| Title:
|
The contact system on the $(m, \ell )$-jet spaces (English) |
| Author:
|
Muñoz, J. |
| Author:
|
Muriel, F. J. |
| Author:
|
Rodríguez, J. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
37 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
291-300 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper is a continuation of [MMR:98], where we give a construction of the canonical Pfaff system $\Omega (M_m^\ell )$ on the space of $(m,\ell )$-velocities of a smooth manifold $M$. Here we show that the characteristic system of $\Omega (M_m^\ell )$ agrees with the Lie algebra of $\operatorname{Aut}({\mathbb R}_m^\ell )$, the structure group of the principal fibre bundle ${\check{M}}_m^\ell \longrightarrow J_m^\ell (M)$, hence it is projectable to an irreducible contact system on the space of $(m,\ell )$-jets ($=\ell $-th order contact elements of dimension $m$) of $M$. Furthermore, we translate to the language of Weil bundles the structure form of jet fibre bundles defined by Goldschmidt and Sternberg in [Gol:Ste:73]. (English) |
| Keyword:
|
near points |
| Keyword:
|
jets |
| Keyword:
|
contact elements |
| Keyword:
|
contact system |
| Keyword:
|
velocities |
| MSC:
|
58A17 |
| MSC:
|
58A20 |
| idZBL:
|
Zbl 1090.58006 |
| idMR:
|
MR1879452 |
| . |
| Date available:
|
2008-06-06T22:29:14Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107807 |
| . |
| Reference:
|
[1] Ehresmann C.: Introduction à la théorie des structures infinitesimales et des pseudo-groupes de Lie.Colloque de Géometrie Différentielle, C.N.R.S. (1953), 97–110. MR 0063123 |
| Reference:
|
[2] Goldschmidt H., Sternberg S.: The Hamilton–Cartan formalism in the calculus of variations.Ann. Inst. Fourier (Grenoble) 23 (1973), 203–267. Zbl 0243.49011, MR 0341531 |
| Reference:
|
[3] Grigore D. R., Krupka D.: Invariants of velocities and higher order Grassmann bundles.J. Geom. Phys. 24 (1998), 244–264. Zbl 0898.53013, MR 1491556 |
| Reference:
|
[4] Jacobson N.: Lie algebras.John Wiley & Sons, Inc., New York, 1962. Zbl 0121.27504, MR 0143793 |
| Reference:
|
[5] Kolář I., Michor P. W., Slovák J.: Natural operations in differential geometry.Springer-Verlag, New York, 1993. Zbl 0782.53013, MR 1202431 |
| Reference:
|
[6] Morimoto A.: Prolongation of connections to bundles of infinitely near points.J. Differential Geom. 11 (1976), 479–498. Zbl 0358.53013, MR 0445422 |
| Reference:
|
[7] Muñoz J., Muriel F. J., Rodríguez J.: Weil bundles and jet spaces.Czechoslovak Math. J. 50 (125) (2000), 721–748. Zbl 1079.58500, MR 1792967 |
| Reference:
|
[8] Muñoz J., Muriel F. J., Rodríguez J.: The contact system on the spaces of $(m,\ell )$-velocities.Proceedings of the 7th International Conference Differential Geometry and Applications (Brno, 1998) (1999), 263–272. |
| Reference:
|
[9] Weil A.,: Théorie des points proches sur les variétés différentiables.Colloque de Géometrie Différentielle, C.N.R.S. (1953), 111–117. Zbl 0053.24903, MR 0061455 |
| . |
