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We generalize the concept of an $(r,s,q)$-jet to the concept of a non-holonomic $(r,s,q)$-jet. We define the composition of such objects and introduce a bundle functor ${\tilde{J}}^{r,s,q}ℱ{ℳ}_{k,l}\times ℱℳ$ defined on the product category of $(k,l)$-dimensional fibered manifolds with local fibered isomorphisms and the category of fibered manifolds with fibered maps. We give the description of such functors from the point of view of the theory of Weil functors. Further, we introduce a bundle functor ${\tilde{J}}_1^{r,s,q}2\text{-}ℱ{ℳ}_{k,l}\to ℱℳ$ defined on the category of $2$-fibered manifolds with $ℱ{ℳ}_{k,l}$-underlying...

Let $M$ be an $m$-dimensional manifold and $A={𝔻}_k^r/I=ℝ\oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T_x^{A}f\in T_x^{A*}M$, $x\in M$ is determined by its values over arbitrary $max\{\mathrm{width}A,m\}$ regular and under the first jet projection linearly independent elements of $T_x^{A}M$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M\simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial result obtained...

We determine all natural operators transforming vector fields on a manifold $M$ to vector fields on $T^{*}{T}_1^{2}M$, $dimM\ge 2$, and all natural operators transforming vector fields on $M$ to functions on $T^{*}T{T}_1^{2}M$, $dimM\ge 3$. We describe some relations between these two kinds of natural operators.

A natural $T$-function on a natural bundle $F$ is a natural operator transforming vector fields on a manifold $M$ into functions on $FM$. For any Weil algebra $A$ satisfying $dimM\ge \mathrm{w}idth\left(A\right)+1$ we determine all natural $T$-functions on $T^{*}{T}^{A}M$, the cotangent bundle to a Weil bundle $T^{A}M$.

We investigate the category of product preserving bundle functors defined on the category of fibered fibered manifolds. We show a bijective correspondence between this category and a certain category of commutative diagrams on product preserving bundle functors defined on the category ℳ f of smooth manifolds. By an application of the theory of Weil functors, the latter category is considered as a category of commutative diagrams on Weil algebras. We also mention the relation with natural transformations...