Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences

Toponogov Victor Andreevich

Chief Scientist

Laboratory of Applied Analysis
Department of Analysis and Geometry
Sobolev Institute of Mathematics
Siberian Branch of the Russian Academy
of Sciences

Scientific Interests
Teaching activity
List of Publications

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Devoted to the 70th anniversary of the birthday
of  Victor Toponogov

Geometry & Applications

2000, March 13-16, Novosibirsk, Russia

see also



   Prof. Victor Andreevich Toponogov, a well-known Russian geometer, was born on the 6th of March, 1930 and grew up in the city of Tomsk, in Russia. During Toponogov’s childhood, his father was subjected to Soviet repression in 1937. After finishing school in 1948 Toponogov entered the department of Mechanics and Mathematics at Tomsk University, and graduated it in 1953 with honours.
   In spite of an active social position and receiving high marks during his studies, the stamp of ‘‘son of the people's enemy’’ did not leave
Toponogov with any hope of continuing his education at the post-graduate course level. However after Joseph Stalin’s death in March of 1953, the situation in the USSR was cardinally changed and Toponogov was included as a post-graduate student at Tomsk University. Toponogov's scientific interests were influenced by his scientific advisor, Prof. A.I.Fet (a well-recognized topologist and specialist in variational calculus in the large, a pupil of L.A.Lusternik) and by the works of academician, A.D.Aleksandrov.
   In 1956, V.A.T
oponogov moved to Novosibirsk where in April of 1957, he became a research scientist at the Institute of Radio-Physics and Electronics, then directed by well-known physicist, Y.B.Rumer. In December of 1958, Toponogov defended his Ph.D. thesis at Moscow State University. In his work the Aleksandrov convexity condition was extended to multi-dimensional Riemannian manifolds. Later on, this theorem was called Toponogov (comparison) theorem about the angles (see, for example, Meyer W.T. Toponogov's Theorem and Applications. Lecture Notes, College on Differential Geometry, Trieste. 1989). In April of 1961, Toponogov moved to the Institute of Mathematics and Computer Center of the Siberian Branch of the Russian Academy of Sciences, at its inception. All his subsequent scientific activity is related to the Institute of Mathematics. In 1968 at this institute, he defended  of his Doctoral thesis on the theme ‘‘Extremal problems for Riemannian spaces with curvature bounded from above’’.
   From 1980 to 1982, T
oponogov was deputy director of the Institute of Mathematics and from 1982  to 2000, was head of one of the laboratories of this same institute. In 2001, he became Chief Scientist of the Department of Analysis and Geometry.

   The first 30 years of Toponogov's scientific life were devoted to one of the most important divisions of modern geometry — Riemannian geometry in the large.
   From the school of mathematics, everybody obtained an impression about so called synthetic methods in geometry, whose sense are the consideration of triangles, conditions of their equality and similarity, etc. From the Archimedean era, analytical methods have come to penetrate geometry
— this is expressed most completely in the Theory of Surfaces, created by K.Gauss. Since that time, these methods have played a leading part in Differential Geometry. In the fundamental works of A.D.Aleksandrov, synthetic methods are again used because objects under study are not smooth enough for applications of the classical analysis methods. In the creative work of  V.A.Toponogov, both of these two methods, synthetic and analytic, are in harmonic correlation. 
   The classic result in this area is the
Toponogov theorem about the angles of a triangle composed from geodesics. This in-depth theorem is the basis of modern investigations of the relations between curvature properties, geodesics behaviour, and the topological structure of Riemannian spaces. In the proof of this theorem, some ideas of A.D.Aleksandrov are combined with the in-depth analytical technique related to the Jacobi differential equation.
   The methods developed by V.A.Toponogov allowed him to obtain a sequence of fundamental results such as characteristics of multi-dimensional sphere by estimates of the Riemannian curvature and diameter, the solution of the Rauch problem  for even dimensional case, and the theorem about the structure of Riemannian space with nonnegative curvature containing a straight line (i.e., the shortest path which may be limitlessly extended in both directions). This and other theorems of V.A.Toponogov are included in monographs and textbooks written by a number of authors. His methods have had a great influence on the modern Riemannian geometry progress.

