The heritage of N. I. Lobachevskij and the activity of Kazan geometers.

*(English. Russian original)*Zbl 0799.01018
Russ. Math. Surv. 48, No. 2, 47-74 (1993); translation from Usp. Mat. Nauk 48, No. 2(290), 47-74 (1993).

With Lobachevskij the famous school of Kazan geometers was initiated. Their main field was of course the further development of non-Euclidean geometry. The authors intend to illustrate the various activities of this school. In direct connection with Lobachevskij there has to be mentioned the edition of Lobachevskij’s work (2 vol. 1883/6; 5 vol. and suppl. 1946-1976) as well as the foundation of a special library containing more than 3000 titles, and the foundation of the Kazan Physics and Mathematics Society (KPMS), due to A. Vasil’ev. This Society publishes a journal since 1895, the Izvestiya of the KPMS. Further on the development of Lobachevskij’s ideas, i.e. of non-Euclidean geometry, was the main aim of the Kazan department of geometry. Among its members there are found many famous names. Their achievements are characterized in the following. P. A. Shirokov who became head of the department in 1933 worked on spaces of constant curvature. On the basis of simple bivectors he introduced their scalar and Plücker-product and an operation called polarization. These bivectors determine screws which are infinitesimal shifts. This idea enabled Kotel’nikov to introduce the concept of sliding vectors. There are also discussed the contributions of N. G. Chebotarev, A. P. Norden who was a student of Kagan and Finikov and who worked on projective differential geometry, and Laptev whose field were Finsler spaces. Among their students are mentioned V. I. Vedernikov, V. V. Shurygin, A. Z. Petrov, A. V. Aminova, P. I. Petrov, I. P. Egorov, and V. G. Kopp.

Reviewer: K.Reich (Stuttgart)

##### MSC:

01A55 | History of mathematics in the 19th century |

01A60 | History of mathematics in the 20th century |