All natural affinors on the $r$-th order cotangent bundle $T^{r*}M$ are determined. Basic affinors of this type are the identity affinor id of $T{T}^{r*}M$ and the $s$-th power affinors $Q_M^s:T{T}^{r*}M\to V{T}^{r*}M$ with $s=1,\cdots ,r$ defined by the $s$-th power transformations $A_s^{r,r}$ of $T^{r*}M$. An arbitrary natural affinor is a linear combination of the basic ones.

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.

We deduce that for $n\ge 2$ and $r\ge 1$, every natural affinor on $J^{r}T$ over $n$-manifolds is of the form $\lambda \delta$ for a real number $\lambda$, where $\delta$ is the identity affinor on $J^{r}T$.

For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle $T^{r,s,q*}Y$ over an (m,n)-dimensional fibered manifold Y.

Let $F$ and $G$ be two natural bundles over $n$-manifolds. We prove that if $F$ is of type (I) and $G$ is of type (II), then any natural differential operator of $F$ into $G$ is of order 0. We give examples of natural bundles of type (I) or of type (II). As an application of the main theorem we determine all natural differential operators between some natural bundles.

We determine all natural functions on $T^{*}{T}^{\left(r\right)}$ and $T^{*}{T}^{r*}$.

We determine all first order natural operators transforming $(0,2)$–tensor fields on a manifold $M$ into $(0,2)$–tensor fields on $TM$.

Natural liftings $D:I\to IT{T}^{*}$ are classified for $n\ge 2$. It is proved that they form a 5-parameter family of operators.

We prove, that $r$-th order gauge natural operators on the bundle of Cartan connections with a target in the gauge natural bundles of the order $(1,0)$ (“tensor bundles”) factorize through the curvature and its invariant derivatives up to order $r-1$. On the course to this result we also prove that the invariant derivations (a generalization of the covariant derivation for Cartan geometries) of the curvature function of a Cartan connection have the tensor character. A modification of the theorem is given for...

For natural numbers n ≥ 3 and r ≥ 1 all natural operators $A:T_{|ℳfₙ}^{(0,0)}⟿T^{(1,1)}{T}^{r*}$ transforming functions from n-manifolds into affinors (i.e. tensor fields of type (1,1)) on the r-cotangent bundle are classified.

The complete description of all natural operators lifting real valued functions to bundle functors on fibered manifolds is given. The full collection of all natural operators lifting projectable real valued functions to bundle functors on fibered manifolds is presented.

Let $A$ be a Weil algebra. The bijection between all natural operators lifting vector fields from $m$-manifolds to the bundle functor $K^A$ of Weil contact elements and the subalgebra of fixed elements $SA$ of the Weil algebra $A$ is determined and the bijection between all natural affinors on $K^A$ and $SA$ is deduced. Furthermore, the rigidity of the functor $K^A$ is proved. Requisite results about the structure of $SA$ are obtained by a purely algebraic approach, namely the existence of nontrivial $SA$ is discussed.

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.