# Article

 Title: Non-existence of some canonical constructions on connections  (English) Author: Mikulski, W. M. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 44 Issue: 4 Year: 2003 Pages: 691-695 . Category: math . Summary: For a vector bundle functor $H:\Cal M f\to \Cal V\Cal B$ with the point property we prove that $H$ is product preserving if and only if for any $m$ and $n$ there is an $\Cal F\Cal M_{m,n}$-natural operator $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $p:Y\to M$ into connections $D(\Gamma)$ on $Hp:HY\to HM$. For a bundle functor $E:\Cal F\Cal M_{m,n}\to \Cal F\Cal M$ with some weak conditions we prove non-existence of $\Cal F\Cal M_{m,n}$-natural operators $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $Y\to M$ into connections $D(\Gamma)$ on $EY\to M$. Keyword: (general) connection Keyword: natural operator MSC: 53C05 MSC: 58A20 MSC: 58A32 idZBL: Zbl 1099.58004 idMR: MR2062885 . Date available: 2009-01-08T19:32:13Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119423 . Reference: [1] Doupovec M., Mikulski W.M: Horizontal extension of connections into $(2)$-connections.to appear. MR 2103898 Reference: [2] Kolář I.: On generalized connections.Beiträge Algebra Geom. 11 (1981), 29-34. MR 0680454 Reference: [3] Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry.Springer-Verlag, Berlin, 1993. MR 1202431 Reference: [4] Kolář I., Mikulski W.M.: Natural lifting of connections to vertical bundles.Suppl. Rend. Circolo Math. Palermo II 63 (2000), 97-102. MR 1758084 Reference: [5] Mikulski W.M.: Non-existence of a connection on $FY\to Y$ canonically dependent on a connection on $Y\to M$.Arch. Math. Brno, to appear. MR 2142138 Reference: [6] Slovák J.: Prolongations of connections and sprays with respect to Weil functors.Suppl. Rend. Circ. Mat. Palermo, Serie II 14 (1987), 143-155. MR 0920852 .

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