# Article

Full entry | PDF   (0.3 MB)
Keywords:
non-holonomic jets and connections; semi-holonomic jets and connections; higher order relative; straight and Cartan connections
Summary:
A Cartan connection associated with a pair $P(M,G^{\prime })\subset P(M,G)$ is defined in the usual manner except that only the injectivity of $\omega :T(P^{\prime })\rightarrow T(G)_{e}$ is required. For an $r$-th order connection associated with a bundle morphism $\Phi :P^{\prime }\rightarrow P$ the concept of Cartan order $q\le r$ is defined, which for $q=r=1, \Phi :P^{\prime }\subset P$, and $\dim {M}=\dim {G/G^{\prime }}$ coincides with the classical definition. Results are obtained concerning the Cartan order of $r$-th order connections that are the product of $r$ first order (Cartan) connections.
References:
[1] Ehresmann C.: Extension du calcul des jets aux jets non holonomes. C.R.A.S. Paris 239(1954), 1762–1764. MR 0066734 | Zbl 0057.15603
[2] Ehresmann C.: Sur les connexions d’ordre supérieur. Atti $V^0$ Cong. Un. Mat. Italiana, Pavia-Torino, 1956, 326–328.
[3] Kobayashi S.: Transformation groups in differential geometry. Ergebnisse der Mathematik 70, Springer Verlag, 1972. MR 0355886 | Zbl 0829.53023
[4] Kobayashi S., Nomizu K.: Foundations of differential geometry, Vol. 1. Wiley-Interscience, 1963. MR 0152974
[5] Kolář I.: Some higher order operations with connections. Czech. Math. J. 24(99) (1974), 311–330. MR 0356114
[6] Kolář I.: On some operations with connections. Math. Nachrichten 69(1975), 297–306. MR 0391157
[7] Kolář I., Michor P. W., Slovák J.: Natural Operations in Differential Geometry. Springer-Verlag, 1993. MR 1202431
[8] Virsik G.: Total connections in Lie groupoids. Arch. Math. (Brno) 31 (1995), 183-200. MR 1368257 | Zbl 0841.53024
[9] Virsik G.: Bunch connections. Diff. Geom. and Applications, Proc. Conf. 1995, Brno, Czech republic, Masaryk University, Brno (1996), 215-229. MR 1406340 | Zbl 0864.53017

Partner of