Lifts of some tensor fields and connections to product preserving functors

Abstract

In this paper we define some lifts of tensor fields of types (1, k) and (0, k) as well as connections to a product preserving functor . We study algebraic properties of introduced lifts and we apply these lifts to prolongation of geometric structures from a manifold M to (M). In particular cases of the tangent bundle of pr -velocities and the tangent bundle of infinitesimal near points our constructions contain all constructions due to Morimoto (see [20]-[23]). In the cases of the tangent bundle our definitions coincide with the definitions of Yano and Kobayashi (see [31]). To construct our lifts and to study its properties we use only general properties of product preserving functors. All lifts verify so-called the naturality condition. It means that for a smooth mapping φ:M → N and for two φ-related geometric objects defined on M and N its lifts to (M) and (N) respectively are (φ)-related. We explain later the term φ-related for considered geometric objects.