   During the last 15 years of his life, V.A.Toponogov devoted himself to Differential geometry of two-dimensional surfaces in the three-dimensional Euclidean space. He obtained the essential progress in the direction related to the Efimov theorem about non-existence of isometric embedding into three-dimensional Euclidean space of a complete Riemannian metric with separated-from-zero negative curvature, and with Milnor hypotheses declaring that an embedding of a complete Riemannian metric with a sum of absolute values of principal curvatures uniformly separated from zero does not exist.
   After starting his work with Y.B.Rumer, Toponogov focused his attention on relations of mathematics with natural sciences. Many efforts of Toponogov are now devoted to the  training of young mathematicians. He was a lecturer in Novosibirsk State University and Novosibirsk State Pedagogical University for more than 45 years. More than 10 of his pupils defended their Ph.D. theses and 7 — Doctoral degrees.
   V.A.Toponogov passed away on 21 November, 2004 and is survived his wife Ljudmila Pavlovna Goncharova and three sons.

Scientific Interests

Riemannian and Differential Geometry

   One of the main objects studied in modern geometry is so called Riemannian space. Riemannian spaces are defined usually by some sufficiently complicated constructions, using concepts of the analysis. Visually Riemannian space may be characterized in such a way that in a small neighbourhood of its arbitrary point the geometry of a space does not differ from the usual Euclidean geometry, and the difference is less when the taken neighbourhood is smaller. The theory of Riemannian spaces is said to be a "Riemannian geometry". The Lobachevskii geometry is an example of Riemannian geometry. Another example – geometry on a sphere of  n-dimensional Euclidean space. These are examples of Riemannian spaces of a constant curvature.

   The works of V.A.Toponogov are devoted to the study of Riemannian geometry "in the large". One of his main results is well-known in the literature as Toponogov comparison theorem. It is intensively used in modern investigations of Riemannian geometry and is referred in many monographs and textbooks on this subject.

   The series of important results in Riemannian geometry in the large belong to his pupils:
V.P. Golubyatnikov
(integral geometry, applications of Hamilton-Jacobi equation), also see non-mathematical page    
Y. Yomdin
Closed trajectories of plane systems of ODE's),
V.B. Marenich
(complete Riemannian manifolds of non-negative curvature),
E.D. Rodionov
(homogeneous Riemannian manifolds),
V.Y. Rovenski (foliations on Riemannian manifolds and submanifolds).
V.A. Sharafutdinov
(Riemannian manifolds of positive curvature, integral geometry of tensor fields),

Teaching activity

   Teaching activity of V.A.Toponogov is related with Novosibirsk State University and Novosibirsk State Pedagogical University. During the years he was a lecturer of modern courses of differential geometry, tensor algebra and tensor analysis on the mathematical and physical faculties.


V.A. Toponogov prepared for publication a textbook:  

Differential Geometry of Curves and Surfaces: A Concise Guide
(in Russian) – 2004. 174 p

In progress in English:
With the editorial assistance of  V.Y. Rovenski.

 From a review:
  ...there are nice exercises; any student working through these  exercises will learn a great deal.  These exercises are meaty, well-   thought-out, and stimulating.


List of Publications

   Scientific projects of V.A.Toponogov are realized in more than 40 scientific papers and training books.

1. On the convexity property of positively curved Riemannian spaces. – Amer. Math. Soc. Transl. Ser. 2. 1964. V. 37, P. 283–285. Zbl 0138.42901

2. Riemannian spaces having their curvature bounded below by a positive number. (in Russian) – Dokl. Akad. Nauk SSSR 1958. V. 120, No. 4,  719–721. Zbl 0086.14904

3. Evaluation of the length of a closed geodesic on a convex surface. (in Russian) – Dokl. Akad. Nauk SSSR 1959. V. 124, No. 2,  282–284. Zbl 0092.14603

