The contact system on the ( m , ℓ ) -jet spaces

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

Archivum Mathematicum (2001)

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

Abstract

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This paper is a continuation of [MMR:98], where we give a construction of the canonical Pfaff system Ω ( M m ℓ ) on the space of ( m , ℓ ) -velocities of a smooth manifold M . Here we show that the characteristic system of Ω ( M m ℓ ) agrees with the Lie algebra of Aut ( ℝ m ℓ ) , the structure group of the principal fibre bundle M ˇ m ℓ ⟶ J m ℓ ( M ) , hence it is projectable to an irreducible contact system on the space of ( m , ℓ ) -jets ( = ℓ -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

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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; velocities; 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; velocities; near points; jets; contact elements; contact system; higher order velocities
UR - http://eudml.org/doc/248750
ER -

References

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  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
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  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 , ℓ ) -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

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