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EN
We construct some extension [...] of the flow operator [...] fibred frame bundle functor. Next using operator [...] we present some construction of general connections[...] depending on classical (not necessarily projectable) linear connections V on Y.
3
Content available remote Some natural operations on functions
EN
Let F : FMm &rarr FM be a bundle functor. We describe all FMm,n- natural operators L transforming functions f : Y &rarr R, Y is an element of Obj(FMm,n), into functions L(f) : FY &rarr R.
4
Content available remote Bundles of contact elements on fibered fibered manifolds and the flow operator
EN
We define the concept of a fibered fibered (k1, k2, l1, l2)- contact element of order (r1, . . . , r8) for r8 > r4 < r5 > r3, r8 > r6 < r7 > r2 and r1 < ri for ri = 2, 3, . . . , 8. For k1 < m1, k2 < m2, l1 < n1, l2 < n2, we define a bundle func tor Kr1,...,r8/ k1,k2,l1,l2 defined on the category FM2 m1,m2,n1,n2 of (m1,m2, n1,n2)-fibere fibered manifolds. We prove that the only natural transformation on the bundle functor Kr1,...r8/ k1,k2,l1,l2 is the identity one. Moreover, we prove that any natural operator lifting projectable vector fields Y to Kr1,...r8/ k1,k2,l1,l2 Y is a constant multiple of the flow operator.
5
Content available remote The natural bundles admitting natural lifting of linear connections
EN
Natural bundles admitting natural lifting of linear connections are characterized. Corollaries are presented. Some other similar results are obtained, too.
6
Content available remote Negative answers to some questions about constructions on connections
EN
Let m and n be natural numbers. For an arbitrary bundle functor G on the category FMm,n of fibred manifolds with m-dimensional bases and n-dimensional fibers and their local fibered diffeomorphisms we prove that there is no FMm,n-natural operator V transforming general connections.
EN
For natural numbers n > 2 and r > 1 all natural operators T* (Jr(*2T*)) lifting vector fields from n-manifolds M into 1-forms on Jr(*2T*) are classified. It's proved that the set of all natural operators A.
EN
We correct the mistake in the proof of Lemma 7 in our paper "Horizontal extension of connections into (2)-connections" published in Demonstratio Math. XXXVII (4) (2004).
EN
It is known a complete description of all bundle functors on Mfm x Mf of finite order in the first factor. The most known example of such a bundle functor is the r-jet bundle functor Jr on Mfm x Mf. In the present note an example of a bundle functor on Mfm x Mf of (essentially) infinite order in the first factor is constructed.
10
Content available remote The natural operators transforming projectable vector fields to vertical bundles
EN
Let F : Mfn -> FM. be a natural bundle. We classify all FMm,n-natural operators D transforming projectable vector fields X on (m, n)-dimensional fibered manifolds Y - M to vector fields D{X) on the F-vertical bundle VFY -> M. We apply this classification result to some more known natural bundles F.
11
Content available remote On naturality of the formal Euler operator
EN
That all natural operators of the type of formal Euler operator from the variational calculus are the constant multiples of the formal Euler operator is deduced.
EN
For any product-preserving bundle functor F denned on the category F2 M of fibered-fibered manifolds, we determine all natural operators transforming projectable-projectable vector fields on Y 6 Ob(F2M) to vector fields on FY. We also determine all natural affinors on FY and prove a composition property analogous to that concerning Weil bundles.
15
Content available remote Horizontal extension of connections into (2)-connections
EN
We discuss the prolongation of connections to to some non product preserving bundles. We introduce the concept of (r)-connection on a fibered manifold Y and for a given connection F on Y we construct its horizontal extension F(2). We also prove that F(2 ) is the unique (2)-connection on Y canonically dependent on F.
16
Content available remote On the contact (k, r)-coelements
17
Content available remote On some natural operators in vector fields
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