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The author proves that for a manifold $M$ of dimension greater than 2 the sets of all natural operators $TM\to (T_k^{r*}M,T_{\ell }^{q*}M)$ and $TM\to T{T}_k^{r*}M$, respectively, are free finitely generated $C^{\infty }\left({\left({ℝ}^k\right)}^r\right)$-modules. The space $T_k^{r*}M=J^r{(M,{ℝ}^k)}_0$, this is, jets with target 0 of maps from $M$ to ${ℝ}^k$, is called the space of all $(k,r)$-covelocities on $M$. Examples of such operators are shown and the bases of the modules are explicitly constructed. The definitions and methods are those of the book of and [Natural operations in differential geometry, Springer-Verlag, Berlin (1993;...

The author studies the problem how a map $L:M\to ℝ$ on an $n$-dimensional manifold $M$ can induce canonically a map $A_M\left(L\right):T^{*}{T}^{\left(r\right)}M\to ℝ$ for $r$ a fixed natural number. He proves the following result: “Let $A:T^{(0,0)}\to T^{(0,0)}\left(T^{*}{T}^{\left(r\right)}\right)$ be a natural operator for $n$-manifolds. If $n\ge 3$ then there exists a uniquely determined smooth map $H:{ℝ}^{S\left(r\right)}\times ℝ\to ℝ$ such that $A=A^{\left(H\right)}$.”The conclusion is that all natural functions on $T^{*}{T}^{\left(r\right)}$ for $n$-manifolds $(n\ge 3)$ are of the form $\{H\circ ({\lambda }_M^{\langle 0,1\rangle },\cdots ,{\lambda }_M^{\langle r,0\rangle })\}$, where $H\in C^{\infty }\left({ℝ}^r\right)$ is a function of $r$ variables.

Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{|ℳfₙ}⇝T*T^{r*}$ transforming vector fields on n-manifolds M to 1-forms on $T^{r*}M=J^r(M,ℝ)₀$ are classified. For n ≥ 3 all natural operators $T_{|ℳfₙ}⇝\Lambda ²T*T^{r*}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r*}M$ are completely described.

We prove that the most interesting operators in the Euler-Lagrange complex from the variational bicomplex in infinite order jet spaces are determined up to multiplicative constant by the naturality requirement, provided the fibres of fibred manifolds have sufficiently large dimension. This result clarifies several important phenomena of the variational calculus on fibred manifolds.

We describe all ${}_m\left(G\right)$-gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^{r}P=P^{r}M{\times }_M{J}^{r}P$ of P → M. In other words, we classify all ${}_m\left(G\right)$-natural transformations $J^{r}LP{\times }_M{W}^{r}P\to T{W}^{r}P=L{W}^{r}P{\times }_M{W}^{r}P$ covering the identity of $W^{r}P$, where $J^{r}LP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all ${}_m\left(G\right)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^{r}LP=L{W}^{r}P$. We formulate axioms which characterize...

Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^r₀⟿Q\left(reg{T}_k^r\to K_k^r\right)$ and $C^r₀⟿Q\left(reg{T}_k^{r*}\to K_k^{r*}\right)$ over n-manifolds is proved. Some generalizations are obtained.

In this paper we extend the variational calculus to fibered-fibered manifolds. Fibered-fibered manifolds are surjective fibered submersions π:Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q with r ≥ 1 we define (r,s,q)th order Lagrangians on fibered-fibered manifolds π:Y → X as base-preserving morphisms $\lambda :J^{r,s,q}Y\to {\bigwedge }^{dimX}T*X$. Then similarly to the fibered manifold case we define critical fibered sections of Y. Setting p=max(q,s) we prove that there exists a canonical “Euler” morphism $\left(\lambda \right):J^{r+s,2s,r+p}Y\to *Y\otimes {\bigwedge }^{dimX}T*X$ of λ satisfying...

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.

Let F:ℳ f →ℱℳ be a bundle functor with the point property F(pt) = pt, where pt is a one-point manifold. We prove that F is product preserving if and only if for any m and n there is an $ℱ{ℳ}_{m,n}$-canonical construction D of general connections D(Γ) on Fp:FY → FM from general connections Γ on fibred manifolds p:Y → M.

A fibered-fibered manifold is a surjective fibered submersion π: Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q an (r,s,q)th order Lagrangian on a fibered-fibered manifold π: Y → X is a base-preserving morphism $\lambda :J^{r,s,q}Y\to {\bigwedge }^{dimX}T*X$. For p= max(q,s) there exists a canonical Euler morphism $\left(\lambda \right):J^{r+s,2s,r+p}Y\to *Y\otimes {\bigwedge }^{dimX}T*X$ satisfying a decomposition property similar to the one in the fibered manifold case, and the critical fibered sections σ of Y are exactly the solutions of the Euler-Lagrange equation $\left(\lambda \right)\circ j^{r+s,2s,r+p}\sigma =0$. In the present paper, similarly...

Admissible fiber product preserving bundle functors F on $ℱ{ℳ}_m$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ{ℳ}_m$ all natural operators $B:T_{proj|ℱ{ℳ}_{m,n}}\to TF$ lifting projectable vector fields to F are classified.

We classify all natural operators lifting linear vector fields on vector bundles to vector fields on vertical fiber product preserving gauge bundles over vector bundles. We explain this result for some known examples of such bundles.

We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^{r}M=invJ{₀}^r({ℝ}^m,M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$. In both cases we deduce that the spaces of all operators in question form free $(m(C_r^{m+r}-1)+1)$-dimensional modules over algebras of all smooth maps $J{₀}^{r-1}T̃{ℝ}^m\to ℝ$ and $J{₀}^{r-1}T{ℝ}^m\to ℝ$ respectively, where $C{ⁿ}_k=n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular, we...

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

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.

Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators $T_{proj|ℱ{ℳ}_{m,n}}⇝T^{(0,0)}\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ transforming projectable vector fields on Y to functions on the dual bundle (FY)* for any $ℱ{ℳ}_{m,n}$-object Y. Next, under some assumption on F we study natural operators $T{*}_{hor|ℱ{ℳ}_{m,n}}⇝T*\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T{*}_{hor|ℱ{ℳ}_{m,n}}⇝T*\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ for some vector bundle functors F on fibered manifolds.