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Natural liftings of foliations to the $$r$$-tangent bundle. (English) Zbl 0848.57025
Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 153-159 (1994).
Let $$F$$ be a $$p$$-dimensional foliation on an $$n$$-manifold $$M$$, and $$T^r M$$ the $$r$$-tangent bundle of $$M$$. The purpose of this paper is to present some reltionship between the foliation $$F$$ and a natural lifting of $$F$$ to the bundle $$T^r M$$. Let $$L^r_q (F)$$ $$(q=0, 1, \dots, r)$$ be a foliation on $$T^r M$$ projectable onto $$F$$ and $$L^r_q= \{L^r_q (F)\}$$ a natural lifting of foliations to $$T^r M$$. The author proves the following theorem: Any natural lifting of foliations to the $$r$$-tangent bundle is equal to one of the liftings $$L^r_0, L^r_1, \dots, L^r_n$$.
The exposition is clear and well organized.
For the entire collection see [Zbl 0823.00015].
MSC:
 57R30 Foliations in differential topology; geometric theory 53C12 Foliations (differential geometric aspects) 53C10 $$G$$-structures