We introduce an exchange natural isomorphism between iterated higher order jet functors depending on a classical linear connection on the base manifold. As an application we study the prolongation of higher order connections to jet bundles.
We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.
We classify all bundle functors $G$ admitting natural operators transforming connections on a fibered manifold $Y\rightarrow M$ into connections on $GY\rightarrow M$. Then we solve a similar problem for natural operators transforming connections on $Y\rightarrow M$ into connections on $GY\rightarrow Y$.
For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with $m$-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors.
By a torsion of a general connection $\Gamma $ on a fibered manifold $Y\rightarrow M$ we understand the Frölicher-Nijenhuis bracket of $\Gamma $ and some canonical tangent valued one-form (affinor) on $Y$. Using all natural affinors on higher order cotangent bundles, we determine all torsions of general connections on such bundles. We present the geometrical interpretation and study some properties of the torsions.