Petr Konstantinovich Rashevsky (on his 50th birthday)

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1958 January-February vol. XIII, item 1(79) of the success of mathematical sciences Peter Konstantinovich Rashevsky (to the fiftieth anniversary of his birth) July 27, 1957 marked the 50th anniversary of the birth of the outstanding council geometer, professor of Moscow University Peter Konstantinovich Rashevsky. Peter Konstantinovich was born in 1907 in Moscow in the family of a famous mathematician, author of a number of original textbooks, Konstantin P. (1919), where the father of the provincial institute Konstantin P. P. Rashev worked for the father of Konstantinsky (181919).

In 1923, after graduating from high school in Ranenburg, P. K. entered the mathematical department of the Faculty of Physics and Mathematics of Moscow State University, and since that time he has been inextricably linked with Moscow University. From the student bench, P. K. becomes one of the most active students of the founder of the Moscow tensor differential-geometric school V. F. Kagan. In September 1927, at one of the first meetings organized by V. F.

Kagan of the seminar on vector and tensor analysis at MSU, developing mainly issues of tensor differential geometry, student Rashevsky talks about his first independent results related to the theory of tensor field, depending not on one but on two points of space [4], and since then P. K. is one of the most active participants in the seminar, and after the death of V. F. Kagan - his leader. In 1928 P. K. graduated from Moscow State University and in 1928-1931. undergoes graduate school under the guidance of V. F. Kagan, to the defense of P. K. V. K.

He has six papers published in the “Reports” of the Paris Academy of Sciences, the “Mathematical Compendium” and “Proceedings of the Seminar on Vector and Tensor Analysis.” These papers deal with very diverse questions of geometry—the theory of subprojective spaces [1,] [6], the theory of rectilinear congruence in tg-dimensional Euclidean space [2], the geometry of two-dimensional affinity spaces [3], the representation of an infinitial group of motions by means of the subprojective spaces [1,], where the success of the X-s (xels) is called 1 ^2.

226 Mathematical Life in the USSR [7], the theories of conformal mapping [8]—and were presented by P. K. as a candidate thesis. In 1934, P. K. Rashevsky was awarded the academic title of professor. In 1936, P. K. defended the doctoral thesis “Polymetric geometry” at Moscow State University.

Very diverse: they cover geometries of spaces that are generalizations of spaces of constant curvature-subprojective [1], [6], lymetric [I], [12], [13], [19], [30], stratified [37], [43], symmetries of spaces of constant curvature-subprojective [40], [44], [45]; theory of geometric objects [41], [53]; theory of congruence in multidimensional space [2], [15]; generalizations of Rieman spaces [46] [46] [14], [34] [24]; theory of geometrics, [35], [34], [24]; theory of geometrics, [24],[24], [15]; theory of geometry of geometry,[24], [24], [15]; and geometry, [15]; theory of geometry,[24]; and geometry of geometry,[24]; [15];[24

The authors also remember the elegant exposition of the spectral theory of linear operators conducted by P. K. in a number of reports, and in recent years P. K. reads an extremely interesting course in mathematical questions of quantum electrodynamics at Moscow State University and is now preparing the publication of this course in the form of a book. The main place in the scientific interests of P. K. is geometry about the spaces that are generalizations of spaces of constant curvature (nons of clidian spaces).

Kaga nom, who was one of the leading experts on non-Euclidean geometry and the foundations of geometry and moved on to tensor differential geometry in connection with his studies in geometrical questions of general relativity. One of the first works of P. K. related to the development of the theory of subprojective spaces created by V. F. Kagan [1], [6], which are one of the simplest generalizations of spaces of constant curvature. Other generalizations of non-Euclidean geometries are bimetric geometries [11], [12], [13], also representing the development of Kaganium.

The systematic presentation of the theory of bimetric systems and their direct generalization of trimetric systems is devoted to the doctoral dissertation of P. K. [301. The main idea of P. K.’s works on polymetric geometry is extremely simple: guided by the analogy with the geometry of the Euclidean (or elliptical) plane, P. K. considers a three-parametric set of elements that can be conditionally called “linear elements”, and in this set two Finsler metrics are “angular metrics” and “lemetics” can be called “the first line”;

In the USSR, there are 227 metrics “points.” In this rather general scheme, P. K. puts much concrete content, considering all sorts of particular types of bimetric systems and proving many theorems that are generalizations of well-known theorems of Euclidean or elliptical plane geometry. These works have been continued in a number of studies of P. K. students, but outside the P. K. school they are not yet appreciated, in our opinion, for example, it can be noted that foreign specialists in integration of the field of geometry, and some of his studies are clearly not familiar with this discipline.

Generalizations of spaces of constant curvature, related to the so-called A-spaces of P. A. Shirokov (often called keler spaces), are stratified spaces in the sense of P. K. Rashevsky [37], [43]. The most important generalization of non-Euclidean spaces are the symmetric spaces of E. Cartan, to which the work is devoted [40], followed by a number of works on the further generalization of these metric spaces in the case of torsion spaces [44], [45].

