Let $F_1$ be a natural bundle of order $r_1$; a basis of the $s$-th order differential operators of $F_1$ with values in $r_2$-th order bundles is an operator $D$ of that type such that any other one is obtained by composing $D$ with a suitable zero-order operator. In this article a basis is found in the following two cases: for $F_1=\text{semi}{F}^{r_1}$ (semi-holonomic $r_1$-th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$-st order frame bundle), $r_2\le s$. The author uses here the so-called method of orbit reduction which provides one with a criterion for checking a basis in terms of the $K_n^{r_1+s,r_2}$-action on the type fiber of the concerned bundle, where $K_n^{r_1+s,r_2}$ denotes the kernel of the projection of the $(n,r_1+s)$ jet group onto the $(n,r_2)$ jet group [see I. Kolár, P. Michor and J. Slovák, ‘Natural operations in differential geometry’ (Springer-Verlag, Berlin) (1993; Zbl 0782.53013) or D. Krupka, Loc!

EUDML-ID : urn:eudml:doc:221792
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@article{701655,
     title = {Natural  operators on frame bundles},
     booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {2000},
     pages = {121-129},
     mrnumber = {MR1758087},
     zbl = {0971.58002},
     url = {http://dml.mathdoc.fr/item/701655}
}

Krupka, Michal. Natural  operators on frame bundles, dans Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books,  (2000), pp. 121-129. http://gdmltest.u-ga.fr/item/701655/