We prove that every natural affinor on ${\left(J^r\left({\odot }^2{T}^{*}\right)\right)}^{*}\left(M\right)$ is proportional to the identity affinor if dim$M\ge 3$.

We determine all natural operators D transforming general connections Γ on fibred manifolds Y → M and torsion free classical linear connections ∇ on M into general connections D(Γ,∇) on the second order jet prolongation J2Y → M of Y → M

Jets of a manifold $M$ can be described as ideals of ${𝒞}^{\infty }\left(M\right)$. This way, all the usual processes on jets can be directly referred to that ring. By using this fact, we give a very simple construction of the contact system on jet spaces. The same way, we also define the contact system for the recently considered $A$-jet spaces, where $A$ is a Weil algebra. We will need to introduce the concept of derived algebra.

This paper is a continuation of [MMR:98], where we give a construction of the canonical Pfaff system $\Omega \left(M_m^{\ell }\right)$ on the space of $(m,\ell )$-velocities of a smooth manifold $M$. Here we show that the characteristic system of $\Omega \left(M_m^{\ell }\right)$ agrees with the Lie algebra of $Aut\left({ℝ}_m^{\ell }\right)$, the structure group of the principal fibre bundle ${\check{M}}_m^{\ell }⟶J_m^{\ell }\left(M\right)$, hence it is projectable to an irreducible contact system on the space of $(m,\ell )$-jets ($=\ell$-th order contact elements of dimension $m$) of $M$. Furthermore, we translate to the language of Weil bundles the structure form...

We study naturality of the Euler and Helmholtz operators arising in the variational calculus in fibered manifolds with oriented bases.

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...

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework...

Let $r$, $s$, $m$, $n$, $q$ be natural numbers such that $s\ge r$. We prove that any $2$-$ℱ{𝕄}_{m,n,q}$-natural operator $A:T_{2-proj}⇝T{J}^{(s,r)}$ transforming $2$-projectable vector fields $V$ on $(m,n,q)$-dimensional $2$-fibred manifolds $Y\to X\to M$ into vector fields $A\left(V\right)$ on the $(s,r)$-jet prolongation bundle $J^{(s,r)}Y$ is a constant multiple of the flow operator ${𝒥}^{(s,r)}$.

For natural numbers $r$ and $n\ge 2$ a complete classification of natural affinors on the natural bundle ${\left(J^r{T}^{*}\right)}^{*}$ dual to $r$-jet prolongation $J^r{T}^{*}$ of the cotangent bundle over $n$-manifolds is given.

We classify all natural affinors on vertical fiber product preserving gauge bundle functors $F$ on vector bundles. We explain this result for some more known such $F$. We present some applications. We remark a similar classification of all natural affinors on the gauge bundle functor $F^{*}$ dual to $F$ as above. We study also a similar problem for some (not all) not vertical fiber product preserving gauge bundle functors on vector bundles.

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s)(Y, R)0)*→R for any Y as above.

For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators $T*{\times }_{ℳfₙ}{T}^{(0,0)}⇝T^{(0,0)}{T}^{\left(r\right)}$ is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators $T{*}_{|ℳfₙ}⇝T{T}^{\left(r\right)}$ is obtained.

Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{|ℳfₙ}⇝T^{(0,0)}\left(F_{|ℳfₙ}\right)*$ transforming vector fields to functions on the dual bundle functor $\left(F_{|ℳfₙ}\right)*$. Next, we study the natural operators $T{*}_{|ℳfₙ}⇝T*\left(F_{|ℳfₙ}\right)*$ lifting 1-forms to $\left(F_{|ℳfₙ}\right)*$. As an application we classify the natural operators $T{*}_{|ℳfₙ}⇝T*\left(F_{|ℳfₙ}\right)*$ for some well known vector bundle functors F.