The contact system on the $(m, \ell )$-jet spaces

J. Muñoz; F. J. Muriel; Josemar Rodríguez

Archivum Mathematicum (2001)

  • Volume: 037, Issue: 4, page 291-300
  • ISSN: 0044-8753

Abstract

top
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].

How to cite

top

Muñoz, J., Muriel, F. J., and Rodríguez, Josemar. "The contact system on the $(m, \ell )$-jet spaces." Archivum Mathematicum 037.4 (2001): 291-300. <http://eudml.org/doc/248750>.

@article{Muñoz2001,
abstract = {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].},
author = {Muñoz, J., Muriel, F. J., Rodríguez, Josemar},
journal = {Archivum Mathematicum},
keywords = {near points; jets; contact elements; contact system; higher order velocities},
language = {eng},
number = {4},
pages = {291-300},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The contact system on the $(m, \ell )$-jet spaces},
url = {http://eudml.org/doc/248750},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Muñoz, J.
AU - Muriel, F. J.
AU - Rodríguez, Josemar
TI - The contact system on the $(m, \ell )$-jet spaces
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 4
SP - 291
EP - 300
AB - 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].
LA - eng
KW - near points; jets; contact elements; contact system; higher order velocities
UR - http://eudml.org/doc/248750
ER -

References

top
  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. (1953) MR0063123
  2. Goldschmidt H., Sternberg S., The Hamilton–Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble) 23 (1973), 203–267. (1973) Zbl0243.49011MR0341531
  3. Grigore D. R., Krupka D., Invariants of velocities and higher order Grassmann bundles, J. Geom. Phys. 24 (1998), 244–264. (1998) Zbl0898.53013MR1491556
  4. Jacobson N., Lie algebras, John Wiley & Sons, Inc., New York, 1962. (1962) Zbl0121.27504MR0143793
  5. Kolář I., Michor P. W., Slovák J., Natural operations in differential geometry, Springer-Verlag, New York, 1993. (1993) Zbl0782.53013MR1202431
  6. Morimoto A., Prolongation of connections to bundles of infinitely near points, J. Differential Geom. 11 (1976), 479–498. (1976) Zbl0358.53013MR0445422
  7. Muñoz J., Muriel F. J., Rodríguez J., Weil bundles and jet spaces, Czechoslovak Math. J. 50 (125) (2000), 721–748. Zbl1079.58500MR1792967
  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. (1998) 
  9. Weil A.,, Théorie des points proches sur les variétés différentiables, Colloque de Géometrie Différentielle, C.N.R.S. (1953), 111–117. (1953) Zbl0053.24903MR0061455

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.