Natural maps depending on reductions of frame bundles

Ivan Kolář

Displaying similar documents to “Natural maps depending on reductions of frame bundles”

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

Similarity:

We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A} M$ , where $T^{A}$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

The natural operators lifting projectable vector fields to some fiber product preserving bundles

W. M. Mikulski (2003)

Annales Polonici Mathematici

Similarity:

Admissible fiber product preserving bundle functors F on $ℱ ℳ_{m}$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ ℳ_{m}$ all natural operators $B : T_{p r o j | ℱ ℳ_{m, n}} \to T F$ lifting projectable vector fields to F are classified.

Non-existence of some canonical constructions on connections

Włodzimierz M. Mikulski (2003)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

For a vector bundle functor $H : ℳ f \to 𝒱 ℬ$ with the point property we prove that $H$ is product preserving if and only if for any $m$ and $n$ there is an $ℱ ℳ_{m, n}$ -natural operator $D$ transforming connections $Γ$ on $(m, n)$ -dimensional fibered manifolds $p : Y \to M$ into connections $D (Γ)$ on $H p : H Y \to H M$ . For a bundle functor $E : ℱ ℳ_{m, n} \to ℱ ℳ$ with some weak conditions we prove non-existence of $ℱ ℳ_{m, n}$ -natural operators $D$ transforming connections $Γ$ on $(m, n)$ -dimensional fibered manifolds $Y \to M$ into connections $D (Γ)$ on $E Y \to M$ .

Fiber product preserving bundle functors as modified vertical Weil functors

Włodzimierz M. Mikulski (2015)

Czechoslovak Mathematical Journal

Similarity:

We introduce the concept of modified vertical Weil functors on the category $ℱ ℳ_{m}$ of fibred manifolds with $m$ -dimensional bases and their fibred maps with embeddings as base maps. Then we describe all fiber product preserving bundle functors on $ℱ ℳ_{m}$ in terms of modified vertical Weil functors. The construction of modified vertical Weil functors is an (almost direct) generalization of the usual vertical Weil functor. Namely, in the construction of the usual vertical Weil functors, we replace...

Bundle functors with the point property which admit prolongation of connections

W. M. Mikulski (2010)

Annales Polonici Mathematici

Similarity:

Let F:ℳ f →ℱℳ be a bundle functor with the point property F(pt) = pt, where pt is a one-point manifold. We prove that F is product preserving if and only if for any m and n there is an $ℱ ℳ_{m, n}$ -canonical construction D of general connections D(Γ) on Fp:FY → FM from general connections Γ on fibred manifolds p:Y → M.

On the fiber product preserving gauge bundle functors on vector bundles

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

Similarity:

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.

The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

Similarity:

Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators $T_{p r o j | ℱ ℳ_{m, n}} ⇝ T^{(0, 0)} (F_{| ℱ ℳ_{m, n}}) *$ 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 $T *_{h o r | ℱ ℳ_{m, n}} ⇝ T * (F_{| ℱ ℳ_{m, n}}) *$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T *_{h o r | ℱ ℳ_{m, n}} ⇝ T * (F_{| ℱ ℳ_{m, n}}) *$ for some vector bundle functors F on fibered manifolds.

The natural operators lifting vector fields to generalized higher order tangent bundles

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

Similarity:

For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{(r), a}$ over $n$ -manifolds such that $T^{(r), 0}$ is the (classical) vector tangent bundle $T^{(r)}$ of order $r$ . For integers $r \geq 1$ and $n \geq 3$ and a real number $a < 0$ we classify all natural operators $T_{| M_{n}} ⇝ T T^{(r), a}$ lifting vector fields from $n$ -manifolds to $T^{(r), a}$ .

Natural affinors on the (r,s,q)-cotangent bundle of a fibered manifold

J. Kurek, W. M. Mikulski (2004)

Annales Polonici Mathematici

Similarity:

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.

On "special" fibred coordinates for general and classical connections

Włodzimierz M. Mikulski (2010)

Annales Polonici Mathematici

Similarity:

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.

The natural operators lifting 1-forms to some vector bundle functors

J. Kurek, W. M. Mikulski (2002)

Colloquium Mathematicae

Similarity:

Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{| ℳ f ₙ} ⇝ T^{(0, 0)} (F_{| ℳ f ₙ}) *$ transforming vector fields to functions on the dual bundle functor $(F_{| ℳ f ₙ}) *$ . Next, we study the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ lifting 1-forms to $(F_{| ℳ f ₙ}) *$ . As an application we classify the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ for some well known vector bundle functors F.