Natural transformations between T²₁T*M and T*T²₁M
We determine all natural transformations T²₁T*→ T*T²₁ where . We also give a geometric characterization of the canonical isomorphism ψ₂ defined by Cantrijn et al.
Torsions of connections on time-dependent Weil bundles
We introduce the concept of a dynamical connection on a time-dependent Weil bundle and we characterize the structure of dynamical connections. Then we describe all torsions of dynamical connections.
Natural transformations of the composition of Weil and cotangent functors
We study geometrical properties of natural transformations depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations can be described in a uniform way by means of a simple geometrical construction.
On the underlying lower order bundle functors
For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with -dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors.
Natural transformations between and
Natural operators transforming vector fields to the second order tangent bundle
Natural liftings of -tensor fields to the tangent bundle
We determine all first order natural operators transforming –tensor fields on a manifold into –tensor fields on .
Invariant subspaces in higher order jet prolongations of a fibred manifold
We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.
Torsions of connections on higher order cotangent bundles
By a torsion of a general connection on a fibered manifold we understand the Frölicher-Nijenhuis bracket of and some canonical tangent valued one-form (affinor) on . 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.
Natural affinors on time-dependent Weil bundles
On the -equivalence of semiholonomic jets
It is well known that the concept of holonomic -jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic -jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.
Natural transformations of separated jets
Given a map of a product of two manifolds into a third one, one can define its jets of separated orders and . We study the functor of separated -jets. We determine all natural transformations of into itself and we characterize the canonical exchange from the naturality point of view.
On involutions of iterated bundle functors
We introduce the concept of an involution of iterated bundle functors. Then we study the problem of the existence of an involution for bundle functors defined on the category of fibered manifolds with m-dimensional bases and of fibered manifold morphisms covering local diffeomorphisms. We also apply our results to prolongation of connections.
Non-existence of exchange transformations of iterated jet functors
We study the problem of the non-existence of natural transformations of iterated jet functors depending on some geometric object on the base of Y.
On prolongation of higher order connections
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.
Gauge natural constructions on higher order principal prolongations
Let be a principal prolongation of a principal bundle P → M. We classify all gauge natural operators transforming principal connections on P → M and rth order linear connections on M into general connections on . We also describe all geometric constructions of classical linear connections on from principal connections on P → M and rth order linear connections on M.
Gauge natural prolongation of connections
The main result is the classification of all gauge bundle functors H on the category which admit gauge natural operators transforming principal connections on P → M into general connections on HP → M. We also describe all gauge natural operators of this type. Similar problems are solved for the prolongation of principal connections to HP → P. A special attention is paid to linear connections.
On iteration of higher order jets and prolongation of connections
We introduce exchange natural equivalences of iterated nonholonomic, holonomic and semiholonomic jet functors, depending on a classical linear connection on the base manifold. We also classify some natural transformations of this type. As an application we introduce prolongation of higher order connections to jet bundles.
On the Courant bracket on couples of vector fields and -forms
If (or ), all natural bilinear operators transforming pairs of couples of vector fields and -forms on -manifolds into couples of vector fields and -forms on are described. It is observed that any natural skew-symmetric bilinear operator as above coincides with the generalized Courant bracket up to three (two, respectively) real constants.
On the existence of prolongation of connections
We classify all bundle functors admitting natural operators transforming connections on a fibered manifold into connections on . Then we solve a similar problem for natural operators transforming connections on into connections on .
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