The natural affinors on the \(r\)-jet prolongations of a vector bundle. (English) Zbl 1064.58003

Mikulski, Włodzimierz M.

Demonstr. Math. 37, No. 3, 709-717 (2004).

In this paper three kinds of \(r\)-jet prolongations of vector bundles are described: the usual one, the vertical \(r\)-jet prolongation and a third one, called the \([r]\)-jet prolongation, which, according the author, is somewhat a generalization of the other two prolongations when the dimension of the base manifold is \(\geq 2\) and \(r\geq 3\).
In a previous paper [W. M. Mikulski, Ann. Pol. Math. 82, No. 2, 155–170 (2003; Zbl 1083.58003)], all natural linear operators lifting vector fields from a vector bundle \(E\) into vector fields on \(F^rE\) are classified (where \(F^rE\) is any of the three former jet prolongations of \(E\)). The aim of this paper is to give a full description of all natural affinors on \(F^rE\), when the dimension of the base manifold is \(\geq 2\); such a description is given in terms of three canonical affinors: the identity affinor together with two affinors \(U\) and \(V\) defined in the paper.

Reviewer: J. Rodriguez (Salamanca)

MSC:

58A20	Jets in global analysis
58A05	Differentiable manifolds, foundations

Keywords:

vector bundle gauge functor; natural affinor; natural operator; jet

Citations:

Zbl 1083.58003