4. Riemannian spaces which contain straight lines. – Amer. Math. Soc. Transl. Ser. 2. 1964. V. 37, P. 287–290. Zbl 0138.42902

5. Riemannian spaces having their curvature bounded below by a positive number. – Amer. Math. Soc. Transl. Ser. 2. 1964. V. 37,  291–336. Zbl 0136.42904

6. Dependence between curvature and topological structure of Riemannian spaces of even dimension. – Soviet Math. Dokl. 1960. V. 1, 943–945. Zbl 0096.15402

7. Abschatzung der Lange einer konvexen Kurve. (in Russian) – Sib. Mat. Zh. 1963. V. 4, No. 5, 1189–1193.  Zbl 0136.18902

8. The metric structure of Riemannian spaces with nonnegative curvature which contain straight lines. – Amer. Math. Soc. Transl. Ser. 2. 1968. V. 70, 225–239. Zbl 0187.43801

9. Estimation of the length of a closed geodesic in a positively curved compact Riemannian space. – Soviet Math. Dokl. 1964. V. 5, 251–254. Zbl 0134.39603

10. One theorem on Riemannian spaces containing straight lines. – Int. Congress of Math.: Theses. Moscow. 1966.  170–171.

11. Some extremal theorems of Riemannian geometry. (in Russian) – Sib. Mat. Zh. 1967. V. 8, No. 5, 1079–1103. Zbl 0162.25501

12. One extremal theorem of Riemannian geometry. (in Russian) – 2-nd Russian Simposium on geometry in the large: theses. Petrozavodsk, 1967. P. 67.

13. An isoperimetric inequality for surfaces whose Gaussian curvature is bounded above. (in Russian) – Sib. Mat. Zh. 1969. V. 10, No. 1, 144–157. Zbl 0186.55801

14. Extremal theorems for Riemannian spaces of curvature bounded above. – Soviet Math. Dokl. 1969. V. 10, 88–90. Zbl 0183.50401

15. Theorems on shortest arcs in non-compact Riemannian spaces of positive curvature. – Soviet Math. Dokl. 1970. V. 11, 412–414. Zbl 0212.26503

16. Non-compact spaces of nonnegative curvature. – Addition to book D. Gromoll, W. Klingenberg and W. Meyer “Riemannian Geometry in the Large” (in Russian) – Moscow: Mir, 1971. 298–337. Translation from: Riemannsche Geometrie im Grosen (in German). Berlin-Heidelberg-New York: Springer–Verlag. VI, 287 PP. 1968. Zbl 0155.30701

17. A characteristic property of a four-dimensional symmetric space of rank 1. – Siberian Math. J. 1973. V. 13, No. 4, 616–628. Zbl 0252.53044

18. On three-dimensional Riemannian spaces with curvature bounded above. – Math. Notes. 1973. V. 13, 526–530. With Yu.D. Burago. Zbl 0277.53025

19. Extremal theorems for Riemannian spaces with curvature bounded above, I. – Siberian. Math. J. 1975. V. 15, No. 6, 954–971. Zbl 0305.53043

20. Riemannian spaces of diameter π– Siberian Math. J. 1975. V. 16, No. 1, 99–105. Zbl 0314.53029

21. Riemannian Geometry. (Russian) – Soviet "Mathematical Encyclopaedia". Moscow. 1984. V. 4, 1003–1009. An updated and annotated translation: Encyclopaedia of mathematics. Volumes 1 – 9  (English). Hazewinkel, Michiel (ed.). Dordrecht: Kluwer Academic Publishers. 1988 – 1993

22. Riemannian Geometry in the Large. (in Russian) – Soviet "Mathematical Encyclopaedia". Moscow. 1984. V. 4, 1009–1013. With Y.D. Burago

23. Immersed Manifolds Theory. (in Russian) – Soviet "Mathematical Encyclopaedia". Moscow. 1984.V. 4, 359–363. With S.Z. Shefel

24. A simple model of branched-chain chemical reaction describing oscillating process. – Letters to kinetics and burning. 1984. V. 25, No. 3–4, 301–304. With V.I. Babushka and V.M. Goldshtein

