| Title:
|
Total connections in Lie groupoids (English) |
| Author:
|
Virsik, George |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
31 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
183-200 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A total connection of order $r$ in a Lie groupoid $\Phi $ over $M$ is defined as a first order connections in the $(r-1)$-st jet prolongations of $\Phi $. A connection in the groupoid $\Phi $ together with a linear connection on its base, ie. in the groupoid $\Pi (M)$, give rise to a total connection of order $r$, which is called simple. It is shown that this simple connection is curvature-free iff the generating connections are. Also, an $r$-th order total connection in $\Phi $ defines a total reduction of the $r$-th prolongation of $\Phi $ to $\Phi \times \Pi (M)$. It is shown that when $r>2$ then this total reduction of a simple connection is holonomic iff the generating connections are curvature free and the one on $M$ also torsion-free. (English) |
| Keyword:
|
Lie groupoids |
| Keyword:
|
semi-holonomic jets |
| Keyword:
|
higher order connections |
| Keyword:
|
total connections |
| Keyword:
|
simple connections |
| MSC:
|
53C05 |
| MSC:
|
58A20 |
| idZBL:
|
Zbl 0841.53024 |
| idMR:
|
MR1368257 |
| . |
| Date available:
|
2008-06-06T21:28:51Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107539 |
| . |
| Reference:
|
[1] Ehresmann, C.: Extension du calcul des jets aux jets non holonomes.C.R.A.S. Paris 239 (1954), 1762–1764. Zbl 0057.15603, MR 0066734 |
| Reference:
|
[2] Ehresmann, C.: Sur les connexions d’ordre supérieur.Atti V$^\circ $ Cong. Un. Mat. Italiana, Pavia - Torino, 1956, 326–328. |
| Reference:
|
[3] Kolř, I.: Some higher order operations with connections.Czechoslovak Math. J. 24 (99) (1974), 311–330. MR 0356114 |
| Reference:
|
[4] Kolř, I.: A generalization of the torsion form.Čas. pěst. mat. 100 (1975), 284–290. MR 0383287 |
| Reference:
|
[5] Kolř, I.: Torsion-free connections on higher order frame bundles.(to appear). |
| Reference:
|
[6] Kolř, I., Michor, P. W., Slov k, J.: Natural Operations in Differential Geometry.Springer-Verlag, 1993. MR 1202431 |
| Reference:
|
[7] Kolř, I., Virsik, G.: Connections in first principal prolongations.(to appear). |
| Reference:
|
[8] Que, N.: Du prolongement des espaces fibrés et des structures infinitésimales.Ann. Inst. Fourier, 17, (1967), 157–223. Zbl 0157.28506, MR 0221416 |
| Reference:
|
[9] Virsik, J. (George): A generalized point of view to higher order connections on fibre bundles.Czechoslovak Math. J. 19 (94) (1969), 110–142. MR 0242187 |
| Reference:
|
[10] Virsik, J. (George): On the holonomity of higher order connections.Cahiers Top. Géom. Diff. 12 (1971), 197–212. MR 0305294 |
| Reference:
|
[11] Yuen P. C.: Higher order frames and linear connections.Cahiers Top. Géom. Diff. 12 (1971), 333–337. Zbl 0222.53033, MR 0307102 |
| . |
