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The linear natural operators transforming affinors to tensor fields of type \((0,p)\) on Weil bundles. (English) Zbl 1029.58001

This paper analyzes a problem of differential geometry in the framework of natural bundles. The main reference for this theory is the book by I. Kolář, P. Michor and J. Slovák [Natural operations in differential geometry. Berlin: Springer-Verlag (corrected electronic version) (1993; Zbl 0782.53013)]. The problem considered in the paper is that of the classification of linear natural operators transforming affinors to tensor fields of type \((0,p)\) on Weil bundles. More precisely, let \(F\colon {\mathcal M}\to {\mathcal FM}\) be a product preserving bundle functor, and \(A=F(\mathbb{R})\) its Weil algebra. Then, the author proves that linear natural operators \(T^{(1,1)}_{ |{\mathcal M}_n} \to T^{(0,p)}F\) are of the form \(\Phi\mapsto (\text{tr}\Phi)^{(\lambda)}\) for \(p=0\), \(\Phi\mapsto d(\text{tr}\Phi)^{(\lambda)}\) for \(p=1\), and \(0\) for \(p\geq 2\). Here, \(\lambda\colon A\to \mathbb{R}\) is a linear map and \((\lambda)\) indicates a lift of functions defined by the same author in a previous paper.

MSC:

58A20 Jets in global analysis
53A55 Differential invariants (local theory), geometric objects

Citations:

Zbl 0782.53013
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