques lagrangiennes, in Algebraic Topology, Barcelone 1986, Springer Lecture Notes in Math. 1298 (1988), 1–15.MathSciNetCrossRefGoogle Scholar
  7. [7]
    A. Banyaga, Sur la structure du groupe de difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174–227.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Banyaga, On isomorphic classical diffeomorphisms II, J. Diff. Geom. 28 (1988), 93–144.MathSciNetGoogle Scholar
  9. [9]
    M. Berger, B. Gostiaux, Differential geometry: manifolds, curves and surfaces, Graduate Texts in Math. #115, Springer, 1988.Google Scholar
  10. [10]
    M. Bialy, L. Polterovich, Hamiltonian diffeomorphisms and Lagrangian distributions, Geometric and Funct. Analysis 2 (1992), 173–210.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J.J. Duistermaat, On the Morse index in variational calculus, Adv. Math. 21 (1976), 173–195.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. B. Fuks, Maslov Arnold characteristic classes, Doklady Akad. Nauk SSSR 178 (1968), 303–306.MathSciNetGoogle Scholar
  13. [13]
    V. Guillemin, S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, RI, 1977.Google Scholar
  14. [14]
    V. Guillemin, S. Sternberg, Symplectic Technics in Physics, Cambridge University Press, 1984.Google Scholar
  15. [15]
    P. Liberman, C. Marle, Symplectic Geometry and Analytical Dynamics, Math. and its Appl., #35, Riedel, Boston, 1987.Google Scholar
  16. [16]
    J.M. Morvan, Une obstruction à la transversalité de deux champs de plans lagrangiens, in Séminaire Sud-Rhodanien de Géométrie I, Travaux en Cours, Hermann, Paris (1984).Google Scholar
  17. [17]
    J. Moser, On the volume elements on α manifold, Trans. Amer. Math. Soc 120 (1965), 286–294.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    L. Polterovich, Monotone Lagrangian submanifolds of linear spaces and the Maslov class in cotangent bundles, Math. Zeitschrift 207 (1991), 217–222.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    I. Vaisman, Symplectic geometry and secondary characteristic classes, Progress in Math. #72, Birkhäuser, 1987.Google Scholar
  20. [20]
    A. Weinstein, Lectures on Symplectic Manifolds, Amer. Math. Soc., BSMS #29, Providence, RI, 1977.Google Scholar
  21. [21]
    E. Zehnder, Remarks on periodic solutions on hypersurfaces, in Periodic solutions of Hamiltonian sytems and related topics, P.H. Rabinowitz Ed., Reidei Publ. Dordrecht (1987), 267–279.CrossRefGoogle Scholar

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© Springer Basel AG 1994

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  • Augustin Banyaga

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