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1
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Natural transformations of higher order cotangent bundle functors

100%
Kurek J.
Annales Polonici Mathematici
|
1993
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tom 58
|
nr 1
29-35
EN
We determine all natural transformations of the rth order cotangent bundle functor $T^{r*}$ into $T^{s*}$ in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of $T^{r*}$ into itself form an r-parameter family linearly generated by the pth power transformations with p =1,...,r.
2
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On prolongations of projectable connections

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2011
|
tom 101
|
nr 3
237-250
EN
We extend the concept of r-order connections on fibred manifolds to the one of (r,s,q)-order projectable connections on fibred-fibred manifolds, where r,s,q are arbitrary non-negative integers with s ≥ r ≤ q. Similarly to the fibred manifold case, given a bundle functor F of order r on (m₁,m₂,n₁,n₂)-dimensional fibred-fibred manifolds Y → M, we construct a general connection ℱ(Γ,Λ):FY → J¹FY on FY → M from a projectable general (i.e. (1,1,1)-order) connection $Γ:Y → J^{1,1,1}Y$ on Y → M by means of an (r,r,r)-order projectable linear connection $Λ:TM → J^{r,r,r}TM$ on M. In particular, for $F = J^{1,1,1}$ we construct a general connection $𝒥^{1,1,1}(Γ,∇): J^{1,1,1}Y → J¹J^{1,1,1}Y$ on $J^{1,1,1}Y → M$ from a projectable general connection Γ on Y → M by means of a torsion-free projectable classical linear connection ∇ on M. Next, we observe that the curvature of Γ can be considered as $𝓡_Γ:J^{1,1,1}Y → T*M ⊗ VJ^{1,1,1}Y$. The main result is that if m₁ ≥ 2 and n₂ ≥ 1, then all general connections $D(Γ,∇):J^{1,1,1}Y → J¹J^{1,1,1}Y$ on $J^{1,1,1}Y → M$ canonically depending on Γ and ∇ form the one-parameter family $𝒥^{1,1,1}(Γ,∇) + t𝓡_Γ$, t ∈ ℝ. A similar classification of all general connections D(Γ,∇):J¹Y → J¹J¹Y on J¹Y → M from (Γ,∇) is presented.
3
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The natural operators of general affine connections into general affine connections

64%
Kurek J., Mikulski W. M.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2017
|
tom 71
|
nr 1
EN
We reduce the problem of describing all \(\mathcal{M} f_m\)-natural operators  transforming general affine connections on \(m\)-manifolds into general affine ones to the known description of all \(GL(\mathbf{R}^m)\)-invariant maps \(\mathbf{R}^{m*}\otimes \mathbf{R}^m\to \otimes^k\mathbf{R}^{m*}\otimes\otimes ^k\mathbf{R}^m\) for \(k=1,3\).
4
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On lifting of connections to Weil bundles

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2012
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tom 103
|
nr 3
319-324
EN
We prove that the problem of finding all $ℳ f_m$-natural operators $B:Q⟿ QT^A$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_M(∇)$ on the Weil bundle $T^{A}M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳ f_m$-natural operators $C:Q ⟿ (T¹_{p-1},T* ⊗ T* ⊗ T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_M(∇):T¹_{p-1}M = ⨁^{p-1}_{M} TM → T*M ⊗ T*M ⊗ TM$.
5
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On naturality of some construction of connections

64%
Kurek J., Mikulski W.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2020
|
tom 74
|
nr 1
EN
Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the  well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.
6
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On the existence of connections with a prescribed skew-symmetric Ricci tensor

64%
Kurek J., Mikulski W.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2018
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tom 72
|
nr 2
EN
We study the so-called inverse problem. Namely, given a prescribed skew-symmetric Ricci tensor we find (locally) a respective linear connection.
7
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On infinitesimal automorphisms of foliated manifolds

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2007
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tom 92
|
nr 1
1-12
EN
Let F:ℱol → ℱℳ be a product preserving bundle functor on the category ℱol of foliated manifolds (M,ℱ) without singularities and leaf respecting maps. We describe all natural operators C transforming infinitesimal automorphisms X ∈ 𝒳(M,ℱ) of foliated manifolds (M,ℱ) into vector fields C(X)∈ 𝒳(F(M,ℱ)) on F(M,ℱ).
8
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On canonical constructions on connections

64%
Kurek J., Mikulski W.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2016
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tom 70
|
nr 1
EN
We study  how a projectable general connection \(\Gamma\) in a 2-fibred manifold \(Y^2\to Y^1\to Y^0\)  and a general vertical connection \(\Theta\) in \(Y^2\to Y^1\to Y^0\) induce a general connection \(A(\Gamma,\Theta)\) in \(Y^2\to Y^1\).
9
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The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

64%
Kurek J., Mikulski W.
Open Mathematics
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2004
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tom 2
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nr 5
801-810
EN
For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s)(Y, R)0)*→R for any Y as above.
10
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Connections from trivializations

64%
Kurek J., Mikulski W.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2016
|
tom 70
|
nr 2
EN
Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.
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