Manifolds over plural numbers and semitangent structures

Abstract

If base manifolds of a multistep submersion are completed with jets of corresponding orders, then a new differentiable manifold having an integrable nilpotent structure of a general form from the standpoint of which this manifold serves as a real realization of a manifold over an algebra of plural numbers arises. We introduce lifts of tensor fields and connections with projectivity property relative to the given submersion. We study invariant objects of a holomorphic geodesic transformation of a complete lift connection and isolate holomorphic geodesic-flat manifolds.