**Stefan**(or

**Stephan**)

**Cohn-Vossen**was born into a Jewish family - his father was Emmanuel Cohn-Vossen. Stefan was brought up and educated in Breslau. The city of around 500,000 inhabitants was, by the time that Cohn-Vossen was growing up there, largely German due chancellor Otto von Bismarck's efforts to Germanize the area. Cohn-Vossen attended Breslau University and his thesis advisor there was Adolf Kneser. He submitted his thesis

*Singuläre Punkte reeller, schlichter Kurvenscharen, deren Differentialgleichung gegeben ist*and graduated with his doctorate in 1924. He moved to the University of Göttingen and published papers such as

*Singularitäten konvexer Flächen*(1927),

*Zwei Sätze über die Starrheit der Eiflachen*(1928), and

*Unstarre geschlossene Flächen*(1929). He habilitated at the University of Göttingen in 1929.

In 1930 Cohn-Vossen was appointed as a privatdocent at the University of Cologne. This was a difficult time in Germany (and in many other countries) as a consequence of the world-wide depression in 1929. This was certainly one of the factors leading to the breakdown of the democratic system in Germany. The Social Democratic government collapsed in 1930 and fears of poverty led many to support the Nazi party. An anti-Semitic campaign led to increased support for the Nazis but Jews like Cohn-Vossen had increasing cause to fear the direction in which the country was moving. On 30 January 1933 Hitler was named chancellor but massive intimidation failed to gain the party an absolute majority in elections on 5 March 1933. This did not prevent the Nazi government taking full control and, on 7 April 1933, the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Cohn-Vossen was dismissed from his teaching position at the University of Cologne.

During the three years that Cohn-Vossen had spent in Cologne he had become well-known with the publication in 1932 of the book *Anschauliche Geometrie* which was a joint work with David Hilbert. The book was based on lectures that Hilbert had given at Göttingen in 1921 but making this course into a book had been Cohn-Vossen's task. Herbert Turnbull writes in [5]:-

Turnbull ends his review with these words:-This remarkable book ... comes as a grateful gesture from Hilbert, who is one of the most eminent of mathematicians, and from his collaborator who has borne the brunt of editing the subject-matter right worthily.

The book was of great importance but this presented problems to the Nazis who had dismissed Cohn-Vossen from his post because of his race. The way that they overcame the difficult conflict between mathematical quality and anti-Semitism was to reduce the role of Cohn-Vossen to that of an amanuensis. In fact Ludwig Bieberbach, who allowed Nazi ideology to completely override his mathematical judgement, took care to emphasise that his only citations toThe book will make a very wide appeal, not only to experts in all branches of pure mathematics, providing as it does a genial connecting link between almost all the maze-like ramifications of the subject; but also to many others who at school or later have felt something of the fascination that geometry ever exercises over the human mind.

*Anschauliche Geometrie*were those that were ascribable solely to Hilbert and definitely not Cohn-Vossen.

We note that the German edition of this book was reprinted in 1973, and there were also translations into many languages including Russian, English (published 1952), Polish (published 1956) and Italian (published 1960). We should comment that the title of the English translation was *Geometry and the Imagination* which is not really a translation of the German title. A more accurate translation of *Anschauliche Geometrie* would be 'Intuitive Geometry'. Donald Coxeter, reviewing this English translation, writes [3]:-

The author of [2] writes of the:-'Auschauliche Geometrie' has been a classic for twenty years. Its breadth of outlook is reminiscent of Klein's 'Elementary Mathematics from an Advanced Standpoint'. Although it deals with elementary topics, it reaches the fringe of our knowledge in many directions.[It contains]an extraordinarily high concentration of interesting ideas and information.

After Cohn-Vossen was dismissed from his position at the University of Cologne in 1933 he left Germany and went to Switzerland. First he went to Locarno, but by 1934 he was teaching at a school in Zürich. Later, still in 1934, he emigrated to Russia and under his influence a school of "geometry in the large" was set up in Moscow and Leningrad. This was to have a major impact on research in Russia as Aleksandr Danilovic Aleksandrov carried on research in this area. Cohn-Vossen had been appointed as a professor at Leningrad University in 1935, working at the Steklov Mathematical Institute of the USSR Academy of Sciences. The Steklov Mathematical Institute moved to Moscow in the following year and Cohn-Vossen moved with the Institute - again he was appointed as a professor at the University in Moscow.... brilliant intuitive and concrete approach to geometry provided by Hilbert and his collaborator Cohn-Vossen ...

He published a number of influential papers during this short period of his life which was brought to a sad end in 1936 when he died of pneumonia. Among these papers, we mention: *Kürzeste Wege und Totalkrümmung auf Flächen* (1935), *Der approximative Sinussatz für kleine Dreiecke auf krummen Flächen* (1936), *Existenz kürzester Wege* (1936), and *Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken* (1936). After his death the paper *Die Kollineationen des n-dimensionalen Raumes* (1938) was published. The importance of this work to the Russian school is seen from the fact that in 1959 a 303-page Russian book *Some problems of differential geometry in the large* (Russian) was published containing Russian translation of seven of Cohn-Vossen's papers. The book contained a survey by Nikolai Vladimirovich Efimov putting Cohn-Vossen's papers into their historical context.

Cohn-Vossen's investigated whether general convex surfaces are uniquely determined by their metric. He proved the rigidity of a closed surface of the class of regularity *C*^{3} with positive Gaussian curvature. His work was a deep development of studies made on the rigidity of a convex polyhedron by Cauchy. After his death it was continued by A D Aleksandrov and then by his student Aleksei Vasilevich Pogorelov.

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