Geometry without Topology as a New Conception of Geometry
Yuri A. Rylov
Institute for Problems in Mechanics, Russian Academy of Sciences,
101-1, Vernadskii Ave., Moscow, 117526, Russia.
email: rylov@ipmnet.ru
March 1, 2001
Abstract
A geometric conception is a method of a geometry construction. The Riemannian
geometric conception and a new T-geometric one are considered. T-geometry
is built only on the basis of information included in the metric (distance
between two points). Such geometric concepts as dimension, manifold, metric
tensor, curve are fundamental in the Riemannian conception of geometry,
and they are derivative in the T-geometric one. T-geometry is the simplest
geometric conception (essentially only finite point sets are investigated)
and simultaneously it is the most general one. It is insensitive to the
space continuity and has a new property -- nondegeneracy. Fitting the T-geometry
metric with the metric tensor of Riemannian geometry, one can compare geometries,
constructed on the basis of different conceptions. The comparison shows
that along with similarity (the same system of geodesics, the same metric)
there is a difference. There is an absolute parallelism in T-geometry,
but it is absent in the Riemannian geometry. In T-geometry any space region
is isometrically embeddable in the space, whereas in Riemannian geometry
only convex region is isometrically embeddable. T-geometric conception
appears to be more consistent logically, than the Riemannian one.
There is Russian and English
versions in Postscript