### 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.

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