# Article

Full entry | PDF   (0.4 MB)
Keywords:
near points; jets; contact elements; contact system; velocities
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].
References:
[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
[2] Goldschmidt H., Sternberg S.: The Hamilton–Cartan formalism in the calculus of variations. Ann. Inst. Fourier (Grenoble) 23 (1973), 203–267. MR 0341531 | Zbl 0243.49011
[3] Grigore D. R., Krupka D.: Invariants of velocities and higher order Grassmann bundles. J. Geom. Phys. 24 (1998), 244–264. MR 1491556 | Zbl 0898.53013
[4] Jacobson N.: Lie algebras. John Wiley & Sons, Inc., New York, 1962. MR 0143793 | Zbl 0121.27504
[5] Kolář I., Michor P. W., Slovák J.: Natural operations in differential geometry. Springer-Verlag, New York, 1993. MR 1202431 | Zbl 0782.53013
[6] Morimoto A.: Prolongation of connections to bundles of infinitely near points. J. Differential Geom. 11 (1976), 479–498. MR 0445422 | Zbl 0358.53013
[7] Muñoz J., Muriel F. J., Rodríguez J.: Weil bundles and jet spaces. Czechoslovak Math. J. 50 (125) (2000), 721–748. MR 1792967 | Zbl 1079.58500
[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.
[9] Weil A.,: Théorie des points proches sur les variétés différentiables. Colloque de Géometrie Différentielle, C.N.R.S. (1953), 111–117. MR 0061455 | Zbl 0053.24903

Partner of