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 S 1 ), 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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  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. 
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  21. [V2] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkhäuser Verlag, 1994. 

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