Lie algebroids and Cartan’s method of equivalence

Blaom, Anthony

doi:10.1090/S0002-9947-2012-05441-9

Lie algebroids and Cartan’s method of equivalence
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by Anthony D. Blaom

Trans. Amer. Math. Soc. 364 (2012), 3071-3135

DOI: https://doi.org/10.1090/S0002-9947-2012-05441-9

Published electronically: February 3, 2012

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Abstract:

Élie Cartan’s general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan’s method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) for objects of finite-type, and are able to interpret these directly as ‘infinitesimal symmetries deformed by curvature’.

Details are developed for transitive structures, but rudiments of the theory include intransitive structures (intransitive symmetry deformations). Detailed illustrations include subriemannian contact structures and conformal geometry.

References

Anthony D. Blaom, Geometric structures as deformed infinitesimal symmetries, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3651–3671. MR 2218993, DOI 10.1090/S0002-9947-06-04057-8
R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148, DOI 10.1007/978-1-4613-9714-4
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
Andreas Čap and A. Rod Gover, Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), no. 4, 1511–1548. MR 1873017, DOI 10.1090/S0002-9947-01-02909-9
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}
Michael Crampin, Cartan connections and Lie algebroids, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 061, 13. MR 2529186, DOI 10.3842/SIGMA.2009.061
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
Robert B. Gardner, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1062197, DOI 10.1137/1.9781611970135
W. K. Hughen. The sub-Riemannian geometry of three-manifolds. PhD thesis, Duke University, 1995.
Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, Providence, RI, 2003. MR 2003610, DOI 10.1090/gsm/061
Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886
Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005. MR 2157566, DOI 10.1017/CBO9781107325883
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994. A basic exposition of classical mechanical systems. MR 1304682, DOI 10.1007/978-1-4612-2682-6
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276, DOI 10.1017/CBO9780511609565
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
Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
Olle Stormark, Lie’s structural approach to PDE systems, Encyclopedia of Mathematics and its Applications, vol. 80, Cambridge University Press, Cambridge, 2000. MR 1771254, DOI 10.1017/CBO9780511569456
Xiang Tang, Deformation quantization of pseudo-symplectic (Poisson) groupoids, Geom. Funct. Anal. 16 (2006), no. 3, 731–766. MR 2238946, DOI 10.1007/s00039-006-0567-6