For every product preserving bundle functor $T^{\mu }$ on fibered manifolds, we describe the underlying functor of any order $(r,s,q),s\ge r\le q$. We define the bundle $K_{k,l}^{r,s,q}Y$ of $(k,l)$-dimensional contact elements of the order $(r,s,q)$ on a fibered manifold $Y$ and we characterize its elements geometrically. Then we study the bundle of general contact elements of type $\mu$. We also determine all natural transformations of $K_{k,l}^{r,s,q}Y$ into itself and of $T\left(K_{k,l}^{r,s,q}Y\right)$ into itself and we find all natural operators lifting projectable vector fields and horizontal...

In this paper a Weil approach to quasijets is discussed. For given manifolds $M$ and $N$, a quasijet with source $x\in M$ and target $y\in N$ is a mapping $T_x^{r}M\to T_y^{r}N$ which is a vector homomorphism for each one of the $r$ vector bundle structures of the iterated tangent bundle $T^r$ [, Casopis Pest. Mat. 111, No. 4, 345-352 (1986; Zbl 0611.58004)]. Let us denote by $Q{J}^r(M,N)$ the bundle of quasijets from $M$ to $N$; the space ${\tilde{J}}^r(M,N)$ of non-holonomic $r$-jets from $M$ to $N$ is embeded into $Q{J}^r(M,N)$. On the other hand, the bundle $Q{T}_m^{r}N$ of $(m,r)$-quasivelocities...

Let $r,s,q,m,n\in \mathbb{N}$ be such that $s\ge r\le q$. Let $Y$ be a fibered manifold with $m$-dimensional basis and $n$-dimensional fibers. All natural affinors on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are classified. It is deduced that there is no natural generalized connection on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$. Similar problems with ${\left(J^{r,s}{(Y,ℝ)}_0\right)}^{*}$ instead of ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are solved.

For natural numbers $r$ and $n\ge 2$ all natural operators $T_{|ℳf_n}⇝T^{*}\left(J^r{T}^{*}\right)$ transforming vector fields from $n$-manifolds $M$ into $1$-forms on $J^r{T}^{*}M=\{j_x^r\left(\omega \right)\mid \omega \in {\Omega }^1\left(M\right),x\in M\}$ are classified. A similar problem with fibered manifolds instead of manifolds is discussed.