| Title:
|
A construction of a connection on $GY\to Y$ from a connection on $Y\to M$ by means of classical linear connections on $M$ and $Y$ (English) |
| Author:
|
Mikulski, W. M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
759-770 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a bundle functor of order $(r,s,q)$, $s\geq r\leq q$, on the category $\Cal F\Cal M_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma$ on an $\Cal F\Cal M_{m,n}$-object $Y\to M$ we construct a general connection $\Cal G(\Gamma,\lambda,\Lambda)$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda$ on $M$ and an $s$-th order linear connection $\Lambda$ on $Y$. Then we construct a general connection $\Cal G (\Gamma,\nabla_1,\nabla_2)$ on $GY\to Y$ by means of auxiliary classical linear connections $\nabla_1$ on $M$ and $\nabla_2$ on $Y$. In the case $G=J^1$ we determine all general connections $\Cal D(\Gamma,\nabla)$ on $J^1Y\to Y$ from general connections $\Gamma$ on $Y\to M$ by means of torsion free projectable classical linear connections $\nabla$ on $Y$. (English) |
| Keyword:
|
general connection |
| Keyword:
|
classical linear connection |
| Keyword:
|
bundle functor |
| Keyword:
|
natural operator |
| MSC:
|
58A05 |
| MSC:
|
58A20 |
| MSC:
|
58A32 |
| idZBL:
|
Zbl 1121.58001 |
| idMR:
|
MR2259506 |
| . |
| Date available:
|
2009-05-05T16:54:59Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119566 |
| . |
| Reference:
|
[1] Doupovec M., Mikulski W.M.: On the existence of prolongation of connections.Czechoslovak Math. J., to appear. Zbl 1164.58300, MR 2280811 |
| Reference:
|
[2] Janyška J., Modugno M.: Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibered manifold.Arch. Math. (Brno) 32 (1996), 281-288. MR 1441399 |
| Reference:
|
[3] Kolář I.: Prolongation of generalized connections.Colloq. Math. Soc. János Bolyai 31. Differential Geometry, Budapest (1979), 317-325. MR 0706928 |
| Reference:
|
[4] Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry.Springer, Berlin, 1993. MR 1202431 |
| Reference:
|
[5] Kolář I., Mikulski W.M.: Natural lifting of connections to vertical bundles.Rend. Circ. Math. Palermo (2), Suppl. no. 63 (2000), 97-102. MR 1758084 |
| Reference:
|
[6] Mikulski W.M.: Non-existence of natural operators transforming connections on $Y\to M$ into connections on $FY\to Y$.Arch. Math. (Brno) 41 1 (2005), 1-4. Zbl 1112.58006, MR 2142138 |
| Reference:
|
[7] Mikulski W.M.: The natural bundles admitting natural lifting of linear connections.Demonstratio Math., to appear. Zbl 1100.58001, MR 2223893 |
| Reference:
|
[8] Vondra A.: Higher-order differential equations represented by connections on prolongations of a fibered manifold.Extracta Math. 15 3 (2000), 421-512. Zbl 0992.34006, MR 1825970 |
| . |
