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.


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