For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{\left(r\right),a}$ over $n$-manifolds such that $T^{\left(r\right),0}$ is the (classical) vector tangent bundle $T^{\left(r\right)}$ of order $r$. For integers $r\ge 1$ and $n\ge 3$ and a real number $a<0$ we classify all natural operators $T_{|M_n}⇝T{T}^{\left(r\right),a}$ lifting vector fields from $n$-manifolds to $T^{\left(r\right),a}$.

Let $G$ be a bundle functor of order $(r,s,q)$, $s\ge r\le q$, on the category $ℱ{ℳ}_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma$ on an $ℱ{ℳ}_{m,n}$-object $Y\to M$ we construct a general connection $𝒢(\Gamma ,\lambda ,\Lambda )$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda$ on $M$ and an $s$-th order linear connection $\Lambda$ on $Y$. Then we construct a general connection $𝒢(\Gamma ,{\nabla }_1,{\nabla }_2)$ on $GY\to Y$ by means of auxiliary classical linear connections ${\nabla }_1$ on $M$ and ${\nabla }_2$ on $Y$. In the case $G=J^1$ we determine all general connections...

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: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.

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.

For natural numbers n ≥ 3 and r ≥ 1 all natural operators $A:T_{|ℳfₙ}^{(0,0)}⟿T^{(1,1)}{T}^{r*}$ transforming functions from n-manifolds into affinors (i.e. tensor fields of type (1,1)) on the r-cotangent bundle are classified.

Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^r₀⟿Q\left(reg{T}_k^r\to K_k^r\right)$ and $C^r₀⟿Q\left(reg{T}_k^{r*}\to K_k^{r*}\right)$ over n-manifolds is proved. Some generalizations are obtained.

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.

Admissible fiber product preserving bundle functors F on $ℱ{ℳ}_m$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ{ℳ}_m$ all natural operators $B:T_{proj|ℱ{ℳ}_{m,n}}\to TF$ lifting projectable vector fields to F are classified.

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.