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Annals of Global Analysis and Geometry

, Volume 14, Issue 4, pp 427–441 | Cite as

Invariants of contact structures and transversally oriented foliations

  • Augustin Banyaga
Article

Abstract

We exhibit new invariants of the contact structure E(α), the contact flow Fα and the transverse symplectic geometry of a contact manifold (M, α). The invariant of contact structures generalizes to transversally oriented foliations. We focus on the particular cases of orientations of smooth manifolds and transverse orientations of foliations. We define the transverse Calabi invariants and determine their kernels.

Key words

Contact structures contact flows characteristic foliations simple foliations transverse symplectic geometry transverse Calabi and Thurston invariants affine representations orientation class 

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References

  1. [1]
    Banyaga, A.: Quelques invariants de la géométrie de contact. C. R. Acad. Sci., Paris, Sér. I, 318 (1994), 671–674.Google Scholar
  2. [2]
    Banyaga, A.: Sur la structure du groupe de difféomorphismes qui préservant une forme symplectique. Comment. Math. Helv. 53 (1978), 174–227.Google Scholar
  3. [3]
    Banyaga, A.: On isomorphic classical diffeomorphism groups I. Proc. Am. Math. Soc. 96 (1986) 1, 113–118.Google Scholar
  4. [4]
    Banyaga, A.: On characteristics of hypersurfaces in symplectic manifolds. Proc. Symp. Pure Math. Vol. 54 Part 2, 1993, 9–17.Google Scholar
  5. [5]
    Banyaga, A.: On the cohomology class of the action functional. Expo. Math. 12 (1994), 371–379.Google Scholar
  6. [6]
    Banyaga, A.; Urwin, R.: Sur la cohomologie du groupe des difféomorphismes. C. R. Acad. Sci., Paris, 294 (1982), 625–627.Google Scholar
  7. [7]
    Banyaga, A.; Urwin, R.: On the cohomology of the diffeomorphism group. Atti Accad. Sci. Torino 117 (1983), 1–34.Google Scholar
  8. [8]
    Banyaga, A.; McInerney, A.: On isomorphic classical diffeomorphism groups III. Ann. Global Anal. Geom. 13 (1995), 117–127.Google Scholar
  9. [9]
    Banyaga, A.; de la Llave, R.; Wayne, E.: Cohomology equations and commutators of germs of contact diffeomorphisms. Trans. Am. Math. Soc. 32 (1989) 2, 755–778.Google Scholar
  10. [10]
    Banyaga, A.; Rukimbira P.: An invitation to contact geometry. To appear.Google Scholar
  11. [11]
    Boothby, W.M.: Transitivity of the automorphisms of certain geometric structures. Trans. Am. Math. Soc. 137 (1969), 93–100.Google Scholar
  12. [12]
    Boothby, W.M.; Wang, H.: On contact manifolds. Ann. Math. 60 (1978), 721–734.Google Scholar
  13. [13]
    Bott, R.: On the characteristic classes of groups of diffeomorphisms. Enseign. Math., II. Sér., 23 (1977) 3–4, 209–220.Google Scholar
  14. [14]
    Calabi, E.: On the group of automorphisms of a symplectic manifold. In: Problems in Analysis., Gunning Ed. Princeton. University Press, Princeton, N. J.Google Scholar
  15. [15]
    Flato, M.; Lichnerowicz, A.: Cohomologie des représentations définies par des dérivations de Lie et à valeurs dans les formes, de l'algébre de Lie des champs de vecteurs d'une variéte différentiable. Premiers espaces de cohomologie. Applications. C. R. Acad. Sci., Paris, 291 (1980), 331–335.Google Scholar
  16. [16]
    Gray, J.: Some global properties of contact structures. Ann. Math. 69 (1959), 421–450.Google Scholar
  17. [17]
    Hilton, P.; Stamback, U.: A Course in Homological Algebra. Grad. Texts Math. 4, Springer Verlag, 1971.Google Scholar
  18. [18]
    Klein, F.: Erlangen Program. Math. Ann. 43 (1893).Google Scholar
  19. [19]
    Kobayashi, S.: Transformation groups in differential geometry. Ergeb. Math. Grenzgeb. 70, Springer 1972.Google Scholar
  20. [20]
    Koszul, J.L.: Homologie des complexes de formes différentielles d' ordre supérieur. Ann. Sci. Éc. Norm. Supér., IV. Sér., 7 (1974), 139–154.Google Scholar
  21. [21]
    Ismagilov, R.S.: On the group of volume preserving diffeomorphisms. Math. USSSR Izvestija 17 (1981), 95–127.Google Scholar
  22. [22]
    Leslie, J.: A remark on the group of automorphisms of a foliation having a dense leaf. J. Differ. Geom. 7 (1972), 597–601.Google Scholar
  23. [23]
    Libermann, P.; Marle, C.M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Co., 1987.Google Scholar
  24. [24]
    Lichnerowicz, A.: Algébre de Lie des automorphismes infinitésimaux d' une structure de contact. J. Math. Pures Appl. 52 (1973), 473–508.Google Scholar
  25. [25]
    Lichnerowicz, A.: Algébres de Lie attachées à un feuilletage. Ann. Fac. Sci. Toulouse, Math. 5 (1979) 1, 45–76.Google Scholar
  26. [26]
    Lychagin, V.V.: Local classification of nonlinear first order partial differential equations. Russ. Math. Surv. 30 (1975), 105–175.Google Scholar
  27. [27]
    Moser, J.: On the volume element on manifolds. Trans. Am. Math. Soc. 120 (1965), 280–296.Google Scholar
  28. [28]
    Omori, H.: Infinite dimensional Lie transformation groups. Lect. Notes Math. 427, Springer 1974.Google Scholar
  29. [29]
    Thomas, C.: Almost regular contact manifolds. J. Differ. Geom. 11 (1978), 521–533.Google Scholar
  30. [30]
    Thurston, W.: On the structure of volume preserving diffeomorphisms. Unpublished.Google Scholar
  31. [31]
    Tondeur, Ph.: Foliations on Riemannian Manifolds. Springer Universitext 1988.Google Scholar
  32. [32]
    Wadsley, A.W.: Geodesic foliations by circles. J. Differ. Geom. 10 (1975), 541–549.Google Scholar
  33. [33]
    Warner, F.: Foundations of differentiable manifolds and Lie groups. Scott, Foresmann and Co., London 1971.Google Scholar
  34. [34]
    Weinstein, A.: Cohomology of symplectomorphism groups and critical values of hamiltonians. Math. Z. 201 (1989), 75–82.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Augustin Banyaga
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUSA

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