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On the natural operators transforming vector fields to the $$r$$-th tensor power. (English) Zbl 0820.58004
Bureš, J. (ed.) et al., The proceedings of the winter school geometry and topology, Srní, Czechoslovakia, January 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 15-20 (1994).
In 1988 the author [Ann. Global Anal. Geom. 6, No. 2, 109-117 (1988; Zbl 0678.58003)] studied the natural operators transforming vector fields on an $$m$$-dimensional manifold $$M$$ into vector fields on an arbitrary Weil $$m$$-manifold. The Weil bundles coincide with product preserving bundle functors on the smooth category. The characterization of natural operators transforming vector fields to the vector fields on the bundle functor of the $$r$$-th order tangent vectors, not product-preserving for $$r > 1$$, was recently obtained by W. M. Mikulski. In this paper another non-product-preserving functor is considered. All natural operators transforming vector fields on $$M$$ into vector fields on the $$r$$-th tensor power $$\bigotimes^ r TM$$ of this tangent bundle are determined. Also some aspects of this problem for natural subbundles of $$\bigotimes^ r TM$$ are considered.
For the entire collection see [Zbl 0794.00022].
##### MSC:
 58A20 Jets in global analysis 58A30 Vector distributions (subbundles of the tangent bundles)
##### Keywords:
natural operator; $$r$$-th tensor power; natural subbundle