Displaying similar documents to “On prolongation of connections”

On prolongations of projectable connections

Jan Kurek, Włodzimierz M. Mikulski (2011)

Annales Polonici Mathematici

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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...

Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections

Anna Bednarska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We classify all 2 m 1 , m 2 , n 1 , n 2 -natural operators A transforming projectable-projectable torsion-free classical linear connections on fibered-fibered manifolds Y of dimension ( m 1 , m 2 , n 1 , n 2 ) into r th order Lagrangians A ( r ) on the fibered-fibered linear frame bundle L f i b - f i b ( Y ) on Y . Moreover, we classify all 2 m 1 , m 2 , n 1 , n 2 -natural operators B transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds Y of dimension  ( m 1 , m 2 , n 1 , n 2 ) into Euler morphism B ( ) on L f i b - f i b ( Y ) . These classifications can be expanded on...

On lifts of projectable-projectable classical linear connections to the cotangent bundle

Anna Bednarska (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We describe all 2 m 1 , m 2 , n 1 , n 2 -natural operators D : Q p r o j - p r o j τ Q T * transforming projectable-projectable classical torsion-free linear connections on fibred-fibred manifolds Y into classical linear connections D ( ) on cotangent bundles T * Y of Y . We show that this problem can be reduced to finding 2 m 1 , m 2 , n 1 , n 2 -natural operators D : Q p r o j - p r o j τ ( T * , p T * q T ) for p = 2 , q = 1 and p = 3 , q = 0 .

The vertical prolongation of the projectable connections

Anna Bednarska (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We prove that any first order 2 m 1 , m 2 , n 1 , n 2 -natural operator transforming projectable general connections on an ( m 1 , m 2 , n 1 , n 2 ) -dimensional fibred-fibred manifold p = ( p , p ) : ( p Y : Y Y ) ( p M : M M ) into general connections on the vertical prolongation V Y M of p : Y M is the restriction of the (rather well-known) vertical prolongation operator 𝒱 lifting general connections Γ ¯ on a fibred manifold Y M into 𝒱 Γ ¯ (the vertical prolongation of Γ ¯ ) on V Y M .

On canonical constructions on connections

Jan Kurek, Włodzimierz Mikulski (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We study  how a projectable general connection Γ in a 2-fibred manifold Y 2 Y 1 Y 0   and a general vertical connection Θ in Y 2 Y 1 Y 0 induce a general connection A ( Γ , Θ ) in Y 2 Y 1 .

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

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We classify all m , n -natural operators : J ² J ² V A transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections ( Γ ) : V A Y J ² V A Y on the vertical Weil bundle V A Y M corresponding to a Weil algebra A.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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We prove that the problem of finding all f m -natural operators B : Q Q T 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 f m -natural operators C : Q ( T ¹ p - 1 , T * T * T ) transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps C M ( ) : T ¹ p - 1 M = M p - 1 T M T * M T * M T M .

On the γ -equivalence of semiholonomic jets

Miroslav Doupovec, Ivan Kolář (2019)

Archivum Mathematicum

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It is well known that the concept of holonomic r -jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic r -jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.

The general rigidity result for bundles of A -covelocities and A -jets

Jiří M. Tomáš (2017)

Czechoslovak Mathematical Journal

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Let M be an m -dimensional manifold and A = 𝔻 k r / I = N A a Weil algebra of height r . We prove that any A -covelocity T x A f T x A * M , x M is determined by its values over arbitrary max { width A , m } regular and under the first jet projection linearly independent elements of T x A M . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T A * M T r * M without coordinate computations, which improves and generalizes the partial...

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let f m be the category of m -dimensional manifolds and local diffeomorphisms and  let T be the tangent functor on f m . Let 𝒱 be the category of real vector spaces and linear maps and let 𝒱 m be the category of m -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors F : 𝒱 m 𝒱 admitting f m -natural operators J ˜ transforming classical linear connections on m -dimensional manifolds M into almost complex structures J ˜ ( ) on F ( T ) M = x M F ( T x M ) .

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let π V denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an m -measurable subset of ℝⁿ with m ( A ) < . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle ( V , v ) | V G ( n , m ) , v V such that, for all P ∈ A, one has m ( n - m ) ( V G ( n , m ) | ( V , π V ( P ) ) Z ) > 0 . One can replace “for all P ∈ A” by “for...

