Regularity conditions for liftings of functions and vector fields to natural bundles. (English) Zbl 0543.58005

Mikulski, W.

Čas. Pěst. Mat. 109, 127-134 (1984).

Let \({\mathcal F}\) be a natural bundle over n-dimensional \(C^{\infty}\)- differentiable manifolds, i.e. a covariant functor from the category of n-dimensional \(C^{\infty}\)-differentiable manifolds and their embeddings into the category of \(C^{\infty}\)-differentiable locally trivial bundles and their bundle mappings, having the well known properties. J. Gancarzewicz [Lifting of functions and vector fields to natural bundles, to appear in Demonstr. Math.] introduced the definitions of lifting of functions and quasi-lifting of vector fields from the base manifolds to natural bundles. Both these definitions contain three conditions, the third of which is called regularity condition. (Roughly speaking, the regularity condition requires the \(C^{\infty}\)-dependence on parameters.) The author in the paper under review proves that in both cases the regularity condition is a consequence of the first two conditions.

Reviewer: J.Vanzura

MSC:

58A99	General theory of differentiable manifolds
58A30	Vector distributions (subbundles of the tangent bundles)

Keywords:

natural bundle; lifting of functions; quasi-lifting of vector fields