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1
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The Euler and Helmholtz operators on fibered manifolds with oriented bases

100%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2006
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tom 105
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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.
2
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Symplectic structures on the tangent bundles of symplectic and cosymplectic manifolds

100%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2003
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tom 82
|
nr 3
273-285
EN
We describe all natural symplectic structures on the tangent bundles of symplectic and cosymplectic manifolds.
3
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Natural affinors on the (r,s,q)-cotangent bundle of a fibered manifold

100%
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.
4
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Lifting vector fields to the rth order frame bundle

100%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2008
|
tom 111
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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.
5
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Bundle functors on all foliated manifold morphisms have locally finite order

100%
Kurek J., Mikulski W.
Annales Polonici Mathematici
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2008
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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.
6
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Constructions on second order connections

100%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2007
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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.
7
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Lifting adapted connections from foliated manifolds to higher order adapted frame bundles

100%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2008
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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 $P^r(M,ℱ)$.
8
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The natural operators lifting 1-forms to some vector bundle functors

100%
Kurek J., Mikulski W.
Colloquium Mathematicum
|
2002
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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.
9
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The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

100%
Kurek J., Mikulski W.
Colloquium Mathematicum
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2003
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tom 97
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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.
10
Content available remote

Lifting to the r-frame bundle by means of connections

100%
Kurek J., Mikulski W.
Annales Polonici Mathematici
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2010
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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.
11
Content available remote

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

100%
Kurek J., Mikulski W.
Colloquium Mathematicum
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2003
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tom 95
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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.
12
Content available remote

Lifting of linear vector fields to fiber product preserving vertical gauge bundles

100%
Kurek J., Mikulski W.
Colloquium Mathematicum
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2005
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tom 103
|
nr 1
33-40
EN
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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