Victor Andreevich Toponogov (russian: Ви́ктор Андре́евич Топоно́гов; March 6, 1930 – November 21, 2004) was an outstanding

Russia Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, noted for his contributions to

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

and so-called

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

"in the large".

Biography

After finishing secondary school in 1948, Toponogov entered the department of Mechanics and Mathematics at

Tomsk State University The National Research Tomsk State University, TSU (russian: Национа́льный иссле́довательский То́мский госуда́рственный университе́т) is a public research university located in Tom ...

, graduated with honours in 1953, and continued as a graduate student there until 1956. He moved to an institution in

Novosibirsk Novosibirsk (, also ; rus, Новосиби́рск, p=nəvəsʲɪˈbʲirsk, a=ru-Новосибирск.ogg) is the largest city and administrative centre of Novosibirsk Oblast and Siberian Federal District in Russia. As of the Russian Census ...

in 1956 and lived in that city for the rest of his career. Since the institution at Novosibirsk had not yet been fully credentialed, he had defended his Ph.D. thesis at Moscow State University in 1958, on a subject in

Riemann space In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...

Novosibirsk State University Novosibirsk State University is a public research university located in Novosibirsk, Russia. The university was founded in 1958, on the principles of integration of education and science, early involvement of students with research activities an ...

was established in 1959. In 1961 Toponogov became a professor at a newly created Institute of Mathematics and Computing in Novosibirsk affiliated with the state university. Toponogov's scientific interests were influenced by his advisor Abram Fet, who taught at Tomsk and later at Novosibirsk. Fet was a well-recognized topologist and specialist in variational calculus in the large. Toponogov's work was also strongly influenced by the work of

Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...

. Later, the class of

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

s known as CAT(''k'') spaces would be named after

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...

, Aleksandrov and Toponogov. Toponogov published over forty papers and some books during his career. His works are concentrated in Riemannian geometry "in the large". A significant number of his students also made notable contributions in this field.

Conjecture on Complete Convex Surfaces

In 1995 Toponogov made the conjecture:

''On a complete convex surface S homeomorphic to a plane the following equality holds:'' ''where $\kappa_1$ and $\kappa_2$ are the principal curvatures of S.''

In words, it states that every complete convex surface homeomorphic to a plane must have an umbilic point which may lie at infinity. As such, it is the natural open analog of the

Carathéodory conjecture In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.''Sitzungsberichte der Berliner Mathematis ...

for closed convex surfaces. In the same paper, Toponogov proved the conjecture under either of two assumptions: the integral of the Gauss curvature is less than

2\pi

, or the Gauss curvature and the gradients of the curvatures are bounded on ''S''. The general case remains open.

References

External links

Toponogov's Biography, including a list of his publications
(in English) 1930 births 2004 deaths 20th-century Russian mathematicians 21st-century Russian mathematicians Moscow State University alumni Tomsk State University alumni People from Tomsk Academic staff of Novosibirsk State University Differential geometers Soviet mathematicians Burials at Yuzhnoye Cemetery (Novosibirsk) {{Russia-mathematician-stub

Biography

Conjecture on Complete Convex Surfaces

See also

References

External links