For a vector bundle functor $H:ℳf\to 𝒱ℬ$ with the point property we prove that $H$ is product preserving if and only if for any $m$ and $n$ there is an $ℱ{ℳ}_{m,n}$-natural operator $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $p:Y\to M$ into connections $D\left(\Gamma \right)$ on $Hp:HY\to HM$. For a bundle functor $E:ℱ{ℳ}_{m,n}\to ℱℳ$ with some weak conditions we prove non-existence of $ℱ{ℳ}_{m,n}$-natural operators $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $Y\to M$ into connections $D\left(\Gamma \right)$ on $EY\to M$.

Let F:ℳ f →ℱℳ be a bundle functor with the point property F(pt) = pt, where pt is a one-point manifold. We prove that F is product preserving if and only if for any m and n there is an $ℱ{ℳ}_{m,n}$-canonical construction D of general connections D(Γ) on Fp:FY → FM from general connections Γ on fibred manifolds p:Y → M.

Let $W_m^{r}P$ be a principal prolongation of a principal bundle P → M. We classify all gauge natural operators transforming principal connections on P → M and rth order linear connections on M into general connections on $W_m^{r}P\to M$. We also describe all geometric constructions of classical linear connections on $W_m^{r}P$ from principal connections on P → M and rth order linear connections on M.

Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some “special” fibred coordinate systems on Y, and then we construct a general connection $\widetilde{ℱ}(\Gamma ,\nabla )$ on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.

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)...

Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators $T_{proj|ℱ{ℳ}_{m,n}}⇝T^{(0,0)}\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ transforming projectable vector fields on Y to functions on the dual bundle (FY)* for any $ℱ{ℳ}_{m,n}$-object Y. Next, under some assumption on F we study natural operators $T{*}_{hor|ℱ{ℳ}_{m,n}}⇝T*\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T{*}_{hor|ℱ{ℳ}_{m,n}}⇝T*\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ for some vector bundle functors F on fibered manifolds.