The vertical prolongation of the projectable connections

Anna Bednarska

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 66 (2012), / Harvested from The Polish Digital Mathematics Library

Résumé

We prove that any first order $\mathcal {F}_2\mathcal {M}_{m_1,m_2,n_1,n_2}$-natural operator transforming projectable general connections on an $(m_1,m_2, n_1, n_2)$-dimensional fibred-fibred manifold $p = (p, p) : (p_Y : Y \rightarrow Y ) \rightarrow (p_M : M \rightarrow M)$ into general connections on the vertical prolongation $V Y \rightarrow M$ of $p: Y \rightarrow M$ is the restriction of the (rather well-known) vertical prolongation operator $\mathcal {V}$ lifting general connections $\overline{\Gamma }$ on a fibred manifold $Y \rightarrow M$ into $\mathcal {V}\overline{\Gamma }$ (the vertical prolongation of $\overline{\Gamma }$) on $V Y \rightarrow M$.

Détail

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:289820

Bibliographie

Doupovec, M., Mikulski, W. M., On the existence of prolongation of connections, Czechoslovak Math. J., 56 (2006), 1323-1334.

Kolar, I., Connections on fibered squares, Ann. Univ. Mariae Curie-Skłodowska Sect. A 59 (2005), 67-76.

Kolar, I., Michor, P. W. and Slovak, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.

Kolar, I., Mikulski, W. M., Natural lifting of connections to vertical bundles, The Proceedings of the 19th Winter School “Geometry and Physics” (Srn´ı, 1999). Rend.

Circ. Mat. Palermo (2) Suppl. No. 63 (2000), 97-102.

Kurek, J., Mikulski, W. M., On prolongations of projectable connections, Ann. Polon. Math, 101 (2011), no. 3, 237-250.

Mikulski, W. M., The jet prolongations of fibered-fibered manifolds and the flow operator, Publ. Math. Debrecen 59 (2001), no. 3-4, 441-458.

Kolar, I., Some natural operations with connections, J. Nat. Acad. Math. India 5 (1987), no. 2, 127-141.