The natural operators lifting horizontal $1$-forms to some vector bundle functors on fibered manifolds
Volume 97 / 2003
Abstract
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.