Symmetric spaces in the Cartan sense can be defined as affinity-free torsion spaces whose curvature tensor is maintained in parallel transfer; in this sense, Cartan symmetric spaces can be called “spaces of constant curvature”, special cases of which are ordinary spaces of constant curvature. Cartan symmetric spaces can be realized as manifolds of certain geometric images in Euclidean, Nee clidian and other simpler spaces, these geometric images being “symmetries”, i.e.

With respect to these images, one can define “reflections” such as reflection from a point, reflection from a straight line, or reflection from a circle (inversion), P. K. introduced a new class of spaces, which are spaces of affinity connection with torsion, in which, in parallel transfer, both the curvature tensor and the torsion tensor are stored; these spaces P. K. called symmetric torsion spaces and showed that these spaces can also be realized in the form of manifolds of certain geometric images, which, however, are no longer the images of Lio’s interesting groups.

Note a number of recent works by P. K. devoted to the representations of Lie groups. These include the work [49], which contains a description of all (finite-dimensional) linear representations of differential groups (i.e., groups of various v times differentiable transformations of coordinate systems considered in the vicinity of a given point with an accuracy of v-order) and a description of all linear differential-geometric objects. It turned out that there is a general scheme analogous to the description of representations of a complete linear group using Jung’s schemes.

In the work [57] the necessary and sufficient condition for (local) realization of arbitrary homogeneous space in the form of a surface of transitivity of projective space is given. 15*

288 MATHEMATIC LIFE IN the USSR With the works of P. K. on tensor differential geometry, P. K.’s interest in the mathematical theory of relativity, the original exposition of which is given to them in the book [46]. Interest in the mathematical theory of relativity naturally led P. K. to the mathematical problems of quantum physics. These questions are related to the algebraic works of P. K. [31], [32], works on the theory of generalized functions [48], [53], quantum statistics [42], the theory of spinors [52] and quantum electrometrics related to geometry.

Thus, in the work [28] proved the fundamental role of the axioms of Desargues and Pascal in projective geometry, which, as it turned out, the only of all configuration axioms are compatible with at least a slight semblance of the axioms of order and continuity. It is especially necessary to note the textbooks written by P. K. on differential geometry (4 editions: [24], [26], [54]), on Riemannian geometry and tensor analysis (two versions: [17], [46]), on the geometric theory of differential civilizational geometry, as well as P.

Their author is characterized by the clarity and exceptional thoughtfulness, which makes them wonderful aids for novice mathematicians. In terms of scientific rigor and clarity of definitions, these books are almost unparalleled in the world differential-geometric literature tour; No wonder A. D. Alexandrov, who spoke several years ago with a detailed criticism of Soviet and foreign textbooks on differential and Riemannian geometry, was forced to make an exception for P. K. The same qualities are inherent in P. K. Lectures. His presentation, which is externally not particularly effective, conquers the internal details of Ped.

P.K. has been working at Moscow State University since 1930 till present, simultaneously conducting teaching work at the Moscow Power Engineering Institute (1930-1934), at the Liebknecht Industrial and Pedagogical Institute, where P.K. headed the Department of Geometry (1931-1941); in 1941-1944 P.K. worked as the Head of the Department of Mathematics of the Tomsk Pedagogical Institute and since 1942 - the Head of the Department of Mathematics and Transportation Institute of Moscow P.T. 1944 (Moscowsk Institute of Electrical Engineering, M.M.T.) - Moscow Institute of Transportation, 1944;

He brought up a large number of geometers of the Soviet Union, including those who successfully defended doctoral theses by B. A. Rosenfeld, N. N. Yanenko, I. P. Egorov, M. A. Javadov, A. 3. Petrov. A. P. Norden B. A. Rosenfeld I. M. Yaglom.

229 list of printed works of P. K. Rashevsky 1930 1. Sur les espaces sous-projectifs, Gompt. Rend. Acad. Sci. (Paris) 191, 547–548. 1931 2. Sur les congruences a plusieurs dimensions, Gompt. Rend. Acad. Sci. (Paris), 192, 137–138. 3. Characteristic feature of the family of 1, tense, 7. and seven. typical analysis of the geo-sor, 7. sor, 7. sor, 7.

Sur G interpretation of infinitesimale du systeme des vecteurs duals, Gompt. Rend. Acad. Sci. (Paris) 197, 217-220. 8. Un criterium caracteristique des representations conformes e2, \nz, -£, Gompt. Rend. Acad. Sci. (Paris) 197, 291-294. 1934 9. Geodesic lines of affinity space infinitely small with a fourth order, DAN 3, 1413-316, infinite space in 357-150, in infinitesimal space, 357-1.

