Czechoslovak Mathematical Journal

, Volume 56, Issue 2, pp 359–376 | Cite as

The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids

  • Jan Kubarski


This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections) saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given.


Lie algebroid Euler class index theorem integration over the fibre flat connection with singularitity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Atiyah: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), 181–207.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    B. Balcerzak, J. Kubarski, and Walas: Primary characteristic homomorphism of pairs of Lie algebroids and Mackenzie algebroid. In: Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publications, Volume 54. Institute of Mathematics, Polish Academy of Science, Warszawa, 2001, pp. 135–173.Google Scholar
  3. [3]
    A. Coste, P. Dazord, and A. Weinstein: Groupoides Symplectiques. Publ. Dep. Math. Université de Lyon 1, 2/A, 1987.Google Scholar
  4. [4]
    P. Dazord, D. Sondaz: Varietes de Poisson—Algebroides de Lie. Publ. Dep. Math. Université de Lyon 1, 1/B, 1988.Google Scholar
  5. [5]
    S. Evens, J.-H. Lu, A. Weinstein: Transverse measures, the modular class, and a cohomology pairing for Lie algebroids. Quarterly J. Math. (1999), 17–434.Google Scholar
  6. [6]
    J. Grabowski: Lie algebroids and Poisson-Nijenhuis structures. Reports on Mathematics Physics, Vol. 40 (1997).Google Scholar
  7. [7]
    J. Grabowski, P. Urbański: Algebroids—general differential calculi on vector bundles. J. Geom. Phys. 31 (1999), 111–141.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    W. Greub, S. Halperin, and R. Vanstone: Connections, Curvature, and Cohomology. Pure and Aplied Mathematics 47, 47-II, 47-III. Academic Press, New York-London, 1971, 1973, 1976.MATHGoogle Scholar
  9. [9]
    J. C. Herz: Pseudo-algèbres de Lie I, II. C. R. Acad. Sci. Paris 263 (1953), 1935–1937, 2289–2291.MathSciNetGoogle Scholar
  10. [10]
    Y. Kosmann-Schwarzbach: Exact Gerstenhaber Algebras and Lie Bialgebroids. Acta Applicandae Mathematicae 41 (1995), 153–165.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Y. Kosmann-Schwarzbach: The Lie Bialgebroid of a Poisson-Nijenhuis Manifold. Letters Mathematical Physics 38 (1996), 421–428.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, Vol. I. Interscience Publishers, New York-London, 1963.Google Scholar
  13. [13]
    J. Kubarski: Pradines-type groupoids over foliations; cohomology, connections and the Chern-Weil homomorphism. Preprint Nr 2. Institute of Mathematics, Technical University of Łódź, August 1986.Google Scholar
  14. [14]
    J. Kubarski: Lie Algebroid of a Principal Fibre Bundle. Publ. Dep. Math. University de Lyon 1, 1/A, 1989.Google Scholar
  15. [15]
    J. Kubarski: The Chern-Weil homomorphism of regular Lie algebroids. Publ. Dep. Math. University de Lyon 1, 1991.Google Scholar
  16. [16]
    J. Kubarski: A criterion for the minimal closedness of the Lie subalgebra corresponding to a connected nonclosed Lie subgroup. Rev. Math. Complut. 4 (1991), 159–176.MATHMathSciNetGoogle Scholar
  17. [17]
    J. Kubarski: Invariant cohomology of regular Lie algebroids. Proceedings of the VIIth International Colloquium on Differential Geometry, Spain, July 26–30, 1994. World Scientific, Singapure, 1995, pp. 137–151.Google Scholar
  18. [18]
    J. Kubarski: Bott’s Vanishing Theorem for Regular Lie Algebroids. Transaction of the A.M.S. Vol. 348, June 1996.Google Scholar
  19. [19]
    J. Kubarski: Fibre integral in regular Lie algebroids. In: New Developments in Differential Geometry, Proceedings of the Conference on Differential Geometry, Budapest, Hungary, July 27–30, 1996. Kluwer Academic Publishers, 1999, pp. 173–202.Google Scholar
