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 $EY\to M$.
@article{119423,
author = {W\l odzimierz M. Mikulski},
title = {Non-existence of some canonical constructions on connections},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {691-695},
zbl = {1099.58004},
mrnumber = {2062885},
language = {en},
url = {http://dml.mathdoc.fr/item/119423}
}
Mikulski, Włodzimierz M. Non-existence of some canonical constructions on connections. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 691-695. http://gdmltest.u-ga.fr/item/119423/
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