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The natural operators lifting \(1\)-forms to some vector bundle functors. (English) Zbl 1020.58003
For a vector bundle functor on the class of \(n\)-dimensional manifolds \(F:{\mathcal M}f_n\to{\mathcal{VB}}\) two classification problems are solved. The first is to describe all natural operators \(T\rightsquigarrow T^{0,0}F^*\) transforming vector fields on \(M\in{\mathcal M}f_n\) to functions on the dual to \(F(M)\). The second concerns the natural operators \(T^*\rightsquigarrow T^*F^*\) lifting 1-forms from \(M\) to \(F(M)\).
An application includes classification of following natural operators: 1) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*T_k^{r*}\); 2) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(\ker\{\pi_1^r:T^{r*}\to T^*\})\); 3) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(J^rT^*)\); 4) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(J^r(\otimes^pT^*))\); 5) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(J^r(\odot^pT^*))\); 6) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(J^r(\Lambda^pT^*))\); 7) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(\otimes^rT^*)\); 8) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(\odot^rT^*)\); 9) \(T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(\Lambda^rT^*)\).

58A20 Jets in global analysis
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