| Title:
|
The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids (English) |
| Author:
|
Kubarski, Jan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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56 |
| Issue:
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2 |
| Year:
|
2006 |
| Pages:
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359-376 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
|
This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either $\mathbb{R}$ or $\mathop {\mathrm sl}(2,\mathbb{R})$ or $\operatorname{so} (3)$ 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 $\mathbb{R}$-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. (English) |
| Keyword:
|
Lie algebroid |
| Keyword:
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Euler class |
| Keyword:
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index theorem |
| Keyword:
|
integration over the fibre |
| Keyword:
|
flat connection with singularitity |
| MSC:
|
53D17 |
| MSC:
|
55R25 |
| MSC:
|
57R19 |
| MSC:
|
57R20 |
| MSC:
|
58H05 |
| idZBL:
|
Zbl 1164.57304 |
| idMR:
|
MR2291742 |
| . |
| Date available:
|
2009-09-24T11:33:45Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128072 |
| . |
| Reference:
|
[1] M. Atiyah: Complex analytic connections in fibre bundles.Trans. Amer. Math. Soc. 85 (1957), 181–207. Zbl 0078.16002, MR 0086359, 10.1090/S0002-9947-1957-0086359-5 |
| Reference:
|
[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. MR 1881644 |
| Reference:
|
[3] A. Coste, P. Dazord, and A. Weinstein: Groupoides Symplectiques.Publ. Dep. Math. Université de Lyon 1, 2/A, 1987. MR 0996653 |
| Reference:
|
[4] P. Dazord, D. Sondaz: Varietes de Poisson—Algebroides de Lie.Publ. Dep. Math. Université de Lyon 1, 1/B, 1988. MR 1040867 |
| Reference:
|
[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. MR 1726784 |
| Reference:
|
[6] J. Grabowski: Lie algebroids and Poisson-Nijenhuis structures.Reports on Mathematics Physics, Vol. 40 (1997). Zbl 1005.53061, MR 1614690 |
| Reference:
|
[7] J. Grabowski, P. Urbański: Algebroids—general differential calculi on vector bundles.J. Geom. Phys. 31 (1999), 111–141. MR 1706624, 10.1016/S0393-0440(99)00007-8 |
| Reference:
|
[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. MR 0400275 |
| Reference:
|
[9] J. C. Herz: Pseudo-algèbres de Lie I, II.C. R. Acad. Sci. Paris 263 (1953), 1935–1937, 2289–2291. Zbl 0050.03201 |
| Reference:
|
[10] Y. Kosmann-Schwarzbach: Exact Gerstenhaber Algebras and Lie Bialgebroids.Acta Applicandae Mathematicae 41 (1995), 153–165. Zbl 0837.17014, MR 1362125, 10.1007/BF00996111 |
| Reference:
|
[11] Y. Kosmann-Schwarzbach: The Lie Bialgebroid of a Poisson-Nijenhuis Manifold.Letters Mathematical Physics 38 (1996), 421–428. Zbl 1005.53060, MR 1421686, 10.1007/BF01815524 |
| Reference:
|
[12] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, Vol. I.Interscience Publishers, New York-London, 1963. MR 0152974 |
| Reference:
|
[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. MR 0946719 |
| Reference:
|
[14] J. Kubarski: Lie Algebroid of a Principal Fibre Bundle.Publ. Dep. Math. University de Lyon 1, 1/A, 1989. MR 1129261 |
| Reference:
|
[15] J. Kubarski: The Chern-Weil homomorphism of regular Lie algebroids.Publ. Dep. Math. University de Lyon 1, 1991. |
| Reference:
|
[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. Zbl 0766.17004, MR 1145691 |
| Reference:
|
[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. Zbl 0996.22005, MR 1414200 |
| Reference:
|
[18] J. Kubarski: Bott’s Vanishing Theorem for Regular Lie Algebroids.Transaction of the A.M.S. Vol. 348, June 1996. Zbl 0858.22009, MR 1357399 |
| Reference:
|
[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. Zbl 0959.58026, MR 1670494 |
| Reference:
|
[20] J. Kubarski: Connections in regular Poisson manifolds over $\mathbb{R}$-Lie foliations.In: Banach Center Publications, Vol. 51 “Poisson geometry”, , Warszawa, 2000, pp. 141–149. MR 1764441 |
| Reference:
|
[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. Zbl 0999.22003, MR 1874430 |
| Reference:
|
[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. MR 1881653 |
| Reference:
|
[23] J. Kubarski: Poincaré duality for transitive unimodular invariantly oriented Lie algebroids.In: Topology and Its Applications, Vol. 121, , , 2002, pp. 333–355. Zbl 1048.58014, MR 1908998 |
| Reference:
|
[24] J. Kubarski: The Euler-Poincaré-Hopf theorem for some regular $\mathbb{R}$-Lie algebroids.In: Proceedings of the Ist Colloquium on Lie Theory and Applications, Universidad de Vigo, 2002, pp. 105–112. MR 1954765 |
| Reference:
|
[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]), , , , pp. . |
| Reference:
|
[26] A. Kumpera, D. C. Spencer: Lie equations, Vol. I: General theory.Annals of Math. Studies, No. 73, Princeton University Press, Princeton, 1972. MR 0380908 |
| Reference:
|
[27] P. Libermann: Pseudogroupes infinitésimaux attachés aux pseudogroupes de Lie.Bull. Soc. Math. France 87 (1959), 409–425. Zbl 0198.26801, MR 0123279, 10.24033/bsmf.1536 |
| Reference:
|
[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. MR 0356117 |
| Reference:
|
[29] K. Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry.Cambridge University Press, Cambridge, 1987. Zbl 0683.53029, MR 0896907 |
| Reference:
|
[30] K. Mackenzie: Lie algebroids and Lie pseudoalgebras.Bull. London Math. Soc. 27 (1995), 97–147. Zbl 0829.22001, MR 1325261, 10.1112/blms/27.2.97 |
| Reference:
|
[31] P. Molino: Etude des feuilletages transversalement complets et applications.Ann. Sci. Ecole Norm. Sup. 10 (1977), 289–307. Zbl 0368.57007, MR 0458446, 10.24033/asens.1328 |
| Reference:
|
[32] P. Molino: Riemannian Foliations. Progress in Mathematics, Vol. 73.Birkhäuser-Verlag, Boston-Basel, 1988. MR 0932463 |
| Reference:
|
[33] L. Maxim-Raileanu: Cohomology of Lie algebroids.An. Sti. Univ. “Al. I. Cuza” Iasi. Sect. I a Mat. XXII f2 (1976), 197–199. Zbl 0361.18013, MR 0438358 |
| Reference:
|
[34] Ngo Van Que: Du prolongement des espaces fibrés et des structures infinitésimales.(1967), Ann. Inst. Fourier, Grenoble, 157–223. |
| Reference:
|
[35] Ngo Van Que: Sur l’espace de prolongement différentiable.J. of. Diff. Geom. 2 (1968), 33–40. Zbl 0164.22604, MR 0235587, 10.4310/jdg/1214501135 |
| Reference:
|
[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. Zbl 0232.22028 |
| Reference:
|
[37] I. Vaisman: Complementary 2-forms of Poisson Structures.Compositio Mathematica 101 (1996), 55–75. Zbl 0853.58056, MR 1390832 |
| Reference:
|
[38] I. Vaisman: The $BV$-algebra of a Jacobi manifolds.Annales Polonici Mathematici 73 (2000), 275–290 (arXiv:math.DG/9904112). MR 1785692, 10.4064/ap-73-3-275-290 |
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