Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 061, 13 pages      arXiv:0906.2554      http://dx.doi.org/10.3842/SIGMA.2009.061
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Cartan Connections and Lie Algebroids

Michael Crampin
Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium
Address for correspondence: 65 Mount Pleasant, Aspley Guise, Beds MK17 8JX, UK

Received March 23, 2009, in final form June 07, 2009; Published online June 14, 2009

Abstract
This paper is a study of the relationship between two constructions associated with Cartan geometries, both of which involve Lie algebroids: the Cartan algebroid, due to [Blaom A.D., Trans. Amer. Math. Soc. 358 (2006), 3651–3671], and tractor calculus [Cap A., Gover A.R., Trans. Amer. Math. Soc. 354 (2001), 1511–1548].

Key words: adjoint tractor bundle; algebroid connection; algebroid representation; Cartan connection; Cartan geometry; Lie algebroid; tractor calculus.

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References

  1. Blaom A.D., Geometric structures as deformed infinitesimal symmetries, Trans. Amer. Math. Soc. 358 (2006), 3651-3671, math.DG/0404313.
  2. Cap A., Infinitesimal automorphisms and deformations of parabolic geometries, J. Eur. Math. Soc. (JEMS) 10 (2008), 415-437, math.DG/0508535.
  3. Cap A., Gover A.R., Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2001), 1511-1548.
  4. Sharpe R.W., Differential geometry. Cartan's generalization of Klein's Erlangen program, with a foreword by S.S. Chern, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.

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