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On a new normalization for tractor covariant derivatives

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

Journal of the European Mathematical Society

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 main result...

On curves and jets of curves on supermanifolds

Andrew James Bruce (2014)

Archivum Mathematicum

In this paper we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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

Anna Bednarska (2013)

Annales UMCS, Mathematica

We describe all F2Mm1,m2,n1,n2-natural operators D: Qτproj-prj ↝QT* 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 F2Mm1,m2,n1,n2-natural operators D: Qτproj-proj ↝ (T*,⊗pT*⊗⊗qT) for p = 2, q = 1 and p = 3, q = 0.

On principal connection like bundles

Włodzimierz M. Mikulski (2014)

Czechoslovak Mathematical Journal

Let 𝒫 m be the category of all principal fibred bundles with m -dimensional bases and their principal bundle homomorphisms covering embeddings. We introduce the concept of the so called ( r , m ) -systems and describe all gauge bundle functors on 𝒫 m of order r by means of the ( r , m ) -systems. Next we present several interesting examples of fiber product preserving gauge bundle functors on 𝒫 m of order r . Finally, we introduce the concept of product preserving ( r , m ) -systems and describe all fiber product preserving gauge...

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 prolongation of higher order connections

Miroslav Doupovec, Włodzimierz M. Mikulski (2011)

Annales Polonici Mathematici

We describe all bundle functors G admitting natural operators transforming rth order holonomic connections on a fibered manifold Y → M into rth order holonomic connections on GY → M. For second order holonomic connections we classify all such natural operators.

On prolongations of projectable connections

Jan Kurek, Włodzimierz M. Mikulski (2011)

Annales Polonici Mathematici

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 (r,r,r)-order...

On "special" fibred coordinates for general and classical connections

Włodzimierz M. Mikulski (2010)

Annales Polonici Mathematici

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 ˜ ( Γ , ) on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.

On special types of nonholonomic 3 -jets

Ivan Kolář (2012)

Archivum Mathematicum

We deduce a classification of all special types of nonholonomic 3 -jets. In the introductory part, we summarize the basic properties of nonholonomic r -jets.

On special types of semiholonomic 3 -jets

Ivan Kolář (2013)

Czechoslovak Mathematical Journal

First we summarize some properties of the nonholonomic r -jets from the functorial point of view. In particular, we describe the basic properties of our original concept of nonholonomic r -jet category. Then we deduce certain properties of the Weil algebras associated with nonholonomic r -jets. Next we describe an algorithm for finding the nonholonomic r -jet categories. Finally we classify all special types of semiholonomic 3 -jets.

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 existence of prolongation of connections

Miroslav Doupovec, Włodzimierz M. Mikulski (2006)

Czechoslovak Mathematical Journal

We classify all bundle functors G admitting natural operators transforming connections on a fibered manifold Y M into connections on G Y M . Then we solve a similar problem for natural operators transforming connections on Y M into connections on G Y Y .

On the functorial prolongations of principal bundles

Ivan Kolář, Antonella Cabras (2006)

Commentationes Mathematicae Universitatis Carolinae

We describe the fundamental properties of the infinitesimal actions related with functorial prolongations of principal and associated bundles with respect to fiber product preserving bundle functors. Our approach is essentially based on the Weil algebra technique and an original concept of weak principal bundle.

On the geometry of vertical Weil bundles

Ivan Kolář (2014)

Archivum Mathematicum

We describe some general geometric properties of the fiber product preserving bundle functors. Special attention is paid to the vertical Weil bundles. We discuss namely the flow natural maps and the functorial prolongation of connections.

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