ON SOME PROPERTIES OF PROJECTIVE FLAT MANIFOLDS WITH AFFINE CONNECTION

Abstract

Aim. We refine the properties of parallel translations of manifolds with affine connection of dimension greater than two, such that for any three points that are sufficiently close, there exists a two-dimensional autoparallel manifold containing them. Methodology. We use the methods of differentiable universal algebras to describe the properties of certain classes of affine-connected spaces. Results. We prove that in this class of projective flat manifolds with affine connection, the “pseudoline” identity is fulfilled, reflecting the properties of parallel translations. The differential-geometric characteristic of a “pseudoline” identity is derived, that is, if the dimension of the manifold is more than two, then the “pseudoline” identity is equivalent to the fact that the corresponding manifolds of affine connection are projective flat and have a common pseudoconnection (the same concurrency) with the manifold of affine connection with zero torsion. Research implications. Differential geometry has numerous applications in theoretical mechanics, Special and General relativity theory, and other fields of natural sciences. This research can be employed to build a specific mathematical model describing the course of physical processes.

Keywords

• projective flat manifolds with affine connection
• curvature and torsion tensors
• parallel translations
• geodesic loop