Sigma-Immanent Form of Euclidean Geometry as a Basis for non-Euclidean Geometry

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences,
101-1, Vernadskii Ave., Moscow, 117526, Russia.

The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). At such a description all information on the geometry properties (such as uniformity, isotropy, continuity and degeneracy) is contained in metric. The Riemannian geometry is constructed by two different ways: (1) by conventional way on the basis of metric tensor, (2) as a result of modification of metric in the sigma-immanent description of the proper Euclidean geometry. The two obtained geometries are compared. The convexity problem in geometry and the problem of collinearity of vectors at distant points are considered. The nonmetric definition of curve is shown to be a concept of the proper Euclidean geometry. This nonmetric concept of the curve appears to be inadequate to any non-Euclidean geometry.

There are text of the paper in English and the figure, the text in Russian and the figure.

November 1, 1999