Natural operators lifting functions to affinors on higher order cotangent bundles

W. M. Mikulski

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Displaying similar documents to “Natural operators lifting functions to affinors on higher order cotangent bundles”

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

W. M. Mikulski (2003)

Annales Polonici Mathematici

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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.

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

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

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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 $T^{(r), a}$ .

The natural operators lifting connections to higher order cotangent bundles

Włodzimierz M. Mikulski (2014)

Czechoslovak Mathematical Journal

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We prove that the problem of finding all $ℳ f_{m}$ -natural operators $C : Q ⇝ Q T^{r *}$ lifting classical linear connections $\nabla$ on $m$ -manifolds $M$ into classical linear connections $C_{M} (\nabla)$ on the $r$ -th order cotangent bundle $T^{r *} M = J^{r} {(M, ℝ)}_{0}$ of $M$ can be reduced to the well known one of describing all $ℳ f_{m}$ -natural operators $D : Q ⇝ ⨂^{p} T \otimes ⨂^{q} T^{*}$ sending classical linear connections $\nabla$ on $m$ -manifolds $M$ into tensor fields $D_{M} (\nabla)$ of type $(p, q)$ on $M$ .

C-semigroup bundles and C-algebras whose irreducible representations are all finite dimensional

Thomas Müller

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We investigate the structure of C*-algebras with a finite bound on the dimensions of their irreducible representations, sometimes called “subhomogeneous”.In the first chapter we develop the theory of C*-semigroup bundles. These are C*-bundles over semigroups together with a “structure map” which links the semigroup structure of the base space lo the bundle. Under suitable conditions we prove the existence of “enough” bounded sections, which arc “compatible” with the C*-semigroup bundle...

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

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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.

A classification theorem on Fano bundles

Roberto Muñoz, Luis E. Solá Conde, Gianluca Occhetta (2014)

Annales de l’institut Fourier

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In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^{2} (X, ℤ) ≅ H^{4} (X, ℤ) ≅ ℤ$ . The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $ℙ (ℰ)$ , that allows us to obtain the cohomological invariants of $X$ and $ℰ$ . As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles

Vadim V. Shurygin, Svetlana K. Zubkova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The second order transverse bundle $T_{}^{2} M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra $𝔻^{2}$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $𝔻^{2}$ -smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $𝔻^{2}$ -smooth foliated diffeomorphism between two second order transverse...

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

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

Colloquium Mathematicae

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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.

Line bundles with partially vanishing cohomology

Burt Totaro (2013)

Journal of the European Mathematical Society

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Define a line bundle $L$ on a projective variety to be $q$ -ample, for a natural number $q$ , if tensoring with high powers of $L$ kills coherent sheaf cohomology above dimension $q$ . Thus 0-ampleness is the usual notion of ampleness. We show that $q$ -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that $q$ -ampleness is a Zariski open condition, which is not clear from the definition. ...

Cyclic actions on $S^{2}$ - and $P^{2}$ -bundles over $S^{1}$

Józef H. Przytycki (1982)

Colloquium Mathematicae

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Non-existence of some natural operators on connections

W. M. Mikulski (2003)

Annales Polonici Mathematici

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Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^{r} ₀ ⟿ Q (r e g T_{k}^{r} \to K_{k}^{r})$ and $C^{r} ₀ ⟿ Q (r e g T_{k}^{r *} \to K_{k}^{r *})$ over n-manifolds is proved. Some generalizations are obtained.

Singular Bundles with Bounded $L^{2}$ -Curvatures

Thiemo Kessel, Tristan Rivière (2008)

Bollettino dell'Unione Matematica Italiana

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We consider calculus of variations of the Yang-Mills functional in dimensions larger than the critical dimension 4. We explain how this naturally leads to a class of - a priori not well-defined - singular bundles including possibly "almost everywhere singular bundles". In order to overcome this difficulty, we suggest a suitable new framework, namely the notion of singular bundles with bounded $L^{2}$ -curvatures.

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

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

Colloquium Mathematicae

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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.

Natural maps depending on reductions of frame bundles

Ivan Kolář (2011)

Annales Polonici Mathematici

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We clarify how the natural transformations of fiber product preserving bundle functors on $ℱ ℳ_{m}$ can be constructed by using reductions of the rth order frame bundle of the base, $ℱ ℳ_{m}$ being the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. The iteration of two general r-jet functors is discussed in detail.

Pseudo-real principal Higgs bundles on compact Kähler manifolds

Indranil Biswas, Oscar García-Prada, Jacques Hurtubise (2014)

Annales de l’institut Fourier

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Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_{G}$ . We define pseudo-real principal $G$ -bundles on $X$ . These are generalizations of real algebraic principal $G$ -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$ -bundles. Their relationships with the usual...

Liftings of functions and vector fields to natural bundles

Jacek Gancarzewicz

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CONTENTS0. Introduction...................................................................................................51. Natural bundles...........................................................................................102. Liftings of functions.....................................................................................153. Liftings of functions to the r-frame bundle...................................................224. A space of liftings of functions.....................................................................265....

Lifting to the r-frame bundle by means of connections

J. Kurek, W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let m and r be natural numbers and let $P^{r} : ℳ f_{m} \to ℱ ℳ$ be the rth order frame bundle functor. Let $F : ℳ f_{m} \to ℱ ℳ$ and $G : ℳ f_{k} \to ℱ ℳ$ be natural bundles, where $k = d i m (P^{r} ℝ^{m})$ . We describe all $ℳ f_{m}$ -natural operators A transforming sections σ of $F M \to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G (P^{r} M) \to P^{r} M$ . We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

Singular principal $G$ -bundles on nodal curves

Alexander Schmitt (2005)

Journal of the European Mathematical Society

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In the present paper, we give a first general construction of compactified moduli spaces for semistable $G$ -bundles on an irreducible complex projective curve $X$ with exactly one node, where $G$ is a semisimple linear algebraic group over the complex numbers.

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...