Prolongation of pairs of connections into connections on vertical bundles. (English) Zbl 1112.58003

Doupovec, Miroslav; Mikulski, Włodzimierz M.

Arch. Math., Brno 41, No. 4, 409-422 (2005).

For every natural bundle \(F\) over \(n\)-manifolds, the vertical prolongation \(V^FY\) of a fibered manifold \(Y\to M\) with \(n\)-dimensional fibers is defined by applying \(F\) fiberwise. The authors determine all natural operators transforming every pair of connections on \(Y\) into a connection on \(V^F Y\to M\). In the case \(F\) where is a Weil bundle \(T^A\), i.e., \(V^F Y\) is the vertical Weil bundle \(V^A Y\) of \(Y\), the general result is geometrized in an interesting way. Special attention is also paid to the cases where \(F\) is the \(r\)-th order tangent bundle or the cotangent bundle or the \(r\)-th order cotangent bundle.

Reviewer: Ivan Kolář (Brno)

MSC:

58A20	Jets in global analysis
58A32	Natural bundles

Keywords:

connection; vertical bundle