Previous |  Up |  Next


second order connection; non-holonomic 2-jet; absolute differentiation; bundle of smooth maps; connection on a functional bundle
We study the second order connections in the sense of C. Ehresmann. On a fibered manifold $Y$, such a connection is a section from $Y$ into the second non-holonomic jet prolongation of $Y$. Our main aim is to extend the classical theory to the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. This requires several new geometric results about the second order connections on $Y$, which are deduced in the first part of the paper.
[1] Cabras, A.: The Ricci identity for general connections. Proc. Conf. Diff. Geom. and Applications, 1995, Masaryk University, Brno 1996, 121-126. MR 1406331 | Zbl 0862.53023
[2] Cabras, A., Kolář, I.: Connections on some functional bundles. Czechoslovak Math. J., 45(1995), 529–548. MR 1344519
[3] Cabras, A., Kolář, I.: On the iterated absolute differentiation on some functional bundles. Arch. Math.(Brno), 33(1997), 23–35. MR 1464298
[4] Cabras, A., Kolář, I.: On the second order absolute differentiation. Rendiconti Circolo Matematico di Palermo, Serie II, Suppl. 59 (1999), 123–133. MR 1692263
[5] Cabras, A., Kolář, I.: The universal connection of an arbitrary system. Annali di Matematica, Vol. CLXXIV (1998), 1–11. MR 1746921
[6] Ehresmann, C.: Extension du calcul des jets aux jets non holonomes. CRAS Paris, 239(1954), 1762–1764. MR 0066734 | Zbl 0057.15603
[7] Ehresmann, C.: Sur les connexions d’ordre supérieur. Atti del V$^\circ $ Cong. dell’Unione Mat. Italiana, 1955, Roma Cremonese 1956, 344–346.
[8] Frölicher, A., Kriegl, A.: Linear spaces and differentiation theory. Pure and Applied Mathematics, J. Wiley, Chichester, 1988. MR 0961256
[9] Jadczyk, A., Janyška, J., Modugno, M.: Galilei general relativistic quantum mechanics revisited. “Geometria, Física-Matemática e outros Ensaios", Homenagem a António Ribeiro Gomes, A. S. Alves, F. J. Craveiro de Carvalho and J. A. Pereira da Silva Eds., Coimbra 1998, 253–313.
[10] Kolář, I.: On some operations with connections. Math. Nachrichten, 69(1975), 279–306. MR 0391157
[11] Kolář, I.: Higher order absolute differentiation with respect to generalized connections. Differential Geometry, Banach Center Publications 12(1984), 153–162. MR 0961078
[12] Kolář, I.: An infinite dimensional motivation in higher order geometry. Proc. Conf. Diff. Geom. and Applications 1995, Masaryk University, Brno 1996, 151–159. MR 1406335
[13] Kolář, I., Modugno, M.: The Frölicher-Nijenhuis bracket on some functional spaces. Ann. Polon. Math., LXVIII(1998), 97–106. MR 1610540
[14] Kolář, I., Virsik, G.: Connections in first principal prolongations. Rendiconti Circolo Matematico di Palermo, Serie II, Suppl. 43(1996), 163–171. MR 1463518
[15] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer-Verlag, 1993. MR 1202431
[16] Modugno, M.: Torsion and Ricci tensor for non-linear connections. Diff. Geom. and its Applications, 1(1991), 177–192. MR 1244443
[17] Pradines, J.: Représentation des jets non holonomes par les morphisms vectoriels doubles soudés. CRAS Paris, série A278, 1974, 1523–1526. MR 0388432
[18] Slovák, J.: Smooth structures of fibre jet spaces. Czechoslovak Math. J., 36(1986), 358–375. MR 0847766
[19] Virsik, J. (George): On the holonomity of higher order connections. Cahiers Topol. Géom. Diff., 12(1971), 197–212. MR 0305294
Partner of
EuDML logo