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1
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Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations

100%
Mikulski W.
Annales Polonici Mathematici
|
2003
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tom 82
|
nr 2
155-170
EN
All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^rE$ of E are given. Similar results are deduced for the r-jet prolongations $J^r_{v}E$ and $J^{[r]}E$ in place of $J^rE$.
2
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Product preserving bundles on foliated manifolds

100%
Mikulski W.
Annales Polonici Mathematici
|
2004
|
tom 84
|
nr 1
67-74
EN
We present a complete description of all product preserving bundle functors on the category ℱol of all foliated manifolds and their leaf respecting maps in terms of homomorphisms of Weil algebras.
3
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On the fiber product preserving gauge bundle functors on vector bundles

100%
Mikulski W.
Annales Polonici Mathematici
|
2003
|
tom 82
|
nr 3
251-264
EN
We present a complete description of all fiber product preserving gauge bundle functors F on the category $𝓥𝓑_m$ of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented.
4
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Product preserving gauge bundle functors on all principal bundle homomorphisms

100%
Mikulski W.
Annales Polonici Mathematici
|
2011
|
tom 101
|
nr 2
163-207
EN
Let 𝓟𝓑 be the category of principal bundles and principal bundle homomorphisms. We describe completely the product preserving gauge bundle functors (ppgb-functors) on 𝓟𝓑 and their natural transformations in terms of the so-called admissible triples and their morphisms. Then we deduce that any ppgb-functor on 𝓟𝓑 admits a prolongation of principal connections to general ones. We also prove a "reduction" theorem for prolongations of principal connections into principal ones by means of Weil functors. We observe that there exist plenty of such prolongations. In Appendix, we classify the natural operators lifting vector-valued 1-forms (or vector-valued maps) to vector-valued 1-forms on Weil bundles.
5
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On prolongation of connections

100%
Mikulski W.
Annales Polonici Mathematici
|
2010
|
tom 97
|
nr 2
101-121
EN
Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. Let r, m,n be positive integers. We present a construction $B^r$ of rth order holonomic connections $B^r(Γ,∇):Y → J^rY$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M. Then we prove that any construction B of rth order holonomic connections $B(Γ,∇):Y → J^rY$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M is equal to $B^r$. Applying $B^r$, for any bundle functor $F:ℱℳ_{m,n} →ℱℳ$ on fibred (m,n)-manifolds we present a construction $ℱ^r_q$ of rth order holonomic connections $ℱ^r_q(Θ,∇):FY → J^r(FY)$ on FY → M from qth order holonomic connections $Θ:Y → J^qY$ on Y → M by means of torsion free classical linear connections ∇ on M (for q=r=1 we have a well-known classical construction ℱ(Γ,∇):FY → J¹(FY)). Applying $B^r$ we also construct a so-called (Γ,∇)-lift of a wider class of geometric objects. In Appendix, we present a direct proof of a (recent) result saying that for r ≥ 3 and m ≥ 2 there is no construction A of rth order holonomic connections $A(Γ):Y → J^rY$ on Y → M from general connections Γ:Y → J¹Y on Y → M.
6
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Continuity of projections of natural bundles

100%
Mikulski W.
Annales Polonici Mathematici
|
1992
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tom 57
|
nr 2
105-120
EN
This paper is a contribution to the axiomatic approach to geometric objects. A collection of a manifold M, a topological space N, a group homomorphism E: Diff(M) → Homeo(N) and a function π: N → M is called a quasi-natural bundle if (1) π ∘ E(f) = f ∘ π for every f ∈ Diff(M) and (2) if f,g ∈ Diff(M) are two diffeomorphisms such that f|U = g|U for some open subset U of M, then E(f)|π^{-1}(U) = E(g)|π^{-1}(U). We give conditions which ensure that π: N → M is continuous. In particular, if (M,N,E,π) is a quasi-natural bundle with N Hausdorff, then π is continuous. Using this result, we classify (quasi) prolongation functors with compact fibres.
7
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On "special" fibred coordinates for general and classical connections

100%
Mikulski W.
Annales Polonici Mathematici
|
2010
|
tom 99
|
nr 1
99-105
EN
Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some "special" fibred coordinate systems on Y, and then we construct a general connection $\widetilde{ℱ}(Γ,∇)$ on Fp:FY → FM for any vector bundle functor F: ℳ f → 𝓥𝓑 of finite order.
8
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Lifting distributions to the cotangent bundle

