**Nikolai Vladimirovich Efimov**'s father, Vladimir Efimov, was a clerk. Nikolai Vladimirovich was born in Orenburg, a city in the south of Russia, near the Kazakhstan border, north of the Caspian Sea but grew up in Rostov-na-Donu, about 30 kilometres from the north east end of the Sea of Azov. He was educated in Rostov-na-Donu, graduating from the high school there in 1927. After leaving school he was employed constructing a depot at the Kamenolomni railway station on the north side of Rostov-na-Donu. Leaving work on this construction site, his next job was as a metal worker in a factory making agricultural machinery in Rostov-na-Donu. One should not get the impression that these jobs were long term, rather they were just to fill in before he entered the Mathematics and Physics section of the Faculty of Education of Rostov University in 1928.

Efimov graduated in 1931, having completed the four-year undergraduate course in three years. His performance had been outstanding and merited the award of a postgraduate scholarship which funded him to undertake research at Moscow State University. Benjamin Fedorovich Kagan held the chair of geometry at Moscow State University at this time and he was running a research seminar on vector and tensor analysis [6]:-

In 1934 Efimov graduated with a Candidate's degree (equivalent to a Ph.D.) and began work at Voronezh State University as a lecturer. However, he continued to work on his geometry research with the aim of submitting it for a doctorate (equivalent to a D.Sc. or habilitation). The year 1934 when Efimov graduated, was highly significant for geometry research in Russia, for late in that year Stefan Cohn-Vossen emigrated there to escape from the Nazis in Germany. Under his influence a school of "geometry in the large" was set up in Moscow and in Leningrad and Efimov's meeting with Cohn-Vossen, who had worked closely with Hilbert, was a major influence on setting the direction of his future research. In 1940 Efimov was awarded a doctorate from Moscow State University after submitting the thesisFrom then on, for almost half a century, Efimov devoted all his energies to geometry, mainly to the qualitative theory of surfaces, and he became one of the founders of the modern aspect of that branch of geometry.

*The curving of surfaces with parabolic points*. During the years he had been working towards his doctorate, he was publishing an impressive collection of papers, in fact his first seven papers were published between 1933 and 1940. These papers [8]:-

Dubnov, mentioned in this quote, is Yakov Semenovich Dubnov. Dubnov had also been a student of Kagan and Efimov worked closely with him during these years. Although World War II started in September 1939 with the German and Russian invasion of Poland, this had little effect in Voronezh, about 500 km south of Moscow. However, on 22 June 1941 the German armies invaded their former ally pushing rapidly east into Soviet lands. They advanced along a broad front from the Baltic to the Black Sea and, in August 1941, Efimov and 80 students from Voronezh State University enlisted as volunteers in the Russian army. In November 1941 he was ordered to go to the Voronezh Aviation Institute and assist the war effort by taking the position of head of the Theoretical Mechanics Department there. Voronezh was by this time close to the front line and far too dangerous a place for research establishments, so the Soviets had relocated the Voronezh Aviation Institute in Tashkent, far away from possible attack. Efimov worked at the Voronezh Aviation Institute in Tashkent until 1943 when he went to Moscow to head the Department of Higher Mathematics at the Moscow State Forest University (it was at that time called the Moscow Institute of Forestry). In fact 1943 was the earliest that Moscow began to operate again as normal after the German attack on the city in late 1941. Although the Germans were driven back in January 1942, the front line remained too close to the city to allow life to return to normal there and it was not until the second half of 1943 that Moscow became totally secure.... concerned an application of tensor methods to the study of various classes of surfaces and the nets of lines in them. In these papers he studied the geometric properties of geodesies, conjugates and other nets; he obtained tensorial criteria for certain classes of surfaces(including quadrics), and jointly with Dubnov he found an invariant characterization of the metric of a Lie surface.

Although Efimov continued as head of the Department of Higher Mathematics at the Moscow State Forest University until 1961 he had several other academic roles within the city. In 1946 he was appointed as a professor in the Faculty of Physics of Moscow State University and he held this post for ten years. His association with Moscow State University did not end in 1956 for in that year he became a professor in the Faculty of Mechanics and Mathematics, appointed to the chair of Mathematical Analysis. He continued to hold this chair for the rest of his life but from 1962 to 1969 he was, in addition, dean of the Faculty of Mechanics and Mathematics [8]:-

As well as this contribution that Efimov made to the teaching programmes of Moscow State University, we must also give an indication of his own skills as a lecturer, his research contributions, and the impressive collection of books that he wrote. First we record that the authors of [6] state that, after his appointment to Moscow State University in 1946, he:-While he was Dean and with his active participation the teaching plans and programmes underwent a radical reorganization, bringing the presentation of mathematics closer to the modern state of the science and its practical requirements.

Many of the references listed below give detailed descriptions of Efimov's important contributions to geometry, but perhaps his achievements are best summed up by quoting from P S Aleksandrov [2]:-... quickly earned the reputation of being one of the best lecturers there. His lectures and discourses before a student audience are always distinguished by their scientific depth, the mastery of exposition, and the attention they give to the problem of training young people.

The theorem which P S Aleksandrov refers to in this quote was one that Efimov became interested in while working for his doctorate but it was not until around 1950 that he began to concentrate all his efforts on obtaining a solution. Only a few mathematicians have the determination shown by Efimov to spend over twelve years of their lives trying to solve a single problem but he did so and published his solution in 1963 inNikolai Vladimirovich Efimov is one of those rare geometers whose international reputation rests not so much on a series of separate results as on one really big discovery. This is that the curvature of a surface of negative curvature without singularities in three-dimensional space cannot be separated from zero, that is, necessarily takes on values of arbitrarily small modulus. The intuitive content of this result is accessible to every mathematician, for one need only look at a one-sheet hyperboloid or a hyperbolic paraboloid to see it. Efimov's result is a generalization in principle of Hilbert's famous theorem that a surface of constant negative curvature in three-dimensional space cannot be without singularities. Efimov's theorem on the qualitative theory of surfaces ranks alongside Hilbert's theorem. I recall with what enthusiasm Heinz Hopf gave his talk at the International Congress of Mathematicians in Moscow in1966on the subject of Efimov's proof.