The canonical constructions of connections on total spaces of fibred manifolds

Włodzimierz M. Mikulski (2024)

Archivum Mathematicum

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We classify classical linear connections A ( Γ , Λ , Θ ) on the total space Y of a fibred manifold Y M induced in a natural way by the following three objects: a general connection Γ in Y M , a classical linear connection Λ on M and a linear connection Θ in the vertical bundle V Y Y . The main result says that if dim ( M ) 3 and dim ( Y ) - dim ( M ) 3 then the natural operators A under consideration form the 17 dimensional affine space.

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

Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital C * -algebras

Kazunori Kodaka (2022)

Mathematica Bohemica

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Let 𝒜 = { A t } t G and = { B t } t G be C * -algebraic bundles over a finite group G . Let C = t G A t and D = t G B t . Also, let A = A e and B = B e , where e is the unit element in G . We suppose that C and D are unital and A and B have the unit elements in C and D , respectively. In this paper, we show that if there is an equivalence 𝒜 - -bundle over G with some properties, then the unital inclusions of unital C * -algebras A C and B D induced by 𝒜 and are strongly Morita equivalent. Also, we suppose that 𝒜 and are saturated and that A ' C = 𝐂 1 . We show that...

Semistability of Frobenius direct images over curves

Vikram B. Mehta, Christian Pauly (2007)

Bulletin de la Société Mathématique de France

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Let X be a smooth projective curve of genus g 2 defined over an algebraically closed field k of characteristic p &gt; 0 . Given a semistable vector bundle  E over X , we show that its direct image F * E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank- p vector bundles  F * L , where L is a line bundle over X .

On a new normalization for tractor covariant derivatives

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)

Journal of the European Mathematical Society

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A regular normal parabolic geometry of type G / P on a manifold M gives rise to sequences D i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ω on the corresponding tractor bundle V , where ω is the normal Cartan connection. The first operator D 0 in the sequence is overdetermined and it is well known that ω yields the prolongation of this operator in the homogeneous case M = G / P . Our first...

Steinness of bundles with fiber a Reinhardt bounded domain

Karl Oeljeklaus, Dan Zaffran (2006)

Bulletin de la Société Mathématique de France

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Let E denote a holomorphic bundle with fiber D and with basis B . Both D and B are assumed to be Stein. For D a Reinhardt bounded domain of dimension d = 2 or 3 , we give a necessary and sufficient condition on D for the existence of a non-Stein such E (Theorem 1 ); for d = 2 , we give necessary and sufficient criteria for E to be Stein (Theorem 2 ). For D a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for E to be Stein (Theorem...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let k be a field of characteristic p &gt; 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

Locally functionally countable subalgebra of ( L )

M. Elyasi, A. A. Estaji, M. Robat Sarpoushi (2020)

Archivum Mathematicum

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Let L c ( X ) = { f C ( X ) : C f ¯ = X } , where C f is the union of all open subsets U X such that | f ( U ) | 0 . In this paper, we present a pointfree topology version of L c ( X ) , named c ( L ) . We observe that c ( L ) enjoys most of the important properties shared by ( L ) and c ( L ) , where c ( L ) is the pointfree version of all continuous functions of C ( X ) with countable image. The interrelation between ( L ) , c ( L ) , and c ( L ) is examined. We show that L c ( X ) c ( 𝔒 ( X ) ) for any space X . Frames L for which c ( L ) = ( L ) are characterized.

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a zero-dimensional space and C c ( X ) be the set of all continuous real valued functions on X with countable image. In this article we denote by C c K ( X ) (resp., C c ψ ( X ) ) the set of all functions in C c ( X ) with compact (resp., pseudocompact) support. First, we observe that C c K ( X ) = O c β 0 X X (resp., C c ψ ( X ) = M c β 0 X υ 0 X ), where β 0 X is the Banaschewski compactification of X and υ 0 X is the -compactification of X . This implies that for an -compact space X , the intersection of all free maximal ideals in C c ( X ) is equal to C c K ( X ) , i.e., M c β 0 X X = C c K ( X ) . By applying...

On the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

Lifting right-invariant vector fields and prolongation of connections

W. M. Mikulski (2009)

Annales Polonici Mathematici

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We describe all m ( G ) -gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation W r P = P r M × M J r P of P → M. In other words, we classify all m ( G ) -natural transformations J r L P × M W r P T W r P = L W r P × M W r P covering the identity of W r P , where J r L P is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all m ( G ) -natural transformations which are similar to the Kumpera-Spencer isomorphism J r L P = L W r P . We formulate axioms which...