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 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.

A complete classification of natural transformations of symplectic structures into Poisson's brackets as well as into Jacobi's brackets is given.

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...