*The impossibility in three-dimensional Euclidean space of a complete regular surface with a negative upper bound of the Gaussian curvature*. We must not give the impression that Efimov did not publish in the years during which he battled with his big problem for nothing would be further from the truth. For example he published five papers in 1961 and six papers in 1962 immediately before his important 1963 paper.

Efimov published many important books, some at undergraduate level, some at research monograph level. They were remarkably influential, not only in Russia but in many countries around the word where translations were published. Another indication of the worth of these books is the large number of editions that some of them went through, not just reprints with minor changes, but often editions containing a wealth of new material. One more general comment before we list a few of these books, and that is that they were all single-authored works. Many others who have written large collections of books have done so in collaboration with their colleagues, so reducing their own workload. Efimov's book with the largest number of editions is *Higher Geometry* (Russian) (1945). Arend Heyting, reviewing this first edition, writes:-

Given Heyting's interests, the 'high standard of rigour' comment is particularly pleasing. Many Russian editions followed and by 1978 a sixth edition was in print. An English translation of this sixth edition was made and published 1980. It was also translated into other languages including Spanish and French. The next book was published in Russian in 1948 and a German translation appeared in 1957. The German editionThis university textbook treats mainly the foundations of geometry. ... The book is written in a clear style and meets a high standard of rigour.

*Flächenverbiegung im Grossen*(1957) collects together Efimov's early work on "geometry in the large" as G Y Rainich explains in a review:-

Another teaching text was published by Efimov in 1963, namelyWhen Cohn-Vossen went to Moscow in1936he took with him, so to say, the theory of bending of surfaces. Many important papers on the subject appeared in Russia in the following years, among them Efimov's review that forms the basis of the present book(which to an extent brings the theory back to Germany). ... The exposition is very clear and detailed, although complicated proofs in some cases are omitted.

*Quadratic forms and matrices*(Russian). This was translated into English and published with the title

*Quadratic forms and matrices. An introductory approach*(1964). The book develops the theory of quadratic forms and matrices, and considers their geometrical interpretations which it presents in a two and three dimensional setting.

Another teaching book, but one containing much not standard material for a work at undergraduate level, was* Linear algebra and multidimensional geometry* (Russian) (1970). It was translated into English and published with the title *Linear algebra and multidimensional geometry* (1975). J Burlak, reviewing the Russian edition, writes:-

Next Efimov publishedAlthough this may be thought of as a book on linear algebra, the treatment is so geometrical that the subject matter merges naturally into higher-dimensional analytic geometry. Thus, although applications(e.g., to rigid body dynamics and elasticity theory)are mentioned and the usual matrix theory is covered(including, e.g., reduction to the Jordan canonical form), there is none of the standard material on the solution of systems of linear equations. On the other hand, the material on such topics as tensor algebra, exterior forms, quadric hypersurfaces and projective space is not often found in elementary books on linear algebra. The book is based on courses of lectures and contains enough material for courses of several kinds.

*Exterior differential forms in Euclidean space*(Russian) (1971) in which he:-

Finally, let us mention the book... gives a clear elementary treatment of the calculus of exterior differential forms inR^{n}, leading up to a proof of Stokes's theorem.

*Introduction to the theory of exterior forms*(Russian) (1977).

Efimov received many honours for his contributions including the Lobachevsky Medal in 1951, the Lenin Prize in 1966 and two orders of the Red Banner of Labour. Perhaps his greatest honour was being an invited plenary speaker at the International Congress of Mathematicians held in Moscow in August 1966. He gave the plenary lecture* Hyperbolic Problems in the Theory of Surfaces* (Russian). He became a corresponding member of the USSR Academy of Sciences in 1979. He was elected a member of the Committee of the Moscow Mathematical Society, was appointed as a member of the Editorial Board of the *Uspekhi Matematicheskikh Nauk*, and was chair of the mathematics section of the Universities' and Educational Institutes' scientific-methodological council in the Ministry of Education of the USSR.

We end this biography by giving two quotes which tell us something of Efimov's character and his interests outside mathematics. First we quote the authors of [6] who, two years before Efimov's death, write:-

The authors of the obituary [8] write:-A scientist with a world-famous name, Efimov is in everyday life and work an extraordinarily conscientious and an unusually modest man. His kindness, his readiness to help by word and deed, his high professional and human qualities have won him the sincere respect of all who come into contact with him. We wish him many years of successful work, good health and happiness.

All who knew Nikolai Vladimirovich Efimov remember his joie de vivre, his exceptional kindness towards people, his thoughtfulness, and extraordinary conscientiousness and thoroughness in all his work. Association with him was always lively and interesting; he was a man of high culture, a great connoisseur and judge of the arts, especially of poetry and the theatre. Not only in his teaching, editing and administrative work, but in all his dealings with the people around him he was able to see what was important; and perhaps this feature of his character appeared particularly brightly in his mathematical research; Efimov's theorems occupy a worthy place in mathematics. Efimov was a remarkable person and a great mathematician. Those who knew him will always cherish his bright memory in their hearts.

**Article by:** *J J O'Connor* and *E F Robertson*

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