|Victor Toponogov is best known for his contribution
to global Riemannian geometry. His results include the famous
angle comparison and splitting theorems, two cornerstones of
modern comparison geometry.
An essay on his scientific and pedagogical activities can be
found in an obituary published in
Russ. Math. Surv. 61, No. 2, 341-345 (2006)
Victor Toponogov (1930-2004)
Victor Andreevich Toponogov was born March 6, 1930 in the city of Tomsk (one of the
oldest Siberian cities). He graduated from the department of Mechanics and Mathematics
of Tomsk University in 1953 with honor and became a PhD student at the same university.
His supervisor was Professor Abram I. Fet. The famous Toponogovís comparison theorem
was the subject of his first paper and of his PhD thesis defended in 1958.
particular, Perelman used the latter theorem in his proof of the Poincare conjecture. Let
us also mention one of Toponogovís extremal theorems. The well known Myers theorem
states that if the Ricci curvature of a complete Riemannian manifold is bounded from
below by a positive constant, then the diameter of the manifold is not more than the
diameter of the corresponding sphere. Toponogov proved: If, under hypotheses of the
Myers theorem, the diameter of the manifold coincides with the same of the sphere, then
the manifold is isometric to the sphere. This result is known as Toponogovís maximal
diameter theorem and was later generalized in several directions.
In 1956, Toponogov moved to Novosibirsk where he worked till his last day for the
Institute of Mathematics of the Siberian Branch of Russian Academy of Sciences. In 1968
he got his Doctor degree (a Russian equivalent for Habilitation) on ďExtremal theorems
for Riemannian spaces of curvature bounded from belowĒ. He was a lecturer at Novosibirsk
State University and Novosibirsk State Pedagogical University for more than 45 years.
More than 15 of his students got their PhDís, and 7 of them obtained Doctor degrees.
The angles comparison theorem for a triangle in a Riemannian manifold with curvature
bounded from below by a non-negative constant remains the most popular of Toponogovís
results since it is widely used in geometry. Toponogovís theorem on splitting a non-
negatively curved Riemannian manifold containing a straight line is also well known.
Victor Toponogov was a versatile person. He was a good chess player. He walked a lot
in the Siberian taiga, rafting along Siberian rivers. He always had many friends.
Victor A. Toponogov passed away November 21, 2004. He is survived by his wife
Lyudmila and three sons.
More details about Victor Toponogovís life and scientific heritage may be found in his
obituary published in Russ. Math. Surv. 61, No. 2, 341-345 (2006).
List of publications of Victor Toponogov