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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 $L^r_0, L^r_1, \dots, L^r_n$. \par The exposition is clear and well organized. (English) |
