Lifts of some tensor fields and connections to product preserving functors. (English) Zbl 0813.53010

Gancarzewicz, Jacek; Mikulski, Włodzimierz; Pogoda, Zdzisław

Nagoya Math. J. 135, 1-41 (1994).

It has been clarified recently that the product preseving bundle functors on the category of manifolds coincide with bundles of infinitely near points introduced by A. Weil. The paper under review presents several important contributions to the application of the technique of Weil algebras in different concrete problems in differential geometry. The introductory part is devoted to a survey of the foundations. Next the lifts of functions and vector fields to a Weil bundle are discussed in detail. Then these results are applied to a rather complete characterization of the lifts of tensor fields of type \((1,k)\) and \((0,k)\). This enables the authors to deduce several interesting results about lifting of different kinds of geometric structures to a Weil bundle. Furthermore, the complete lifts of connections are studied from different points of view. Finally, some properties of the lifts of Riemannian metrics and symplectic structures are determined.

Reviewer: I.Kolář (Brno)

MSC:

53A55	Differential invariants (local theory), geometric objects
58A20	Jets in global analysis

Keywords:

product preseving bundle functors; Weil algebras; Weil bundle; Riemannian metrics; symplectic structures

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