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Natural operations of Hamiltonian type on the cotangent bundle

Doupovec, MiroslavKurek, Jan — 1997

Proceedings of the 16th Winter School "Geometry and Physics"

The authors study some geometrical constructions on the cotangent bundle T * M from the viewpoint of natural operations. First they deduce that all natural operators transforming functions on T * M into vector fields on T * M are linearly generated by the Hamiltonian vector field with respect to the canonical symplectic structure of T * M and by the Liouville vector field of T * M . Then they determine all natural operators transforming pairs of functions on T * M into functions on T * M . In this case, the main generator is...

Liftings of 1-forms to some non product preserving bundles

Doupovec, MiroslavKurek, Jan — 1998

Proceedings of the 17th Winter School "Geometry and Physics"

Summary: The article is devoted to the question how to geometrically construct a 1-form on some non product preserving bundles by means of a 1-form on an original manifold M . First, we will deal with liftings of 1-forms to higher-order cotangent bundles. Then, we will be concerned with liftings of 1-forms to the bundles which arise as a composition of the cotangent bundle with the tangent or cotangent bundle.

Natural transformations of higher order cotangent bundle functors

Jan Kurek — 1993

Annales Polonici Mathematici

We determine all natural transformations of the rth order cotangent bundle functor T r * into T s * in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of T r * into itself form an r-parameter family linearly generated by the pth power transformations with p =1,...,r.

Natural affinors on higher order cotangent bundle

Jan Kurek — 1992

Archivum Mathematicum

All natural affinors on the r -th order cotangent bundle T r * M are determined. Basic affinors of this type are the identity affinor id of T T r * M and the s -th power affinors Q M s : T T r * M V T r * M with s = 1 , , r defined by the s -th power transformations A s r , r of T r * M . An arbitrary natural affinor is a linear combination of the basic ones.

Connections from trivializations

Jan KurekWłodzimierz Mikulski — 2016

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.

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

Jan KurekWłodzimierz Mikulski — 2014

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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 KurekWłodzimierz Mikulski — 2016

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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 .

Torsions of connections on higher order cotangent bundles

Miroslav DoupovecJan Kurek — 2003

Czechoslovak Mathematical Journal

By a torsion of a general connection Γ on a fibered manifold Y M we understand the Frölicher-Nijenhuis bracket of Γ and some canonical tangent valued one-form (affinor) on Y . Using all natural affinors on higher order cotangent bundles, we determine all torsions of general connections on such bundles. We present the geometrical interpretation and study some properties of the torsions.

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

Jan KurekWłodzimierz M. Mikulski — 2015

Annales UMCS, Mathematica

If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) between the tangent TM and the cotangent T∗ M bundles of M. In the present note, we generalize this isomorphism to the one T(r)M ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in...

Canonical symplectic structures on the r-th order tangent bundle of a symplectic manifold.

Jan KurekWlodzimierz M. Mikulski — 2006

Extracta Mathematicae

We describe all canonical 2-forms Λ(ω) on the r-th order tangent bundle TM = J (;M) of a symplectic manifold (M, ω). As a corollary we deduce that all canonical symplectic structures Λ(ω) on TM over a symplectic manifold (M, ω) are of the form Λ(ω) = Σ αω for all real numbers α with α ≠ 0, where ω is the (k)-lift (in the sense of A. Morimoto) of ω to TM.

On prolongations of projectable connections

Jan KurekWłodzimierz 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 lifting of connections to Weil bundles

Jan KurekWłodzimierz 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 infinitesimal automorphisms of foliated manifolds

Jan KurekWłodzimierz M. Mikulski — 2007

Annales Polonici Mathematici

Let F:ℱol → ℱℳ be a product preserving bundle functor on the category ℱol of foliated manifolds (M,ℱ) without singularities and leaf respecting maps. We describe all natural operators C transforming infinitesimal automorphisms X ∈ 𝒳(M,ℱ) of foliated manifolds (M,ℱ) into vector fields C(X)∈ 𝒳(F(M,ℱ)) on F(M,ℱ).

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