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On some directions in the development of jet calculus

Miroslav Kureš — 2011

Banach Center Publications

Two significant directions in the development of jet calculus are showed. First, jets are generalized to so-called quasijets. Second, jets of foliated and multifoliated manifold morphisms are presented. Although the paper has mainly a survey character, it also includes new results: jets modulo multifoliations are introduced and their relation to (R,S,Q)-jets is demonstrated.

Natural operators lifting vector fields to bundles of Weil contact elements

Miroslav KurešWłodzimierz M. Mikulski — 2004

Czechoslovak Mathematical Journal

Let A be a Weil algebra. The bijection between all natural operators lifting vector fields from m -manifolds to the bundle functor K A of Weil contact elements and the subalgebra of fixed elements S A of the Weil algebra A is determined and the bijection between all natural affinors on K A and S A is deduced. Furthermore, the rigidity of the functor K A is proved. Requisite results about the structure of S A are obtained by a purely algebraic approach, namely the existence of nontrivial S A is discussed.

On the simplicial structure of some Weil bundles

Kureš, Miroslav — 2000

Proceedings of the 19th Winter School "Geometry and Physics"

The author considers the problem to give explicit descriptions for several types of bundles on smooth manifolds, naturally related with the bundle of k -dimensional velocities, or k -jets. In fact, this kind of bundles are very natural objects in differential geometry, mechanics and Lagrangian dynamics. For this the author considers Weil bundles that arose from Weil algebras. If a suitable combinatorial data is provided by a simplicial coloured structure, then the author describes the corresponding...

Torsions of connections on tangent bundles of higher order

Kureš, Miroslav — 1998

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

The torsions of a general connection Γ on the r th-order tangent bundle of a manifold M are defined as the Frölicher-Nijenhuis bracket of Γ with the natural affinors. The author deduces the basic properties of these torsions. Then he compares them with the classical torsion of a principal connection on the r th-order frame bundle of M .

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