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Let Y be a fibered square of dimension (m1, m2, n1, n2). Let V be a finite dimensional vector space over. We describe all 21,m2,n1,n2 - canonical V -valued 1-form Θ TPrA (Y) → V on the r-th order adapted frame bundle PrA(Y).

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...

We describe all F2Mm1,m2,n1,n2-natural operators D: Qτproj-prj ↝QT* transforming projectable-projectable classical torsion-free linear connections ∇ on fibred-fibred manifolds Y into classical linear connections D(∇) on cotangent bundles T*Y of Y . We show that this problem can be reduced to finding F2Mm1,m2,n1,n2-natural operators D: Qτproj-proj ↝ (T*,⊗pT*⊗⊗qT) for p = 2, q = 1 and p = 3, q = 0.

We prove that any first order F2 Mm1,m2,n1,n2-natural operator transforming projectable general connections on an (m1,m2, n1, n2)-dimensional fibred-fibred manifold p = (p, p) : (pY : Y → Y) → (pM : M → M) into general connections on the vertical prolongation V Y → M of p: Y → M is the restriction of the (rather well-known) vertical prolongation operator V lifting general connections Γ on a fibred manifold Y → M into VΓ (the vertical prolongation of Γ) on V Y → M.

We describe all M fm-natural operators S: Q ↝ Symp P1 transforming classical linear connections ∇ on m-dimensional manifolds M into almost symplectic structures S(∇) on the linear frame bundle P1M over M.

We describe all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D\left(\nabla \right)$ on cotangent bundles $T^{*}Y$ of $Y$. We show that this problem can be reduced to finding ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{*},{\otimes }^p{T}^{*}\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.

We prove that any first order ${ℱ}_2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operator transforming projectable general connections on an $(m_1,m_2,n_1,n_2)$-dimensional fibred-fibred manifold $p=(p,p):(p_Y:Y\to Y)\to (p_M:M\to M)$ into general connections on the vertical prolongation $VY\to M$ of $p:Y\to M$ is the restriction of the (rather well-known) vertical prolongation operator $𝒱$ lifting general connections $\overline{\Gamma }$ on a fibred manifold $Y\to M$ into $𝒱\overline{\Gamma }$ (the vertical prolongation of $\overline{\Gamma }$) on $VY\to M$.

Let $ℳf_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳf_m$. Let $𝒱$ be the category of real vector spaces and linear maps and let  ${𝒱}_m$ be the category of  $m$-dimensional real vector spaces and linear isomorphisms. Let $w$ be a polynomial in one variable with real coefficients. We describe all regular covariant functors $F:{𝒱}_m\to 𝒱$ admitting $ℳf_m$-natural operators $\tilde{P}$ transforming classical linear connections $\nabla$ on $m$-dimensional manifolds $M$ into almost polynomial $w$-structures ...

We classify all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\nabla$ on fibered-fibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into $r$th order Lagrangians $A\left(r\right)$ on the fibered-fibered linear frame bundle $L^{fib-fib}\left(Y\right)$ on $Y$. Moreover, we classify all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into Euler morphism $B\left(\nabla \right)$ on $L^{fib-fib}\left(Y\right)$. These classifications can be expanded on the $k$th order...