Geodesic
Prolongation of second order connections to vertical Weil bundles
Archivum mathematicum, Tome 37 (2001) no. 4, pp. 333-347 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We study systematically the prolongation of second order connections in the sense of C. Ehresmann from a fibered manifold into its vertical bundle determined by a Weil algebra $A$. In certain situations we deduce new properties of the prolongation of first order connections. Our original tool is a general concept of a $B$-field for another Weil algebra $B$ and of its $A$-prolongation.
We study systematically the prolongation of second order connections in the sense of C. Ehresmann from a fibered manifold into its vertical bundle determined by a Weil algebra $A$. In certain situations we deduce new properties of the prolongation of first order connections. Our original tool is a general concept of a $B$-field for another Weil algebra $B$ and of its $A$-prolongation.
Classification : 53C05, 58A20, 58A32
Keywords: non-holonomic jet; Weil bundle; Weil field; second order connection; prolongation of connections 
@article{ARM_2001_37_4_a9,
     author = {Cabras, Antonella and Kol\'a\v{r}, Ivan},
     title = {Prolongation of second order connections to vertical {Weil} bundles},
     journal = {Archivum mathematicum},
     pages = {333--347},
     year = {2001},
     volume = {37},
     number = {4},
     mrnumber = {1879456},
     zbl = {1090.58003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2001_37_4_a9/}
}
TY  - JOUR
AU  - Cabras, Antonella
AU  - Kolář, Ivan
TI  - Prolongation of second order connections to vertical Weil bundles
JO  - Archivum mathematicum
PY  - 2001
SP  - 333
EP  - 347
VL  - 37
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2001_37_4_a9/
LA  - en
ID  - ARM_2001_37_4_a9
ER  - 
%0 Journal Article
%A Cabras, Antonella
%A Kolář, Ivan
%T Prolongation of second order connections to vertical Weil bundles
%J Archivum mathematicum
%D 2001
%P 333-347
%V 37
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2001_37_4_a9/
%G en
%F ARM_2001_37_4_a9
Cabras, Antonella; Kolář, Ivan. Prolongation of second order connections to vertical Weil bundles. Archivum mathematicum, Tome 37 (2001) no. 4, pp. 333-347. http://geodesic.mathdoc.fr/item/ARM_2001_37_4_a9/

[1] Cabras, A., Kolář, I.: Prolongation of tangent valued forms to Weil bundles. Arch. Math. (Brno) 31 (1995), 139–145. | MR

[2] Cabras, A., Kolář, I.: On the second order absolute differentiation. Suppl. Rendiconti Circolo Mat. Palermo, Serie II 59 (1999), 123–133. | MR

[3] Cabras, A., Kolář, I.: Second order connections on some functional bundles. Arch. Math. (Brno) 35 (1999), 347–365. | MR

[4] Cabras, A., Kolář, I.: Prolongation of projectable tangent valued forms. to appear in Rendiconti Palermo. | MR

[5] Doupovec M., Kolář, I.: Iteration of fiber product preerving bundle functors. to appear. | MR

[6] Ehresmann, C.: Extension du calcul des jets aux jets non holonomes. CRAS Paris 239 (1954), 1762–1764. | MR | Zbl

[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] Goldschmidt, H., Sternberg, S.: The Hamilton formalism in the calculus of variations. Ann. Inst. Fourier (Grenoble) 23 (1973), 203–267. | MR

[9] Kolář, I.: Higher order absolute differentiation with respect to generalized connections. Differential Geometry, Banach Center Publications 12 (1984), 153–162. | MR

[10] Kolář, I.: An infinite dimensional motivation in higher order geometry. Proc. Conf. Diff. Geom. and Applications 1995, Masaryk University, Brno 1996, 151–159. | MR

[11] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer--Verlag, 1993. | MR

[12] Kolář, I., Mikulski, W. M.: Natural lifting of connections to vertical bundles. Suppl. Rendiconti Circolo Mat. Palermo, Serie II 63 (2000), 97–102. | MR

[13] Libermann, P.: Introduction to the theory of semi-holonomic jets. Arch. Math. (Brno) 33 (1997), 173–189. | MR | Zbl

[14] Mangiarotti, L., Modugno, M.: Graded Lie algebras and connections on a fibered space. Journ. Math. Pures et Appl. 83 (1984), 111–120. | MR

[15] Pradines, J.: Représentation des jets non holonomes par les morphisms vectoriels doubles soudés. CRAS Paris, série A278 (1974), 1523–1526. | MR

[16] Tomáš, J.: On quasijet bundles. to appear in Rendiconti Palermo. | MR

[17] Virsik, J.: On the holonomity of higher order connections. Cahiers Topol. Géom. Diff. 12 (1971), 197–212. | MR | Zbl

[18] Weil, A.: Théorie des points proches sur les variétés différentielles. Colloque de topol. et géom. diff., Strasbourg (1953), 111-117. | MR