On the geometric results of A. V. Pogorelov

Abstract

We present a survey of the principal results of A. V. Pogorelov in the field of geometry.On May 13, 2005, the memorial plaque of Academician A. V. Pogorelov was unveiled during the meeting held at the Kharkov Institute for Low-Temperature Physics and Engineering of the Ukrainian National Academy of Sciences. In their speeches, Academicians V. A. Marchenko and V. V. Eremenko, A. A. Borisenko (Corresponding Member of the Ukrainian National Academy of Sciences), representatives of the other institutions and Kharkov administration, Consul of Russian Federation in Kharkov, and others described the outstanding contribution made by Acad. Pogorelov to various fields of science, his life, and remarkable features of his mind and character. Among the great number of participants of the meeting, we can especially mention colleagues and disciples of A. V. Pogorelov, researchers of the institute, professors and students of the Karazin Kharkov National University, his son Leonid, and many others. After the meeting, it became absolutely clear that everything created by Pogorelov-his scientific results, ideas, monographs, textbooks, and theories-remain with us in our memory and scientific life. The materials of this seminar, including the talk of Academician V. A. Marchenko about some interesting facts from Pogorelov's life, can be found in [1].The fields of scientific activity of Acad. Pogorelov were fairly large and mainly concentrated in the following four directions: geometry, mechanics, secondary and higher education, and cryogenic engineering. The aim of the present paper is to give a survey of the principal geometric achievements of this prominent mathematician. The results of Pogorelov are remarkable for their completeness, profound ideas, and enviable simplicity of formulations. His reputation was quite high and generally recognized by the world's scientific community.In 1951, Pogorelov obtained his first outstanding result by proving that the general convex surfaces in E 3 are uniquely defined by their metric. The history of this problem goes back to Cauchy who proved that closed convex polyhedra are uniquely defined. Later, this problem was studied by Liebmann, Hilbert, Cohn-Vossen, and Herglotz. Most likely, Pogorelov was acquainted with this problem by his brilliant teachers -A. D. Aleksandrov and N. V. Efimov. Friendly relations between Efimov and Pogorelov preserved for the whole life can serve as an excellent example of proper relations between a teacher and a disciple. Their great friendship successfully passed the tests of time and distance. Note that Efimov had great positive and productive influence upon many people.In the 1930s, a school of geometers "in the large" was formed in Moscow and Leningrad, to a significant extent under the influence of Cohn-Vossen who emigrated from fascist Germany to the USSR in 1934. CohnVossen was a coauthor of Hilbert in their famous book "Visual Geometry." He proved the nonflexibility of a closed surface of the class of regularity C 3 with positive Gaussian curvature.