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Natural liftings of foliations to the $r$ -tangent bunde

Mikulski, Włodzimierz M. — 1994

Proceedings of the Winter School "Geometry and Physics"

Let $F$ be a $p$ -dimensional foliation on an $n$ -manifold $M$ , and $T^{r} M$ the $r$ -tangent bundle of $M$ . The purpose of this paper is to present some reltionship between the foliation $F$ and a natural lifting of $F$ to the bundle $T^{r} M$ . Let $L_{q}^{r} (F)$ $(q = 0, 1, \dots, r)$ be a foliation on $T^{r} M$ projectable onto $F$ and $L_{q}^{r} = {L_{q}^{r} (F)}$ a natural lifting of foliations to $T^{r} M$ . The author proves the following theorem: Any natural lifting of foliations to the $r$ -tangent bundle is equal to one of the liftings $L_{0}^{r}, L_{1}^{r}, \dots, L_{n}^{r}$ . The exposition is clear and well organized.

Natural transformations of Weil functors into bundle functors

Mikulski, Włodzimierz M. — 1990

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor $T^{A}$ of A-velocities [, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of $T^{A}$ into F is finite and is less than or equal to $dim (F_{0} ℛ^{k})$ . The spaces of all natural transformations of Weil functors into linear functors...

Natural transformations of foliations into foliations on the cotangent bundle

Mikulski, Włodzimierz M. — 1993

Proceedings of the Winter School "Geometry and Topology"

Natural lifting of connections to vertical bundles

Kolář, Ivan; Mikulski, Włodzimierz M. — 2000

Proceedings of the 19th Winter School "Geometry and Physics"

One studies the flow prolongation of projectable vector fields with respect to a bundle functor of order $(r, s, q)$ on the category of fibered manifolds. As a result, one constructs an operator transforming connections on a fibered manifold $Y$ into connections on an arbitrary vertical bundle over $Y$ . It is deduced that this operator is the only natural one of finite order and one presents a condition on vertical bundles over $Y$ under which every natural operator in question has finite order.

Properties of product preserving functors

Gancarzewicz, Jacek; Mikulski, Włodzimierz; Pogoda, Zdzisław — 1994

Proceedings of the Winter School "Geometry and Physics"

A product preserving functor is a covariant functor $ℱ$ from the category of all manifolds and smooth mappings into the category of fibered manifolds satisfying a list of axioms the main of which is product preserving: $ℱ (M_{1} \times M_{2}) = ℱ (M_{1}) \times ℱ (M_{2})$ . It is known that any product preserving functor $ℱ$ is equivalent to a Weil functor $T^{A}$ . Here $T^{A} (M)$ is the set of equivalence classes of smooth maps $ϕ : ℝ^{n} \to M$ and $ϕ, ϕ^{'}$ are equivalent if and only if for every smooth function $f : M \to ℝ$ the formal Taylor series at 0 of $f \circ ϕ$ and $f \circ ϕ^{'}$ are equal in $A = ℝ [[x_{1}, \dots, x_{n}]] / 𝔞$ . In this paper all...

Liftings of linear vector fields to product preserving gauge bundle functors on vector bundles.

Kureš, Miroslav; Mikulski, Włodzimierz M. — 2003

Lobachevskii Journal of Mathematics

The natural operators lifting $k$ -projectable vector fields to product-preserving bundle functors on $k$ -fibered manifolds.

Mikulski, Włodzimierz M.; Tomáš, Jiři M. — 2003

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

On regular local operators on smooth maps

Włodzimierz Mikulski — 2015

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Let X, Y, Z, W be manifolds and π : Z → X be a surjective submersion. We characterize π-local regular operators A : C∞(X,Y) → C∞(Z,W) in terms of the corresponding maps Ã : J∞(X,Y) ×XZ → W satisfying the so-called local finite order factorization property.

Continuity of projections of natural bundles

Włodzimierz M. Mikulski — 1992

Annales Polonici Mathematici

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)...

Product preserving gauge bundle functors on vector bundles

Włodzimierz M. Mikulski — 2001

Colloquium Mathematicae

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 $d i m_{ℝ} (V) < \infty$ . Some applications of this result are presented.

