### 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.
email: rylov@ipmnet.ru
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