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The author studies the problem how a map $L:M\to\bbfR$ on an $n$-dimensional manifold $M$ can induce canonically a map $A_M(L):T^* T^{(r)}M\to \bbfR$ for $r$ a fixed natural number. He proves the following result: ``Let $A: T^{(0,0)}\to T^{(0,0)}(T^* T^{(r)})$ be a natural operator for $n$-manifolds. If $n\ge 3$ then there exists a uniquely determined smooth map $H: \bbfR^{S(r)}\times \bbfR\to\bbfR$ such that $A= A^{(H)}$.''\par The conclusion is that all natural functions on $T^* T^{(r)}$ for $n$-manifolds $(n\ge 3)$ are of the form $\{H\circ(\lambda^{\langle 0,1\rangle}_M,\dots, \lambda^{\langle r,0\rangle}_M)\}$, where $H\in C^\infty(\bbfR^r)$ is a function of $r$ variables. (English) |
