Higher order valued reduction theorems for general linear connections

Abstract

The reduction theorems for general linear and classical connections are  generalized for operators with values in higher order gauge-natural bundles.  We prove that natural operators depending on the $s_1$-jets of classical  connections, on the $s_2$-jets of general linear connections and on the  $r$-jets of tensor fields with values in gauge-natural bundles of order $k\ge 1$, $s_1+2\ge s_2$, $s_1,s_2\ge r-1\ge k-2$, can be factorized through the  $(k-2)$-jets of both connections, the $(k-1)$-jets of the tensor fields and  sufficiently high covariant differentials of the curvature tensors and the  tensor fields.