Institute for Problems in Mechanics, Russian
Academy of Sciences
101-1 ,Vernadskii Ave., Moscow, 119526, Russia
Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm
or mirror Web site: http://gas-dyn.ipmnet.ru/~rylov/yrylov.htm
Updated September 27, 2006
The reasons of the crisis in the contemporary (Riemannian) geometry are discussed. The conventional method of the generalized geometries construction, based on a use of the topology, leads to a overdetermination of the Riemannian geometry. In other words, at the Riemannian geometry construction one uses the needless information (topology), which disagrees with other original axioms. The crisis manifests in the fact, that the mathematical community cannot see and does not want to see the overdetermination of the Riemannian geometry. The most geometers-topologists deny the alternative method of the generalized geometry construction, which does not uses the topology, because it does not contain theorems. Most geometers see the geometry presentation as a set of definitions and theorems. They cannot imagine the geometry presentation without customary theorems. As a result the most clever topologists, which have acknowledged the negligible role of the topology in the geometry construction and inconsistency of the conventional method of the generalized geometry construction, appear in the difficult situation (conflict with the mathematical community).