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.

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 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.

We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A}M$, where $T^A$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{|ℳfₙ}⇝T^{(0,0)}\left(F_{|ℳfₙ}\right)*$ transforming vector fields to functions on the dual bundle functor $\left(F_{|ℳfₙ}\right)*$. Next, we study the natural operators $T{*}_{|ℳfₙ}⇝T*\left(F_{|ℳfₙ}\right)*$ lifting 1-forms to $\left(F_{|ℳfₙ}\right)*$. As an application we classify the natural operators $T{*}_{|ℳfₙ}⇝T*\left(F_{|ℳfₙ}\right)*$ for some well known vector bundle functors F.

We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^{r}M=invJ{₀}^r({ℝ}^m,M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$. In both cases we deduce that the spaces of all operators in question form free $(m(C_r^{m+r}-1)+1)$-dimensional modules over algebras of all smooth maps $J{₀}^{r-1}T̃{ℝ}^m\to ℝ$ and $J{₀}^{r-1}T{ℝ}^m\to ℝ$ respectively, where $C{ⁿ}_k=n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular,...

All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r}E$ of E are given. Similar results are deduced for the r-jet prolongations $J_v^{r}E$ and $J^{\left[r\right]}E$ in place of $J^{r}E$.

Let $r$, $s$, $m$, $n$, $q$ be natural numbers such that $s\ge r$. We prove that any $2$-$ℱ{𝕄}_{m,n,q}$-natural operator $A:T_{2-proj}⇝T{J}^{(s,r)}$ transforming $2$-projectable vector fields $V$ on $(m,n,q)$-dimensional $2$-fibred manifolds $Y\to X\to M$ into vector fields $A\left(V\right)$ on the $(s,r)$-jet prolongation bundle $J^{(s,r)}Y$ is a constant multiple of the flow operator ${𝒥}^{(s,r)}$.

We prove that the problem of finding all $ℳf_m$-natural operators $B:Q⟿Q{T}^A$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_M\left(\nabla \right)$ on the Weil bundle $T^{A}M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳf_m$-natural operators $C:Q⟿(T{¹}_{p-1},T*\otimes T*\otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_M\left(\nabla \right):T{¹}_{p-1}M={⨁}_M^{p-1}TM\to T*M\otimes T*M\otimes TM$.

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

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 $EY\to M$.