Liftings of 1-forms to $(J^{r} T ) $

Włodzimierz M. Mikulski — 2002

Colloquium Mathematicae

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.

The natural operators $T^{(0, 0)} ⇝ T^{(1, 1)} T^{(r)}$

Włodzimierz M. Mikulski — 2003

Colloquium Mathematicae

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A (f) : T T^{(r)} M \to T T^{(r)} M$ 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 $A : T_{| ℳ f ₙ}^{(0, 0)} ⇝ T^{(1, 1)} T^{(r)}$ . 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 $^{\infty} (T^{(r)} ℝ)$ and we construct explicitly a basis of this module.

Prolongation of linear semibasic tangent valued forms to product preserving gauge bundles of vector bundles.

Wlodzimierz M. Mikulski — 2006

Extracta Mathematicae

Let A be a Weil algebra and V be an A-module with dim V < ∞. Let E → M be a vector bundle and let TE → TM be the vector bundle corresponding to (A,V). We construct canonically a linear semibasic tangent valued p-form Tφ : T E → ΛT*TM ⊗ TTE on TE → TM from a linear semibasic tangent valued p-form φ : E → ΛT*M ⊗ TE on E → M. For the Frolicher-Nijenhuis bracket we prove that [[Tφ, Tψ]] = T ([[φ,ψ]]) for any linear semibasic tangent valued p- and q-forms φ and ψ on E → M. We apply these results...

On "special" fibred coordinates for general and classical connections

Włodzimierz M. Mikulski — 2010

Annales Polonici Mathematici

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 $\tilde{ℱ} (Γ, \nabla)$ on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.

Product preserving gauge bundle functors on all principal bundle homomorphisms

Włodzimierz M. Mikulski — 2011

Annales Polonici Mathematici

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....

Lifting distributions to the cotangent bundle

Włodzimierz M. Mikulski — 2008

Annales Polonici Mathematici

A classification of all $ℳ f_{m}$ -natural operators $A : G r_{p} ⟿ G r_{q} T *$ 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.

Product preserving bundles on foliated manifolds

Włodzimierz M. Mikulski — 2004

Annales Polonici Mathematici

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.

On the fiber product preserving gauge bundle functors on vector bundles

Włodzimierz M. Mikulski — 2003

Annales Polonici Mathematici

We present a complete description of all fiber product preserving gauge bundle functors F on the category $_{m}$ of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented.

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

Włodzimierz M. Mikulski — 2003

Annales Polonici Mathematici

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 $J^{r} E$ of E are given. Similar results are deduced for the r-jet prolongations $J_{v}^{r} E$ and $J^{[r]} E$ in place of $J^{r} E$ .

On prolongation of connections

Włodzimierz M. Mikulski — 2010

Annales Polonici Mathematici

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 $B^{r} (Γ, \nabla) : Y \to J^{r} Y$ 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 $B (Γ, \nabla) : Y \to J^{r} Y$ 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...

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Currently displaying 1 – 20 of 83

Natural liftings of foliations to the $r$ -tangent bunde

Natural transformations of Weil functors into bundle functors

Natural transformations of foliations into foliations on the cotangent bundle

Natural lifting of connections to vertical bundles

Properties of product preserving functors

Liftings of linear vector fields to product preserving gauge bundle functors on vector bundles.

The natural operators lifting $k$ -projectable vector fields to product-preserving bundle functors on $k$ -fibered manifolds.

On regular local operators on smooth maps

Continuity of projections of natural bundles

Product preserving gauge bundle functors on vector bundles

Liftings of 1-forms to $(J^{r} T ) $

The natural operators $T^{(0, 0)} ⇝ T^{(1, 1)} T^{(r)}$

Prolongation of linear semibasic tangent valued forms to product preserving gauge bundles of vector bundles.

On "special" fibred coordinates for general and classical connections

Product preserving gauge bundle functors on all principal bundle homomorphisms

Lifting distributions to the cotangent bundle

Product preserving bundles on foliated manifolds

On the fiber product preserving gauge bundle functors on vector bundles

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

On prolongation of connections