Deformation principle and problem of parallelism in geometry and physics

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Science
101-1, Vernadskii Ave., Moscow, 119526, Russia.
Updated February 1, 2002


The deformation principle admits one to obtain a very broad class of non-uniform 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.


English version of the paper in Postscript.
Russian  version of the paper in Postscript.