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Homogeneous Cartan geometries

Matthias Hammerl

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 5, page 431-442
  • ISSN: 0044-8753

Abstract

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We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.

How to cite

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Hammerl, Matthias. "Homogeneous Cartan geometries." Archivum Mathematicum 043.5 (2007): 431-442. <http://eudml.org/doc/250150>.

@article{Hammerl2007,
abstract = {We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.},
author = {Hammerl, Matthias},
journal = {Archivum Mathematicum},
keywords = {Cartan geometry; homogeneous space; infinitesimal automorphism; holonomy; conformal geometry; Cartan geometry; homogeneous space; infinitesimal automorphism; holonomy; conformal geometry},
language = {eng},
number = {5},
pages = {431-442},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Homogeneous Cartan geometries},
url = {http://eudml.org/doc/250150},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Hammerl, Matthias
TI - Homogeneous Cartan geometries
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 5
SP - 431
EP - 442
AB - We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.
LA - eng
KW - Cartan geometry; homogeneous space; infinitesimal automorphism; holonomy; conformal geometry; Cartan geometry; homogeneous space; infinitesimal automorphism; holonomy; conformal geometry
UR - http://eudml.org/doc/250150
ER -

References

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  8. Hammerl M., Homogeneous Cartan geometries, Diploma thesis, 2006. http://www.mat.univie.ac.at/~cap/files/Hammerl.pdf. Zbl1199.53021MR2381786
  9. Leitner F., Conformal holonomy of bi-invariant metrics, 2004. math.DG/0406299. Zbl1138.53041
  10. Michor P., Topics in Differential Geometry, Book in preparation. http://www.mat.univie.ac.at/~michor/dgbook.ps. Zbl1175.53002MR2428390
  11. Tanaka N., On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J. (1979), 23–84. (1979) MR0533089
  12. Wang H.-Ch., On invariant connections over a principal fibre bundle, Nagoya Math. J. 13 (1958), 1–19. (1958) Zbl0086.36502MR0107276
  13. Yamaguchi K., Differential systems associated with simple graded Lie algebras, Adv. Stud. Pure Math. (1993), 413–494. (1993) Zbl0812.17018MR1274961

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