  20. [20]
    J. Kubarski: Connections in regular Poisson manifolds over ℝ-Lie foliations. In: Banach Center Publications, Vol. 51 “Poisson geometry”. Warszawa, 2000, pp. 141–149.MATHMathSciNetGoogle Scholar
  21. [21]
    J. Kubarski: Gysin sequence and Euler class of spherical Lie algebroids. In: Publicationes Mathematicae Debrecen, Tomus 59. vol. 3–4, 2001, pp. 245–269.MathSciNetGoogle Scholar
  22. [22]
    J. Kubarski: Weil algebra and secondary characteristic homomorphism of regular Lie algebroids. In: Lie algebroids and related topics in differential geometry, Banach Center Publications, Vol. 51. Inst. of Math. Polish Academy of Sciences, Warszawa, 2001, pp. 135–173.Google Scholar
  23. [23]
    J. Kubarski: Poincaré duality for transitive unimodular invariantly oriented Lie algebroids. In: Topology and Its Applications, Vol. 121. 2002, pp. 333–355.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    J. Kubarski: The Euler-Poincaré-Hopf theorem for some regular ℝ-Lie algebroids. In: Proceedings of the Ist Colloquium on Lie Theory and Applications. Universidad de Vigo, 2002, pp. 105–112.Google Scholar
  25. [25]
    A. Kumpera: An introduction to Lie groupoids. Duplicated notes, Núcleo de Estudos e Pesquiese Cientificas, Rio de Janeiro, 1971. (The greater part of these Notes appear as the Appendix of [26]).Google Scholar
  26. [26]
    A. Kumpera, D. C. Spencer: Lie equations, Vol. I: General theory. Annals of Math. Studies, No. 73. Princeton University Press, Princeton, 1972.Google Scholar
  27. [27]
    P. Libermann: Pseudogroupes infinitésimaux attachés aux pseudogroupes de Lie. Bull. Soc. Math. France 87 (1959), 409–425.MATHMathSciNetGoogle Scholar
  28. [28]
    P. Libermann: Sur les prolongements des fibrés principaux et groupoides différentiables banachiques. In: Seminaire de mathématiques superieures—élé 1969, Analyse Globale. Les presses de l’Université de Montréal, Montréal, 1971, pp. 7–108.Google Scholar
  29. [29]
    K. Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry. Cambridge University Press, Cambridge, 1987.MATHGoogle Scholar
  30. [30]
    K. Mackenzie: Lie algebroids and Lie pseudoalgebras. Bull. London Math. Soc. 27 (1995), 97–147.MATHMathSciNetGoogle Scholar
  31. [31]
    P. Molino: Etude des feuilletages transversalement complets et applications. Ann. Sci. Ecole Norm. Sup. 10 (1977), 289–307.MATHMathSciNetGoogle Scholar
  32. [32]
    P. Molino: Riemannian Foliations. Progress in Mathematics, Vol. 73. Birkhüser-Verlag, Boston-Basel, 1988.MATHGoogle Scholar
  33. [33]
    L. Maxim-Raileanu: Cohomology of Lie algebroids. An. Sti. Univ. “Al. I. Cuza” Iasi. Sect. I a Mat. XXII f2 (1976), 197–199.MathSciNetGoogle Scholar
  34. [34]
    Ngo Van Que: Du prolongement des espaces fibrés et des structures infinitésimales. Ann. Inst. Fourier, Grenoble.Google Scholar
  35. [35]
    Ngo Van Que: Sur l’espace de prolongement différentiable. J. of. Diff. Geom. 2 (1968), 33–40.MATHGoogle Scholar
  36. [36]
    J. Pradines: Théorie de Lie pour les groupoides differentiables. In: Atti del Convegno Internazionale di Geometria Differenziale, Bologna, September 28–30, 1967. 1970, pp. 233–236.Google Scholar
  37. [37]
    I. Vaisman: Complementary 2-forms of Poisson Structures. Compositio Mathematica 101 (1996), 55–75.MATHMathSciNetGoogle Scholar
  38. [38]
    I. Vaisman: The BV-algebra of a Jacobi manifolds. Annales Polonici Mathematici 73 (2000), 275–290 (arXiv:math.DG/9904112).MATHMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2006

Authors and Affiliations

  • Jan Kubarski
    • 1
  1. 1.Institute of MathematicsTechnical University of ŁódźŁódźPoland

Personalised recommendations