25.Extreme case of the comparison theorem of the angles of a triangle. (in Russian) – Sib. Mat. Zh. 1985. V. 26, No. 1(149), 206–209. With A.I. Vedenyapin and E.D. Mazaev. Zbl 0567.53016

26. Open manifolds of nonnegative Ricci curvature with rapidly growing volume. (in Russian) – Sib. Mat. Zh. 1985. V. 26, No. 4(152), 191–194. With V.B. Marenich. Zbl 0578.53030

27. Geometric characteristics of a Complex Projective Space. – Notes of Moscow State University, Series “Mathematics”. 1986. No. 5. Abstract (of  Report on Extended Seminar of Moscow State University in memory of N. Efimov, 1985). With V.Y. Rovenski

28. Comparison theorem of angles of a triangle for a class of Riemannian manifolds. (in Russian) – Tr. Inst. of Mathematics of USSR Siberian Branch. 1987. V. 9, 16–25. With S.A. Akbarov. Zbl 0642.53043

29. A geometric characterization of the complex projective space. (in Russian) – In: Geometry and topology of homogeneous spaces, Interuniv. Collect. Barnaul. 1988. 98–104. With V.Y. Rovenski. Zbl 0725.53046

30. Open manifolds of nonnegative curvature. – J. Soviet Math. 1991. V. 55, No. 6, 2115–2130. With V.B. Marenich. Zbl 0729.53041

31. The effectiveness of nonstationary control for a class of catalytic reactions. (in Russian) – K.I. Zamaraev (ed.) et al. Mathematical problems in chemical kinetics. Novosibirsk: Nauka, Sibirsk. Otdel. 1989. 319–331. With I.A. Zolotarskij. Zbl 0967.92502

32. A condition sufficient for nonexistence of a cycle in a two-dimensional system quadratic in one of the variables. – Siberian Math. J. 1993. V. 34, No. 2, 350–352. Zbl 0835.34032

33. Surfaces of generalized constant width. – Siberian Math. J. 1993. V. 34, No. 3. P. 555–565. Zbl 0815.53004

34. A uniqueness theorem for a surface with principal curvatures connected by the relation (1–k1d)(1–k2d)=-1. –
 Siberian Math. J. 1993. V. 34, No. 4, 767–769. Zbl 0818.53006

35. Cylinder theorems for convex hypersurfaces. – Siberian Math. J. 1994. V. 35, No. 4, 815–817. Zbl 0854.53003

36. On conditions for existence of umbilical points on a convex surface. – Siberian Math. J. 1995. V. 36, No. 4, 780–786. Zbl 0856.53006

37. On conditions for existence of periodic solutions to a system of ordinary differential equations given integral characteristics. – Siberian Math. J. 1995. V. 36, No. 5, 999–1008. Zbl 0858.34036

38. Tensor algebra and tensor analysis: lecture course notes for the students of Physical Department. (in Russian) – Novosibirsk, Novosibirsk State University. 1995. 50 P.

39. Practical works on Foundations of Geometry: Methodical instructions. (in Russian) – Novosibirsk, Novosibirsk State University, 1995. 24 P. With N.A. Burova.

40. A uniqueness theorem for convex surfaces with no umbilical points and interrelated principal curvatures. – Siberian Math. J. 1996. V. 37, No. 5, 1037–1040. Zbl 0874.53004

41. Great sphere foliations and manifolds with curvature bounded above. – Article dg-ga/9609007, 1996. 13 P.
With V.Y. Rovenski. ArXiv: math.

42. The Cheeger–Gromoll theorem for one class of open Riemannian manifolds with curvature nonnegative in the intergal sense. – Siberian Math. J. 1997. V. 38, No. 1, 179–185. Zbl pre01830113

42. Great sphere foliations and manifolds with curvature bounded above. – Appendix A to the book by V. Rovenski “Foliations on Riemannian manifolds and submanifolds.” Boston etc.: Birkhauser. 1998. 218–234. With V. Rovenski. Zbl 0958.53021.


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