About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Hammerl, Matthias
Homogeneous Cartan geometries. (English). Archivum Mathematicum, vol. 43 (2007), issue 5, pp. 431-442
MSC: 53A30, 53B15, 53C29, 53C30, 53Cxx | MR 2381786 | Zbl 1199.53021
Keywords:
Cartan geometry; homogeneous space; infinitesimal automorphism; holonomy; conformal geometry
Summary:
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.
References:
[1] Ambrose W., Singer I. M.: A theorem on holonomy. Trans. Amer. Math. Soc. 75 (1953), 428–443. MR 0063739 | Zbl 0052.18002
[2] Armstrong S.: Definite signature conformal holonomy: a complete classification. 2005. math.DG/0503388.
[3] Čap A.: Infinitesimal automorphisms and deformations of parabolic geometries. 2005, to appear in J. Europ. Math. Soc. math.DG/0508535. MR 2390330 | Zbl 1161.32020
[4] Čap A.: On left invariant CR structures on SU(2). 2006. math.DG/0603730. MR 2322406 | Zbl 1164.32304
[5] Čap A., Schichl H.: Parabolic geometries and canonical Cartan connections. Hokkaido Math. J. 29(3) (2000), 453–505. MR 1795487 | Zbl 0996.53023
[6] Čap A., and Slovák J.: Parabolic Geometries. Book in preparation.
[7] Cartan É.: Les espaces à connexion conforme. Ann. Soc. Pol. Math. (2) (1923), 172–202.
[8] Hammerl M.: Homogeneous Cartan geometries. Diploma thesis, 2006. http://www.mat.univie.ac.at/~cap/files/Hammerl.pdf MR 2381786 | Zbl 1199.53021
[9] Leitner F.: Conformal holonomy of bi-invariant metrics. 2004. math.DG/0406299. Zbl 1138.53041
[10] Michor P.: Topics in Differential Geometry. Book in preparation. http://www.mat.univie.ac.at/~michor/dgbook.ps MR 2428390 | Zbl 1175.53002
[11] Tanaka N.: On the equivalence problem associated with simple graded Lie algebras. Hokkaido Math. J. (1979), 23–84. MR 0533089
[12] Wang H.-Ch.: On invariant connections over a principal fibre bundle. Nagoya Math. J. 13 (1958), 1–19. MR 0107276 | Zbl 0086.36502
[13] Yamaguchi K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. (1993), 413–494. MR 1274961 | Zbl 0812.17018
Partner of
EuDML logo