The classifying Lie algebroid of a geometric structure I: Classes of coframes

Loja Fernandes, Rui; Struchiner, Ivan

doi:10.1090/S0002-9947-2014-05973-4

The classifying Lie algebroid of a geometric structure I: Classes of coframes
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by Rui Loja Fernandes and Ivan Struchiner PDF

Trans. Amer. Math. Soc. 366 (2014), 2419-2462 Request permission

Abstract:

We present a systematic study of symmetries, invariants and moduli spaces of classes of coframes. We introduce a classifying Lie algebroid to give a complete description of the solution to Cartan’s realization problem that applies to both the local and the global versions of this problem.

References

Robert L. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), no. 3, 623–715. MR 1824987, DOI 10.1090/S0894-0347-01-00366-6
Ana Cannas da Silva and Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, vol. 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999. MR 1747916
Élie Cartan, Sur la structure des groupes infinis de transformation, Ann. Sci. École Norm. Sup. (3) 21 (1904), 153–206 (French). MR 1509040
Marius Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721. MR 2016690, DOI 10.1007/s00014-001-0766-9
M. Crainic and R. L. Fernandes, Secondary characteristic classes of Lie algebroids, Quantum field theory and noncommutative geometry, Lecture Notes in Phys., vol. 662, Springer, Berlin, 2005, pp. 157–176. MR 2179182, DOI 10.1007/11342786_{9}
—, Lectures on integrability of Lie brackets, Geometry & Topology Monographs 17 (2010), 1–94.
Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. MR 1973056, DOI 10.4007/annals.2003.157.575
Sam Evens, Jiang-Hua Lu, and Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436. MR 1726784, DOI 10.1093/qjmath/50.200.417
R. L. Fernandes and I. Struchiner, The classifying Lie algebroid of a geometric structure II: $G$-structures and examples, preprint in preperation.
Rui Loja Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), no. 1, 119–179. MR 1929305, DOI 10.1006/aima.2001.2070
Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886
Y. Kosmann-Schwarzbach, C. Laurent-Gengoux, and A. Weinstein, Modular classes of Lie algebroid morphisms, Transform. Groups 13 (2008), no. 3-4, 727–755. MR 2452613, DOI 10.1007/s00031-008-9032-y
Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276, DOI 10.1017/CBO9780511609565
Peter J. Olver, Non-associative local Lie groups, J. Lie Theory 6 (1996), no. 1, 23–51. MR 1406004
Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957), iii+123. MR 121424
Proceedings of the “Rencontres Mathématiques de Glanon”, Symplectic connections via integration of Poisson structures, 2002.
R. W. Sharpe, Differential geometry, Graduate Texts in Mathematics, vol. 166, Springer-Verlag, New York, 1997. Cartan’s generalization of Klein’s Erlangen program; With a foreword by S. S. Chern. MR 1453120
I. M. Singer and Shlomo Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1–114. MR 217822, DOI 10.1007/BF02787690
Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
I. Struchiner, The classifying Lie algebroid of a geometric structure, Ph.D. thesis, University of Campinas - UNICAMP, 2009.