**DIPS 1/2006**[tex source, PostScript, PDF file, dvi]-
**Title:**Iterated Differential Forms I: Tensors.**Authors:**A. M. Vinogradov and L. VitaglianoThis note is the first in a series of short communications dedicated to general theory and some applications of iterated differential forms. Both are developed either in the "classical" context, or in the "quantistic" one, i.e., of Secondary Calculus. Detailed expositions containing proofs of the announced results will be appearing in due course.

With iterated forms we solve the problem of a*conceptual foundation of tensor calculus*. In particular, we show that covariant tensors are differential forms over a certain graded commutative algebra called the algebra of iterated differential forms. From one side, this interpretation extends noteworthy frames of the traditional tensor calculus and enriches it by numerous new natural operators. On the other side, it allows various generalizations of tensor calculus, the most important of which is that to secondary ("quantized") calculus. In particular, this leads to an unified solution of the secondarization ("quantization") problem for arbitrary tensors.

In this communication the algebra of iterated (differential) forms over an arbitrary (graded) commutative algebra is defined. It is also shown how tensors on a (smooth) manifold $M$ are naturally interpreted as iterated differential forms over the algebra C^{¥}(M).

See also arXiv: math.DG/0605113.

9 pages, LaTeX-2e.

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