| Title:
|
Homogeneous Cartan geometries (English) |
| Author:
|
Hammerl, Matthias |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
43 |
| Issue:
|
5 |
| Year:
|
2007 |
| Pages:
|
431-442 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Cartan geometry |
| Keyword:
|
homogeneous space |
| Keyword:
|
infinitesimal automorphism |
| Keyword:
|
holonomy |
| Keyword:
|
conformal geometry |
| MSC:
|
53A30 |
| MSC:
|
53B15 |
| MSC:
|
53C29 |
| MSC:
|
53C30 |
| MSC:
|
53Cxx |
| idZBL:
|
Zbl 1199.53021 |
| idMR:
|
MR2381786 |
| . |
| Date available:
|
2008-06-06T22:52:05Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108082 |
| . |
| Reference:
|
[1] Ambrose W., Singer I. M.: A theorem on holonomy.Trans. Amer. Math. Soc. 75 (1953), 428–443. Zbl 0052.18002, MR 0063739 |
| Reference:
|
[2] Armstrong S.: Definite signature conformal holonomy: a complete classification.2005. math.DG/0503388. |
| Reference:
|
[3] Čap A.: Infinitesimal automorphisms and deformations of parabolic geometries.2005, to appear in J. Europ. Math. Soc. math.DG/0508535. Zbl 1161.32020, MR 2390330 |
| Reference:
|
[4] Čap A.: On left invariant CR structures on SU(2).2006. math.DG/0603730. Zbl 1164.32304, MR 2322406 |
| Reference:
|
[5] Čap A., Schichl H.: Parabolic geometries and canonical Cartan connections.Hokkaido Math. J. 29(3) (2000), 453–505. Zbl 0996.53023, MR 1795487 |
| Reference:
|
[6] Čap A., and Slovák J.: Parabolic Geometries.Book in preparation. |
| Reference:
|
[7] Cartan É.: Les espaces à connexion conforme.Ann. Soc. Pol. Math. (2) (1923), 172–202. |
| Reference:
|
[8] Hammerl M.: Homogeneous Cartan geometries.Diploma thesis, 2006. http://www.mat.univie.ac.at/~cap/files/Hammerl.pdf. Zbl 1199.53021, MR 2381786 |
| Reference:
|
[9] Leitner F.: Conformal holonomy of bi-invariant metrics.2004. math.DG/0406299. Zbl 1138.53041 |
| Reference:
|
[10] Michor P.: Topics in Differential Geometry.Book in preparation. http://www.mat.univie.ac.at/~michor/dgbook.ps. Zbl 1175.53002, MR 2428390 |
| Reference:
|
[11] Tanaka N.: On the equivalence problem associated with simple graded Lie algebras.Hokkaido Math. J. (1979), 23–84. MR 0533089 |
| Reference:
|
[12] Wang H.-Ch.: On invariant connections over a principal fibre bundle.Nagoya Math. J. 13 (1958), 1–19. Zbl 0086.36502, MR 0107276 |
| Reference:
|
[13] Yamaguchi K.: Differential systems associated with simple graded Lie algebras.Adv. Stud. Pure Math. (1993), 413–494. Zbl 0812.17018, MR 1274961 |
| . |
