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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 , , 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 , , 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 k ) . The spaces of all natural transformations of Weil functors into linear functors...

Natural lifting of connections to vertical bundles

Kolář, IvanMikulski, 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.

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)...

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 T T ( r ) M on the vector r-tangent bundle T ( r ) M = ( J r ( M , ) ) * over M. This problem is reflected in the concept of natural operators A : T | f ( 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 ( T ( 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...

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....

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 , ) : Y 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 , ) : Y 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...

On symmetrization of jets

Włodzimierz M. Mikulski — 2011

Czechoslovak Mathematical Journal

Let F = F ( A , H , t ) and F 1 = F ( A 1 , H 1 , t 1 ) be fiber product preserving bundle functors on the category ℱℳ m of fibred manifolds Y with m -dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism ( A , H , t ) ( A 1 , H 1 , t 1 ) to be a G L ( m ) -invariant algebra homomorphism ν : A A 1 with t 1 = ν t . The main result is that there exists an ℱℳ m -natural transformation F Y F 1 Y depending on a classical linear connection on the base of Y if and only if there exists a quasi-morphism ( A , H , t ) ( A 1 , H 1 , t 1 ) . As applications, we study existence problems of symmetrization (holonomization)...

On the Kolář connection

Włodzimierz M. Mikulski — 2013

Archivum Mathematicum

Let Y M be a fibred manifold with m -dimensional base and n -dimensional fibres and E M be a vector bundle with the same base M and with n -dimensional fibres (the same n ). If m 2 and n 3 , we classify all canonical constructions of a classical linear connection A ( Γ , Λ , Φ , Δ ) on Y from a system ( Γ , Λ , Φ , Δ ) consisting of a general connection Γ on Y M , a torsion free classical linear connection Λ on M , a vertical parallelism Φ : Y × M E V Y on Y and a linear connection Δ on E M . An example of such A ( Γ , Λ , Φ , Δ ) is the connection ( Γ , Λ , Φ , Δ ) by I. Kolář.

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