The impact of Lobachevsky's ideas on the development of differential geometry (Q1358341)

After giving a summary of the celebrations on the occasion of the 200th anniversary of the birth of Nikolai Lobachevskij (1792-1856), the intent of the article is formulated: ``to trace the development of new ideas in geometry since Lobachevskij to our days, with particular emphasis placed on the role played by the Moscow geometrical school''. This is given in the following sections: Discovery of non-Euclidean geometry. Beltrami's interpretation; origination of Riemannian geometry. Klein's Erlangen Programme; development of group-theoretical investigation methods. Further development of the theory of generalized spaces. The role of the Moscow school of differential geometry. In the latter, especially the role of S. P. Finikov's seminar on classical differential geometry is emphasized in developing and refining of Cartan's method. Both the authors belong to the participants of this seminar. Most prominent geometricians, later leaders of the seminar, were G. F. Laptev (1909-1972) and A. M. Vasil'ev (1923-1987). A short review of their contributions (with references) is given. [See also: \textit{A. M. Vasil'ev}, Differential geometry (Russian), in: Mathematics at Moscow University, 174-192 (1992); \textit{S. Arutyunyan}, \textit{P. Grushko}, \textit{L. Evtushik} and \textit{Ü. Lumiste}, Differential-algebraic methods of geometric research in works of A. M. Vasil'ev and his scientific school, in: Problems of Geometry (Russian), Itogi Naoki Tekhniki (VINITI, Moskva) (1988)].