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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) $$T^*|_{{\mathcal M}f_n}\rightsquigarrow T^*(\Lambda^rT^*)$$.

##### MSC:
 58A20 Jets in global analysis
##### Keywords:
bundle functor; natural operator
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