We describe all bundle functors G admitting natural operators transforming rth order holonomic connections on a fibered manifold Y → M into rth order holonomic connections on GY → M. For second order holonomic connections we classify all such natural operators.

We extend the concept of r-order connections on fibred manifolds to the one of (r,s,q)-order projectable connections on fibred-fibred manifolds, where r,s,q are arbitrary non-negative integers with s ≥ r ≤ q. Similarly to the fibred manifold case, given a bundle functor F of order r on (m₁,m₂,n₁,n₂)-dimensional fibred-fibred manifolds Y → M, we construct a general connection ℱ(Γ,Λ):FY → J¹FY on FY → M from a projectable general (i.e. (1,1,1)-order) connection $\Gamma :Y\to J^{1,1,1}Y$ on Y → M by means of an (r,r,r)-order...

We deal with a $(1,1)$-tensor field $\alpha$ on the tangent bundle $TM$ preserving vertical vectors and such that $J\alpha =-\alpha J$ is a $(1,1)$-tensor field on $M$, where $J$ is the canonical almost tangent structure on $TM$. A connection ${\Gamma }_{\alpha }$ on $TM$ is constructed by $\alpha$. It is shown that if $\alpha$ is a $VB$-almost complex structure on $TM$ without torsion then ${\Gamma }_{\alpha }$ is a unique linear symmetric connection such that $\alpha \left({\Gamma }_{\alpha }\right)={\Gamma }_{\alpha }$ and ${\nabla }_{{\Gamma }_{\alpha }}\left(J\alpha \right)=0$.

Two significant directions in the development of jet calculus are showed. First, jets are generalized to so-called quasijets. Second, jets of foliated and multifoliated manifold morphisms are presented. Although the paper has mainly a survey character, it also includes new results: jets modulo multifoliations are introduced and their relation to (R,S,Q)-jets is demonstrated.

We deduce a classification of all special types of nonholonomic $3$-jets. In the introductory part, we summarize the basic properties of nonholonomic $r$-jets.

First we summarize some properties of the nonholonomic $r$-jets from the functorial point of view. In particular, we describe the basic properties of our original concept of nonholonomic $r$-jet category. Then we deduce certain properties of the Weil algebras associated with nonholonomic $r$-jets. Next we describe an algorithm for finding the nonholonomic $r$-jet categories. Finally we classify all special types of semiholonomic $3$-jets.

Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category ${\mathrm{ℱℳ}}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$ to be a $GL\left(m\right)$-invariant algebra homomorphism $\nu :A\to A^1$ with $t^1=\nu \circ t$. The main result is that there exists an ${\mathrm{ℱℳ}}_m$-natural transformation $FY\to F^{1}Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization)...

We classify all bundle functors $G$ admitting natural operators transforming connections on a fibered manifold $Y\to M$ into connections on $GY\to M$. Then we solve a similar problem for natural operators transforming connections on $Y\to M$ into connections on $GY\to Y$.

We present a complete description of all fiber product preserving gauge bundle functors F on the category ${}_m$ of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented.