Displaying similar documents to “Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations”

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 .

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.

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 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 | f T ( 0 , 0 ) ( F | f ) * transforming vector fields to functions on the dual bundle functor ( F | f ) * . Next, we study the natural operators T * | f T * ( F | f ) * lifting 1-forms to ( F | f ) * . As an application we classify the natural operators T * | f T * ( F | f ) * for some well known vector bundle functors F.

Liftings of 1-forms to ( J r T * ) *

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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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 linear operators T * T T ( r )

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

Colloquium Mathematicae

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For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators T * × f T ( 0 , 0 ) T ( 0 , 0 ) T ( r ) is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators T * | f T T ( r ) is obtained.

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 .

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...

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 Q ( r e g T k r K k r ) and C r Q ( r e g T k r * K k r * ) over n-manifolds is proved. Some generalizations are obtained.

The natural operators T ( 0 , 0 ) T ( 1 , 1 ) T ( r )

Włodzimierz M. Mikulski (2003)

Colloquium Mathematicae

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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.

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 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 .

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 .

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...

Lifting vector fields to the rth order frame bundle

J. Kurek, W. M. Mikulski (2008)

Colloquium Mathematicae

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We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle L r M = i n v J r ( m , M ) over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on L r M . In both cases we deduce that the spaces of all operators in question form free ( m ( C r m + r - 1 ) + 1 ) -dimensional modules over algebras of all smooth maps J r - 1 T ̃ m and J r - 1 T m respectively, where C k = n ! / ( n - k ) ! k ! . We explicitly construct bases of these modules. In particular,...

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold X have a pseudo-effective (instead of generically nef) determinant. A first consequence is that X is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of X is not covered by compact positive-dimensional analytic...

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...

Linear natural operators lifting p -vectors to tensors of type ( q , 0 ) on Weil bundles

Jacek Dębecki (2016)

Czechoslovak Mathematical Journal

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We give a classification of all linear natural operators transforming p -vectors (i.e., skew-symmetric tensor fields of type ( p , 0 ) ) on n -dimensional manifolds M to tensor fields of type ( q , 0 ) on T A M , where T A is a Weil bundle, under the condition that p 1 , n p and n q . The main result of the paper states that, roughly speaking, each linear natural operator lifting p -vectors to tensor fields of type ( q , 0 ) on T A is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

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 ) .

On prolongation of connections

Włodzimierz M. Mikulski (2010)

Annales Polonici Mathematici

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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...

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...

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If ( M , g ) is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism T M = ˜ T * M given by v g ( v , - ) between the tangent T M and the cotangent T * M bundles of M . In the present note, we generalize this isomorphism to the one T ( r ) M = ˜ T r * M between the r -th order vector tangent T ( r ) M = ( J r ( M , R ) 0 ) * and the r -th order cotangent T r * M = J r ( M , R ) 0 bundles of M . Next, we describe all base preserving  vector bundle maps C M ( g ) : T ( r ) M T r * M depending on a Riemannian metric g in terms of natural (in g ) tensor fields on 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 .

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...

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.

Grauert's line bundle convexity, reduction and Riemann domains

Viorel Vâjâitu (2016)

Czechoslovak Mathematical Journal

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We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X . This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H 0 ( X , L ) separates each point of X , then X can be realized as a Riemann domain over the complex projective...

The natural operators T | f T * T r * and T | f Λ ² T * T r *

W. M. Mikulski (2002)

Colloquium Mathematicae

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Let r and n be natural numbers. For n ≥ 2 all natural operators T | f T * T r * transforming vector fields on n-manifolds M to 1-forms on T r * M = J r ( M , ) are classified. For n ≥ 3 all natural operators T | f Λ ² T * T r * transforming vector fields on n-manifolds M to 2-forms on T r * M are completely described.