Deformation principle and problem of parallelism in geometry and physics

TL;DR

This paper examines the deformation principle in geometry, highlighting inconsistencies in Riemannian geometry's approach to parallelism and proposing a comparison of different definitions to address these issues.

Contribution

It investigates the deformation principle's application to geometry, compares two parallelism definitions, and reveals inconsistencies in Riemannian geometry's treatment of parallelism.

Findings

The second definition of parallelism in Riemannian geometry is inconsistent.

Riemannian geometry lacks absolute parallelism, affecting space-time descriptions.

The deformation principle can generate a broad class of geometries.

Abstract

The deformation principle admits one to obtain a very broad class of nonuniform geometries as a result of deformation of the proper Euclidean geometry. The Riemannian geometry is also obtained by means of a deformation of the Euclidean geometry. Application of the deformation principle appears to be not consecutive, and the Riemannian geometry appears to be not completely consistent. Two different definitions of two vectors parallelism are investigated and compared. The first definitions is based on the deformation principle. The second definition is the conventional definition of parallelism, which is used in the Riemannian geometry. It is shown, that the second definition is inconsistent. It leads to absence of absolute parallelism in Riemannian geometry and to discrimination of outcome outside the framework of the Riemannian geometry at description of the space-time geometry.