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2000 | 51 | 1 | 141-149
Tytuł artykułu

Connections in regular Poisson manifolds over ℝ-Lie foliations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The subject of this paper is the notion of the connection in a regular Poisson manifold M, defined as a splitting of the Atiyah sequence of its Lie algebroid. In the case when the characteristic foliation F is an ℝ-Lie foliation, the fibre integral operator along the adjoint bundle is used to define the Euler class of the Poisson manifold M. When M is oriented 3-dimensional, the notion of the index of a local flat connection with singularities along a closed transversal is defined. If, additionally, F has compact leaves (then F is a fibration over $S^{1}$), an analogue of the Euler-Poincaré-Hopf index theorem for flat connections with singularities along closed transversals is obtained.
Rocznik
Tom
51
Numer
1
Strony
141-149
Opis fizyczny
Daty
wydano
2000
Twórcy
autor
  • Institute of Mathematics, Technical University of, Łódź, Al. Politechniki 11, PL-90-924, Łódź, Poland
Bibliografia
  • [A] M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207.
  • [C-D-W] A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques, Publ. Dep. Math. Université de Lyon 1, 2/A (1987).
  • [C-H] F. A. Cuesta and G. Hector, Intégration symplectique des variétés de Poisson régulières, Israel J. Math. 90 (1995), 125-165.
  • [D-S] P. Dazord and D. Sondaz, Variétés de Poisson - Algébroï des de Lie, Publ. Dep. Math. Université de Lyon 1, 1/B (1988).
  • [He] G. Hector, Une nouvelle obstruction à l'intégrabilité des variétés de Poisson régulières, Hokkaido Math. J. 21 (1992), 159-185.
  • [H-H] G. Hector and U. Hirsh, Introduction to the Geometry of Foliations, Part A and B, Braunschweig, 1981, 1983.
  • [K1] J. Kubarski, Pradines-type groupoides over foliations; cohomology, connections and the Chern-Weil homomorphism, Preprint Nr 2, August 1986, Institute of Mathematics, Technical University of Łódź.
  • [K2] J. Kubarski, Characteristic classes of some Pradines-type groupoids and a generalization of the Bott Vanishing Theorem, in: Differential Geometry and Its Applications, Proceedings of the Conference August 24-30, 1986, Brno, Czechoslovakia.
  • [K3] J. Kubarski, Lie algebroid of a principal fibre bundle, Publ. Dep. Math. Université de Lyon 1, 1/A, 1989.
  • [K4] J. Kubarski, About Stefan's definition of a foliation with singularities: a reduction of the axioms, Bull. Soc. Math. France 118 (1990), 391-394.
  • [K5] J. Kubarski, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dep. Math. Université de Lyon 1, 1991.
  • [K6] J. Kubarski, Fibre integral in regular Lie algebroids, in: New Developments in Differential Geometry (Budapest, 1996), Kluwer, 1999.
  • [K7] J. Kubarski, Euler class and Gysin sequence of spherical Lie algebroids, in preparation.
  • [K8] J. Kubarski, An analogue of the Euler-Poincaré-Hopf theorem in topology of some 3-dimensional Poisson manifolds.
  • [M] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, 1987.
  • [M2] K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97-147.
  • [M-S] C. C. Moore and C. Schochet, Global Analysis on Foliated Spaces, Math. Sci. Res. Inst. Publ. 9, Springer-Verlag, 1988.
  • [P1] J. Pradines, Théorie de Lie pour les groupoï des différentiables dans la catégorie des groupoï des, Calcul différentiel dans la catégorie des groupoï des infinitésimaux, C. R. Acad. Sci. Sér. A-B Paris 264 (1967), 245-248.
  • [P2] J. Pradines, Théorie de Lie pour les groupoï des différentiables, Atti Conv. Intern. Geom. 7 Diff. Bologna, 1967, Bologna-Amsterdam.
  • [V1] I. Vaisman, Remarks on the Lichnerowicz-Poisson cohomology, Ann. Inst. Fourier Grenoble 40 (1990), 951-963.
  • [V2] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkhäuser Verlag, 1994.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv51z1p141bwm
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