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Abstract

We study the problem of how a map $f:M\to {{\mathbb R}}$ on an $n$-manifold $M$ induces canonically an affinor $A(f):TT^{(r)}M\to TT^{(r)}M$ on the vector $r$-tangent bundle $T^{(r)}M=(J^r(M,{{\mathbb R}})_0)^*$ over $M$. This problem is reflected in the concept of natural operators $A:T^{(0,0)}_{| {\cal M} f_n} \rightsquigarrow T^{(1,1)}T^{(r)}$. For integers $r\geq 1$ and $n\geq 2$ we prove that the space of all such operators is a free $(r+1)^2$-dimensional module over ${\cal C}^\infty (T^{(r)}{{\mathbb R}})$ and we construct explicitly a basis of this module. \par