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1
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Non-existence of some natural operators on connections

100%
Mikulski W.
Annales Polonici Mathematici
|
2003
|
tom 81
|
nr 2
157-166
EN
Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^r₀⟿ Q(reg T^r_k → K^r_k)$ and $C^r₀ ⟿ Q(reg T^{r*}_k → K^{r*}_k)$ over n-manifolds is proved. Some generalizations are obtained.
2
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The natural operators lifting projectable vector fields to some fiber product preserving bundles

100%
Mikulski W.
Annales Polonici Mathematici
|
2003
|
tom 81
|
nr 3
261-271
EN
Admissible fiber product preserving bundle functors F on $ℱℳ _m$ are defined. For every admissible fiber product preserving bundle functor F on $ℱℳ _m$ all natural operators $B:T_{proj|ℱℳ _{m,n}} → TF$ lifting projectable vector fields to F are classified.
3
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Natural operators lifting functions to affinors on higher order cotangent bundles

100%
Mikulski W.
Annales Polonici Mathematici
|
2004
|
tom 83
|
nr 1
49-55
EN
For natural numbers n ≥ 3 and r ≥ 1 all natural operators $A:T^{(0,0)}_{|ℳ fₙ} ⟿ T^{(1,1)}T^{r*}$ transforming functions from n-manifolds into affinors (i.e. tensor fields of type (1,1)) on the r-cotangent bundle are classified.
4
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Bundle functors with the point property which admit prolongation of connections

100%
Mikulski W.
Annales Polonici Mathematici
|
2010
|
tom 97
|
nr 3
253-256
EN
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 $ℱℳ _{m,n}$-canonical construction D of general connections D(Γ) on Fp:FY → FM from general connections Γ on fibred manifolds p:Y → M.
5
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Lifting right-invariant vector fields and prolongation of connections

100%
Mikulski W.
Annales Polonici Mathematici
|
2009
|
tom 95
|
nr 3
243-252
EN
We describe all $𝓟𝓑_m(G)$-gauge-natural operators 𝓐 lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields 𝓐(X) on the rth order principal prolongation $W^rP=P^rM×_MJ^rP$ of P → M. In other words, we classify all $𝓟𝓑_m(G)$-natural transformations $J^rLP×_M W^rP→ TW^rP=LW^rP×_MW^rP$ covering the identity of $W^rP$, where $J^rLP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all $𝓟𝓑_m(G)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^rLP=LW^rP$. We formulate axioms which characterize the flow operator of the gauge-bundle $W^rP → M$. We apply the flow operator to prolongations of connections.
6
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Uniqueness results for operators in the variational sequence

100%
Mikulski W.
Annales Polonici Mathematici
|
2009
|
tom 95
|
nr 2
125-133
EN
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.
7
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The natural operators $T_{|ℳ fₙ} ⇝ T*T^{r*}$ and $T_{|ℳ fₙ} ⇝ Λ²T*T^{r*}$

100%
Mikulski W.
Colloquium Mathematicum
|
2002
|
tom 93
|
nr 1
55-65
EN
Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{|ℳ fₙ} ⇝ T*T^{r*}$ 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 $T_{|ℳ fₙ} ⇝ Λ²T*T^{r*}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r*}M$ are completely described.
8
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On the Helmholtz operator of variational calculus in fibered-fibered manifolds

100%
Mikulski W.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 1
59-76
EN
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 $λ: J^{r,s,q}Y → ⋀^{dim X}T*X$. For p= max(q,s) there exists a canonical Euler morphism $𝓔(λ): J^{r+s,2s,r+p}Y → 𝓥*Y⊗ ⋀^{dim X}T*X$ 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 $𝓔(λ) ∘ j^{r+s,2s,r+p}σ = 0$. In the present paper, similarly to the fibered manifold case, for any morphism $B:J^{r,s,q}Y → 𝓥*Y ⊗ ⋀^{m}T*X$ over Y, s ≥ r ≤ q, we define canonically a Helmholtz morphism $𝓗(B): J^{s+p,s+p,2p}Y → 𝓥*J^{r,s,r}Y ⊗ 𝓥*Y ⊗ ⋀^{dim X}T*X$, and prove that a morphism $B:J^{r+s,2s,r+p} Y → 𝓥*Y ⊗ ⋀ T*M$ over Y is locally variational (i.e. locally of the form B = 𝓔(λ) for some (r,s,p)th order Lagrangian λ) if and only if 𝓗(B) = 0, where p = max(s,q). Next, we study naturality of the Helmholtz morphism 𝓗(B) on fibered-fibered manifolds Y of dimension (m₁,m₂,n₁,n₂). We prove that any natural operator of the Helmholtz morphism type is c𝓗(B), c ∈ ℝ, if n₂≥ 2.
9
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On the variational calculus in fibered-fibered manifolds

100%
Mikulski W.
Annales Polonici Mathematici
|
2006
|
tom 89
|
nr 1
1-12
EN
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 $λ:J^{r,s,q}Y →⋀^{dimX}T*X$. 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 $𝓔(λ):J^{r+s,2s,r+p}Y → 𝓥*Y ⊗ ⋀^{dimX}T*X$ of λ satisfying a decomposition property similar to the one in the fibered manifold case, and we deduce that critical fibered sections σ are exactly the solutions of the "Euler-Lagrange" equations $𝓔(λ)∘ j^{r+s,2s,r+p}σ=0$. Next we study the naturality of the "Euler" morphism. We prove that any natural operator of the "Euler" morphism type is c𝓔(λ), c ∈ ℝ, provided dim X ≥ 2.
10
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The Euler and Helmholtz operators on fibered manifolds with oriented bases

64%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2006
|
tom 105
|
nr 2
171-177
EN
We study naturality of the Euler and Helmholtz operators arising in the variational calculus in fibered manifolds with oriented bases.
11
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Symplectic structures on the tangent bundles of symplectic and cosymplectic manifolds

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2003
|
tom 82
|
nr 3
273-285
EN
We describe all natural symplectic structures on the tangent bundles of symplectic and cosymplectic manifolds.
12
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Natural affinors on the (r,s,q)-cotangent bundle of a fibered manifold

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2004
|
tom 83
|
nr 1
57-64
EN
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.
13
Content available remote

Lifting vector fields to the rth order frame bundle

64%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2008
|
tom 111
|
nr 1
51-58
EN
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.
14
Content available remote

Bundle functors on all foliated manifold morphisms have locally finite order

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 1
69-74
EN
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.
15
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Constructions on second order connections

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2007
|
tom 92
|
nr 3
215-223
EN
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.
16
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Lifting adapted connections from foliated manifolds to higher order adapted frame bundles

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 2
163-169
EN
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 $P^r(M,ℱ)$.
17
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The natural operators lifting 1-forms to some vector bundle functors

64%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2002
|
tom 93
|
nr 2
259-265
EN
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.
18
Content available remote

The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

64%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2003
|
tom 97
|
nr 2
141-149
EN
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.
19
Content available remote

Lifting to the r-frame bundle by means of connections

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2010
|
tom 97
|
nr 1
63-71
EN
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.
20
Content available remote

The natural linear operators $T* ⇝ TT^{(r)}$

64%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2003
|
tom 95
|
nr 1
37-47
EN
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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