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Some natural operations on functions

Mikulski W.

Demonstratio Mathematica

2007

Vol. 40, nr 3

745-749

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.

Bundles of contact elements on fibered fibered manifolds and the flow operator

Mikulski W., Tomas J.

Demonstratio Mathematica

2007

Vol. 40, nr 2

485-493

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.

The natural operators transforming projectable vector fields to vertical bundles

Mikulski W.

Demonstratio Mathematica

2005

Vol. 38, nr 3

753--760

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.

On the product preserving bundle functors on k-fibered manifolds

Mikulski W.

Demonstratio Mathematica

2001

Vol. 34, nr 3

693-700

A complete description of all product preserving bundle functors F on k-fibered manifolds in terms of sequences Gk- Gk-1- ....- G0 of natural transformations of product preserving bundle functors on manifolds is given.