We classify all F2Mm1, m2, n1, n2-natural operators Atransforming projectable-projectable torsion-free classical linear connections ∇ on fibered-fibered manifolds Y of dimension (m1,m2, n1, n2) into rth order Lagrangians A(∇) on the fibered-fibered linear frame bundle Lfib-fib(Y) on Y. Moreover, we classify all F2Mm1, m2, n1, n2-natural operators B transforming projectable-projectable torsion-free classical linear connections ∇ on fiberedfibered manifolds Y of dimension (m1, m2, n1, n2) into Euler...

Let $r,n$ be fixed natural numbers. We prove that for $n$-manifolds the set of all linear natural operators $T^{*}\to T^{*}{T}^{\left(r\right)}$ is a finitely dimensional vector space over $R$. We construct explicitly the bases of the vector spaces. As a corollary we find all linear natural operators $T^{*}\to T^{r*}$.

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A}M$, where $T^A$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

We define equivariant tensors for every non-negative integer $p$ and every Weil algebra $A$ and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type $(p,0)$ on an $n$-dimensional manifold $M$ to tensor fields of type $(p,0)$ on $T^{A}M$ if $1\le p\le n$. Moreover, we determine explicitly the equivariant tensors for the Weil algebras ${𝔻}_k^r$, where $k$ and $r$ are non-negative integers.