The natural operators lifting horizontal $1$-forms to some vector bundle functors on fibered manifolds

J. Kurek ¹   ; W. M. Mikulski ²

¹ Institute of Mathematics Maria Curie-Skłodowska University Pl. Marii Curie-Skłodowskiej 1 20-031 Lublin, Poland
² Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Colloquium Mathematicum, Tome 97 (2003) no. 2, pp. 141-149

Résumé

Let $F:{\mathcal F}{\mathcal M}\rightarrow {\mathcal V}{\mathcal B}$ be a vector bundle functor. First we classify all natural operators $T_{{\rm proj}| {\mathcal F}{\mathcal M}_{m,n}}\rightsquigarrow T^{(0,0)} (F_{| {\mathcal F}{\mathcal M}_{m,n}})^*$ transforming projectable vector fields on $Y$ to functions on the dual bundle $(FY)^*$ for any ${\mathcal F}{\mathcal M}_{m,n}$-object $Y$. Next, under some assumption on $F$ we study natural operators $T^*_{{\rm hor} | {\mathcal F}{\mathcal M}_{m,n}}\rightsquigarrow T^*(F_{| {\mathcal F}{\mathcal M}_{m,n}})^*$ lifting horizontal $1$-forms on $Y$ to $1$-forms on $(FY)^*$ for any $Y$ as above. As an application we classify natural operators $T^*_{{\rm hor}| {\mathcal F}{\mathcal M}_{m,n}}\rightsquigarrow T^*(F_{| {\mathcal F}{\mathcal M}_{m,n}})^*$ for some vector bundle functors $F$ on fibered manifolds.

Détail

DOI : 10.4064/cm97-2-1

Keywords: mathcal mathcal rightarrow mathcal mathcal vector bundle functor first classify natural operators proj mathcal mathcal rightsquigarrow mathcal mathcal * transforming projectable vector fields functions dual bundle * mathcal mathcal object under assumption study natural operators * hor mathcal mathcal rightsquigarrow * mathcal mathcal * lifting horizontal forms forms * above application classify natural operators * hor mathcal mathcal rightsquigarrow * mathcal mathcal * vector bundle functors fibered manifolds