# Connections in regular Poisson manifolds over ℝ-Lie foliations

Banach Center Publications (2000)

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

## Access Full Article

top## Abstract

top## How to cite

topKubarski, 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

top- [A] M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207. Zbl0078.16002
- [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. Zbl0756.58017
- [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. Zbl0724.57021
- [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. Zbl0999.22003
- [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. Zbl0683.53029
- [M2] K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97-147. Zbl0829.22001
- [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. Zbl0154.21704
- [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. Zbl0708.58010
- [V2] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkhäuser Verlag, 1994.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.