Une geometrie metrique duale, fondee sur les espaces de Gartan generalises, Compt. Rend. Acad. Sci. (Paris) 201, 921-923. 13. Systeme bimetrique dual, Gompt. Rend. Acad. Sci. (Paris; 201, 1088-1090. 14. Representations dual conformes zn, ez, In'z au point de vue de geometrie conforme, Mathem. 42, 157-168. 15. Imeg., Imacad., 2nd. , 2nd., 2nd. , 2nd. , 2nd, and , 2nd , 2, 2nd s, 2, 2nd , and s. , , , 2nd. s. , 2nd. s. s. , 2nd. s. s. s. , 2nd. , 2nd. s. , 2nd. , 2nd. , 2nd. ,

Un schema unifiant la theorie des groupes abstraits avec la theorie de groupes infinitesimaux de Lie, Compt. Rend. Acad. Sci. (Paris) 202, 1012–1013. 19. Systemes trimetriques et la metrique de Finsler generalised, Compt. Rend. Acad Sci. (Paris) 202, 1237–1239. 20. Abstract-algebrac dia of the main theorems of the theory of continuous groups, Processes of the Second All-Union Mathematical Congress, 19., 13-114, Vol., Vol., Vol., Vol., Vol. 14, Vol., Vol., Vol., Vol., Vol. 14, Vol., Vol., Vol. 14, Vol., Vol.

230 Mathematical Life in the USSR 21. On one generalization of elliptic geometry that preserves the sine theorem, Scholar. s.p. Ped. in-that named after Liebknecht, ser. physical-mate. 1, 50–72. 23. Geometry of the cone of the nz^left directions.Scientific. in-that named after Liebknecht, ser. phys. 1, 73–93. 24.1938. course of differential geometry, 2nd., 2nd. ON, 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., 2nd., ., 2nd., 2nd., ., ., ., ., 2nd., ., ., ., . . . . . . . . . . . . . . . .

Sur 1 'unicite de la geometrie projective dans le plan, Matem. 8 (50), 107-120. 29. Sur une geometrie projective avec les nouveaux axiomes de configuration, Matem. Sb. 8 (50), 183-204. 1941 30. Polymetric geometry, Proceedings of Seminary and Tens. analysis, pp. 5, 21-147. 31. Les simple problems les pluss de "I'alge quasiques de lassieurs, 911," and 4xxxxxxxxxxxxxxxxxxx.

On the equilibrium of elastic bodies with screw symmetry, Matem. 15 (57), 55-70. 1947 34. Geometric theory of partial-derivative equations, M.-L., Gostehizdat, 354 pages 1948 35. "Foundations of geometry" by Hilbert and their place in the historical development of the question, Introductory article to D. Hilbert's book "Foundations of Geometry", M.-L., 7-52. 36. Notes to "Ostehizdatika, 30.25, Geometry, 3894 years, M. - M.L., 37. - 8. - 8. - 8.28 years of geometry, M. - M.

3rd, M.-L., Gostehizdat, 428 pp. 40. Notes to E. Cartan’s book “The Geometry of Lie Groups and Symmetric Spaces”, M., IL, 293-330. 41. Galois’ Theory in the Bodies of Geometric Objects, Proceedings of Semin. on the Veteran. and Tenz. Ana lyse, pp. 7, 167-186. 42. Statistics of Bose-Einstein and Fermi-Dirac from the Tensor Point of View, 7, 380, and seven points of view, 3, 380, 3.

Mathematical Life in the USSR 231 1950 43. On the pair of connections on ^-dimensional surfaces in 2d-dimensional layered space, Proceedings of Seminary. on vector and tens. analysis, item 8, 301-313. 1951 44. On the geometry of homogeneous spaces, DAN 80, 169-171. 1952 45. On the geometry of homogeneous spaces, Proceedings of Seminary and Tens. analysis, item 9, 49-74. differenti.1953, 48-97.00, Limatoreatrical, 4.00, 4.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

On linear representations of semi-simple Lie groups with a nilpotent radical, DAN 97, 781–783. 51. Internal algebraic groups Lie, DAN 98, 539–540. 1955 52. Theory of spinors, UMN X, pp. 2, 3–110. 53. Multidimensional ^-functions and differential-geometric objects, UMN X, ot. 4, 145–152. 1954. Course of differential geometry, ed. 4-e, III., M., II., AN., 64., , 517, T., T., T., T.S., 540, T. in the USSR, 4., 5, T. T. T. T. T. T., 5, 5, 5, T., T. T. T. T. T. T. T. T. T. T. T., T., T. T., 5, 5, 5, 5, 5, 5, T. T. T., T., 5,

On linear representations of homogeneous spaces, Proceedings of the III All-Union Mathematical Congress, Moscow, June-July 1956, Vol. II, M., ed. of the Academy of Sciences of the USSR. 142. 1957 58. On linear representations of differential groups and Lie groups with a nilpotent radical, Proceedings of the Moscow Mathematical Islands 6, 337-370.