International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 12, Pages 733-760
Geometry without topology as a new conception of geometry
Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1, Vernadskii Avenue, Moscow 119526, Russia
Received 3 March 2001; Revised 4 July 2001
Copyright © 2002 Yuri A. Rylov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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: the nondegeneracy. Fitting the T-geometry metric with the metric tensor of Riemannian geometry, we 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.