| Title:
|
On the Kolář connection (English) |
| Author:
|
Mikulski, Włodzimierz M. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
49 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
223-240 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $Y\rightarrow M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\rightarrow M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma $ on $Y\rightarrow M$, a torsion free classical linear connection $\Lambda $ on $M$, a vertical parallelism $\Phi \colon Y\times _ME\rightarrow VY$ on $Y$ and a linear connection $\Delta $ on $E\rightarrow M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář. (English) |
| Keyword:
|
general connection |
| Keyword:
|
linear connection |
| Keyword:
|
classical linear connection |
| Keyword:
|
vertical parallelism |
| Keyword:
|
natural operators |
| MSC:
|
53C05 |
| MSC:
|
58A32 |
| idZBL:
|
Zbl 1299.53071 |
| idMR:
|
MR3159312 |
| DOI:
|
10.5817/AM2013-4-223 |
| . |
| Date available:
|
2014-01-16T11:13:05Z |
| Last updated:
|
2015-09-09 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143548 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/144429 |
| . |
| Reference:
|
[1] Doupovec, M., Mikulski, W. M.: Reduction theorems for principal and classical connections.Acta Math. Sinica 26 (1) (2010), 169–184. Zbl 1186.53036, MR 2584996, 10.1007/s10114-010-7333-2 |
| Reference:
|
[2] Gancarzewicz, J.: Horizontal lifts of linear connections to the natural vector bundles.Differential geometry (Santiago de Compostela, 1984), vol. 131, Pitman, Boston, MA, 1985, pp. 318–341. |
| Reference:
|
[3] Janyška, J., Vondra, J.: Natural principal connections on the principal gauge prolongation of a principal bundle.Rep. Math. Phys. 64 (3) (2009), 395–415. Zbl 1195.53040, MR 2602937, 10.1016/S0034-4877(10)00002-9 |
| Reference:
|
[4] Kolář, I.: Induced connections on total spaces of fibred bundles.Int. J. Geom. Methods Mod. Phys. (2010), 705–711. MR 2669064, 10.1142/S021988781000452X |
| Reference:
|
[5] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry.Springer Verlag, 1993. |
| . |
