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Some natural operations on vector fields. (English) Zbl 0766.58005
For every bundle functor $$H$$, the author establishes a bijection between the fibre $$H_ 0R$$ of $$H$$ over $$0\in R$$ and the set $$\text{Trans}(H\mid{\mathcal M}_ n)$$ of all natural transformations which transform vector fields into sections of $$H$$.
A classification theorem describes explicitly the set $$\text{Trans}_{Ex}(T^ r\mid{\mathcal M}_ n)$$ of all natural base- extending transformations transforming vector fields on $$n$$-manifolds into vector fields on the linear taangent bundle functor $$T^ r\mid{\mathcal M}_ n$$ of order $$r$$ over $$n$$-manifolds $$(n\geq 2)$$, generalizing thus the similar results obtained for $$r=1$$ and $$r=2$$ respectively, by M. Sekizawa and M. Doupovec.

##### MSC:
 58A20 Jets in global analysis 53A55 Differential invariants (local theory), geometric objects