We consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle $PE$ into itself.

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 $n\ge q$.

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor $T^A$ of A-velocities [I. Kolař, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of $T^A$ into F is finite and is less than or equal to $dim\left(F_0{ℛ}^k\right)$. The spaces of all natural transformations of Weil functors into linear...

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...