100%
Mikulski W.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 3
211-215
EN
A classification of all $ℳ f_m$-natural operators $A: Gr_p ⟿ Gr_qT*$ lifting p-dimensional distributions D ⊂ TM on m-manifolds M to q-dimensional distributions A(D) ⊂ TT*M on the cotangent bundle T*M is given.
9
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The natural operators $T^{(0,0)} ⇝ T^{(1,1)}T^{(r)}$

100%
Mikulski W.
Colloquium Mathematicum
|
2003
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tom 96
|
nr 1
5-16
EN
We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A(f): TT^{(r)}M → TT^{(r)}M$ on the vector r-tangent bundle $T^{(r)}M = (J^r(M,ℝ)₀)*$ over M. This problem is reflected in the concept of natural operators $A:T^{(0,0)}_{|ℳ fₙ} ⇝ T^{(1,1)}T^{(r)}$. For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over $𝓒^{∞}(T^{(r)}ℝ)$ and we construct explicitly a basis of this module.
10
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Liftings of 1-forms to $(J^{r}T*)*$

100%
Mikulski W.
Colloquium Mathematicum
|
2002
|
tom 91
|
nr 1
69-77
EN
Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.
11
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Product preserving gauge bundle functors on vector bundles

100%
Mikulski W.
Colloquium Mathematicum
|
2001
|
tom 90
|
nr 2
277-285
EN
A complete description is given of all product preserving gauge bundle functors F on vector bundles in terms of pairs (A,V) consisting of a Weil algebra A and an A-module V with $dim_{ℝ}(V) < ∞$. Some applications of this result are presented.
12
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Non-existence of exchange transformations of iterated jet functors

64%
Doupovec M., Mikulski W.
Banach Center Publications
|
2007
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tom 76
|
nr 1
497-504
EN
We study the problem of the non-existence of natural transformations $J^{r}J^{s}Y → J^{s}J^{r}Y$ of iterated jet functors depending on some geometric object on the base of Y.
13
Content available remote

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.
14
Content available remote

On iteration of higher order jets and prolongation of connections

64%
Doupovec M., Mikulski W.
Annales Polonici Mathematici
|
2011
|
tom 100
|
nr 1
55-75
EN
We introduce exchange natural equivalences of iterated nonholonomic, holonomic and semiholonomic jet functors, depending on a classical linear connection on the base manifold. We also classify some natural transformations of this type. As an application we introduce prolongation of higher order connections to jet bundles.
15
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On lifting of connections to Weil bundles

64%
Kurek J., Mikulski W.
Annales Polonici Mathematici
|
2012
|
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 $C_M(∇):T¹_{p-1}M = ⨁^{p-1}_{M} TM → T*M ⊗ T*M ⊗ TM$.
16
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On prolongation of higher order connections

64%
Doupovec M., Mikulski W.
Annales Polonici Mathematici
|
2011
|
tom 102
|
nr 3
279-292
EN
We describe all bundle functors G admitting natural operators transforming rth order holonomic connections on a fibered manifold Y → M into rth order holonomic connections on GY → M. For second order holonomic connections we classify all such natural operators.
17
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On involutions of iterated bundle functors

64%
Doupovec M., Mikulski W.
Colloquium Mathematicum
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2006
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tom 106
|
nr 1
135-145
EN
We introduce the concept of an involution of iterated bundle functors. Then we study the problem of the existence of an involution for bundle functors defined on the category of fibered manifolds with m-dimensional bases and of fibered manifold morphisms covering local diffeomorphisms. We also apply our results to prolongation of connections.
18
Content available remote

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).
19
Content available remote

Gauge natural constructions on higher order principal prolongations

64%
Doupovec M., Mikulski W.
Annales Polonici Mathematici
|
2007
|
tom 92
|
nr 1
87-97
EN
Let $W^r_mP$ be a principal prolongation of a principal bundle P → M. We classify all gauge natural operators transforming principal connections on P → M and rth order linear connections on M into general connections on $W^r_mP → M$. We also describe all geometric constructions of classical linear connections on $W^r_mP$ from principal connections on P → M and rth order linear connections on M.
20
Content available remote

Gauge natural prolongation of connections

64%
Doupovec M., Mikulski W.
Annales Polonici Mathematici
|
2009
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tom 95
|
nr 1
37-50
EN
The main result is the classification of all gauge bundle functors H on the category $𝓟ℬ_m(G)$ which admit gauge natural operators transforming principal connections on P → M into general connections on HP → M. We also describe all gauge natural operators of this type. Similar problems are solved for the prolongation of principal connections to HP → P. A special attention is paid to linear connections.
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