Connections in regular Poisson manifolds over ℝ-Lie foliations

Jan Kubarski

Banach Center Publications (2000)

  • Volume: 51, Issue: 1, page 141-149
  • ISSN: 0137-6934

Abstract

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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 ), an analogue of the Euler-Poincaré-Hopf index theorem for flat connections with singularities along closed transversals is obtained.

How to cite

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Kubarski, Jan. "Connections in regular Poisson manifolds over ℝ-Lie foliations." Banach Center Publications 51.1 (2000): 141-149. <http://eudml.org/doc/209025>.

@article{Kubarski2000,
abstract = {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.},
author = {Kubarski, Jan},
journal = {Banach Center Publications},
keywords = {Lie algebroid; ℝ-Lie foliation; Poisson manifold; closed transversal; flat connection with singularites along closed transversals; connection; regular Poisson manifold; Atiyah sequence; characteristic foliation; Euler class; index theorem},
language = {eng},
number = {1},
pages = {141-149},
title = {Connections in regular Poisson manifolds over ℝ-Lie foliations},
url = {http://eudml.org/doc/209025},
volume = {51},
year = {2000},
}

TY - JOUR
AU - Kubarski, Jan
TI - Connections in regular Poisson manifolds over ℝ-Lie foliations
JO - Banach Center Publications
PY - 2000
VL - 51
IS - 1
SP - 141
EP - 149
AB - 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.
LA - eng
KW - Lie algebroid; ℝ-Lie foliation; Poisson manifold; closed transversal; flat connection with singularites along closed transversals; connection; regular Poisson manifold; Atiyah sequence; characteristic foliation; Euler class; index theorem
UR - http://eudml.org/doc/209025
ER -

References

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  1. [A] M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207. Zbl0078.16002
  2. [C-D-W] A. Coste, P. Dazord and A. Weinstein, Groupoï des symplectiques, Publ. Dep. Math. Université de Lyon 1, 2/A (1987). 
  3. [C-H] F. A. Cuesta and G. Hector, Intégration symplectique des variétés de Poisson régulières, Israel J. Math. 90 (1995), 125-165. 
  4. [D-S] P. Dazord and D. Sondaz, Variétés de Poisson - Algébroï des de Lie, Publ. Dep. Math. Université de Lyon 1, 1/B (1988). 
  5. [He] G. Hector, Une nouvelle obstruction à l'intégrabilité des variétés de Poisson régulières, Hokkaido Math. J. 21 (1992), 159-185. Zbl0756.58017
  6. [H-H] G. Hector and U. Hirsh, Introduction to the Geometry of Foliations, Part A and B, Braunschweig, 1981, 1983. 
  7. [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. 
  8. [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. 
  9. [K3] J. Kubarski, Lie algebroid of a principal fibre bundle, Publ. Dep. Math. Université de Lyon 1, 1/A, 1989. 
  10. [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. Zbl0724.57021
  11. [K5] J. Kubarski, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dep. Math. Université de Lyon 1, 1991. 
  12. [K6] J. Kubarski, Fibre integral in regular Lie algebroids, in: New Developments in Differential Geometry (Budapest, 1996), Kluwer, 1999. 
  13. [K7] J. Kubarski, Euler class and Gysin sequence of spherical Lie algebroids, in preparation. Zbl0999.22003
  14. [K8] J. Kubarski, An analogue of the Euler-Poincaré-Hopf theorem in topology of some 3-dimensional Poisson manifolds. 
  15. [M] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, 1987. Zbl0683.53029
  16. [M2] K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97-147. Zbl0829.22001
  17. [M-S] C. C. Moore and C. Schochet, Global Analysis on Foliated Spaces, Math. Sci. Res. Inst. Publ. 9, Springer-Verlag, 1988. 
  18. [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. Zbl0154.21704
  19. [P2] J. Pradines, Théorie de Lie pour les groupoï des différentiables, Atti Conv. Intern. Geom. 7 Diff. Bologna, 1967, Bologna-Amsterdam. 
  20. [V1] I. Vaisman, Remarks on the Lichnerowicz-Poisson cohomology, Ann. Inst. Fourier Grenoble 40 (1990), 951-963. Zbl0708.58010
  21. [V2] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkhäuser Verlag, 1994. 

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