We determine all natural transformations T²₁T*→ T*T²₁ where $T_k^{r}M=J_0^r({ℝ}^k,M)$. We also give a geometric characterization of the canonical isomorphism ψ₂ defined by Cantrijn et al.

We determine all natural transformations of the rth order cotangent bundle functor $T^{r*}$ into $T^{s*}$ in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of $T^{r*}$ into itself form an r-parameter family linearly generated by the pth power transformations with p =1,...,r.

Let ${\overline{J}}^3$ be the functor of semi-holonomic $3$-jets and ${\overline{J}}^{3,2}$ be the functor of those semi-holonomic $3$-jets, which are holonomic in the second order. We deduce that the only natural transformations ${\overline{J}}^3\to {\overline{J}}^3$ are the identity and the contraction. Then we determine explicitely all natural transformations ${\overline{J}}^{3,2}\to {\overline{J}}^{3,2}$, which form two $5$-parameter families.

Given a map of a product of two manifolds into a third one, one can define its jets of separated orders $r$ and $s$. We study the functor $J$ of separated $(r;s)$-jets. We determine all natural transformations of $J$ into itself and we characterize the canonical exchange $J\to J^{s;r}$ from the naturality point of view.

We study geometrical properties of natural transformations $T^{A}T*\to T*T^A$ depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations $T^{A}T*\to T*T^A$ can be described in a uniform way by means of a simple geometrical construction.

In this paper are determined all natural transformations of the natural bundle of $(g,r)$-covelocities over $n$-manifolds into such a linear natural bundle over $n$-manifolds which is dual to the restriction of a linear bundle functor, if $n\ge q$.

A classification of natural transformations transforming functions (or vector fields) to functions on such natural bundles which are restrictions of bundle functors is given.

A classification of natural transformations transforming vector fields on $n$-manifolds into affinors on the extended $r$-th order tangent bundle over $n$-manifolds is given, provided $n\ge 3$.

Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)u^H+\beta \left(h\left(u\right)\right)u^V,$$ where $u^V$ is the vertical lift of $u\in T_{x}M$, $u^H$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta$ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma }_1\left(h\left(u\right)\right)\Lambda (g,K)+{\gamma }_2\left(h\left(u\right)\right)u^H\wedge u^V,$$ where ${\gamma }_1$, ${\gamma }_2$ are smooth real functions defined...

We study the problem of the non-existence of natural transformations $J^r{J}^{s}Y\to J^s{J}^{r}Y$ of iterated jet functors depending on some geometric object on the base of Y.

Under some weak assumptions on a bundle functor $F:F{M}_{m,n}\to FM$ we prove that there is no $F{M}_{m,n}$-natural operator transforming connections on $Y\to M$ into connections on $FY\to Y$.

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 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.