For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle $T^{r,s,q*}Y$ over an (m,n)-dimensional fibered manifold Y.
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We describe all natural operators 𝓐 lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields 𝓐(X) on the rth order frame bundle $L^{r}M = inv J₀^{r}(ℝ^m, M)$ over M. Next, we describe all natural operators 𝓐 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 𝓐 in question form free $(m(C^{m+r}_{r}-1) + 1)$-dimensional modules over algebras of all smooth maps $J₀^{r-1}T̃ℝ^m → ℝ$ and $J₀^{r-1}Tℝ^m → ℝ$ respectively, where $Cⁿ_k = n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular, we find that the vector space over ℝ of all natural linear operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$ is $(m²C^{m+r-1}_{r-1}(C^{m+r}_r - 1) + 1)$-dimensional.
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We prove that any bundle functor F:ℱol → ℱℳ on the category ℱ olof all foliated manifolds without singularities and all leaf respecting maps is of locally finite order.
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We classify all $ℱℳ_{m,n}$-natural operators $𝓓 :J² ⟿ J²V^{A}$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $𝓓(Γ): V^{A}Y → J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y → M$ corresponding to a Weil algebra A.
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Let (M,ℱ) be a foliated manifold. We describe all natural operators 𝓐 lifting ℱ-adapted (i.e. projectable in adapted coordinates) classical linear connections ∇ on (M,ℱ) into classical linear connections 𝓐(∇) on the rth order adapted frame bundle $P^r(M,ℱ)$.
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Let F:ℳ f→ 𝓥ℬ be a vector bundle functor. First we classify all natural operators $T_{|ℳ fₙ} ⇝ T^{(0,0)} (F_{|ℳ fₙ})*$ transforming vector fields to functions on the dual bundle functor $(F_{|ℳ fₙ})*$. Next, we study the natural operators $T*_{|ℳ fₙ} ⇝ T*(F_{|ℳ fₙ})*$ lifting 1-forms to $(F_{|ℳ fₙ})*$. As an application we classify the natural operators $T*_{|ℳ fₙ} ⇝ T*(F_{|ℳ fₙ})*$ for some well known vector bundle functors F.
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Let F:ℱ ℳ → 𝓥ℬ be a vector bundle functor. First we classify all natural operators $T_{proj|ℱ ℳ _{m,n}} ⇝ T^{(0,0)}(F_{|ℱ ℳ_{m,n}})*$ 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 $T*_{hor|ℱ ℳ _{m,n}} ⇝ T*(F_{|ℱ ℳ _{m,n}})*$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T*_{hor|ℱ ℳ _{m,n}} ⇝ T*(F_{|ℱ ℳ _{m,n}})*$ for some vector bundle functors F on fibered manifolds.
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Let m and r be natural numbers and let $P^r:ℳ f_m → ℱℳ$ be the rth order frame bundle functor. Let $F:ℳ f_m → ℱℳ$ and $G:ℳ f_k → ℱℳ$ be natural bundles, where $k=dim (P^rℝ^m)$. We describe all $ℳ f_m$-natural operators A transforming sections σ of $FM → M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G(P^rM) → P^rM$. We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.
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For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators $T*×_{ℳ fₙ} T^{(0,0)} ⇝ T^{(0,0)}T^{(r)}$ is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators $T*_{|ℳ fₙ} ⇝ TT^{(r)}$ is obtained.
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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.
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