Vrănceanu was awarded an Adamachi scholarship to study at Iasi University, which he entered in 1919, and there he was taught mathematics by Alexandru, Vera Myller, Simeon Sanielevici, Victor Valcovici and Simeon Stoilow, all famous Romanian mathematicians of that time. Vrănceanu was very highly regarded by his professors and, on 15 February 1922, he graduated from the University of Iasi. However, while still a student in his third year, Vrănceanu was appointed, on 1 December 1921, as an assistant to the mathematics seminar at the request of S Sanielevici. After a brilliant undergraduate career Vrănceanu went first to Göttingen in 1923 where he studied under Hilbert, then he went to Rome to study for his doctorate in mathematics.
In Rome Vrănceanu studied under Levi-Civita, obtaining his doctorate on 5 November 1924 for a dissertation Sopra una teorema di Weierstrass e le sue applicazioni alla stabilita which gave a new proof of a theorem on the decomposition of analytical functions of more variables and also studied applications of the theorem to mechanics. The examining board consisted of 11 professors, headed by Volterra. His doctoral thesis, and all his earlier publications, concerned applications of analysis to mechanics. Vrănceanu returned to Iasi and, in 1926, still developing ideas suggested by Levi-Civita, Vrănceanu discovered the notion of a non-holonomic space. Today this concept is named after Vrănceanu.
Although only 26 years old when he made this remarkable discovery, it quickly turned him into a celebrity. He was appointed a lecturer at Iasi University and then, during 1927-1928, he was awarded a Rockefeller scholarship to study in France and in the United States. In Paris he worked with Élie Cartan and then he went to the United States where he studied at Harvard University and Princeton University. He met Birkhoff and Veblen and later they became good friends. When his scholarship came to an end he was offered a position as a professor but he preferred to return to Romania, taking up his post in Iasi.
In 1929 Vrănceanu moved to Cernauti University where he was appointed professor of analytical geometry, then still at Cernauti he was appointed professor of Differential and Integral Geometry in the following year. After 10 years of great mathematical activity at Cernauti University, he was asked to fill in the professorship at Bucharest University which had become vacant on the death of Gheorghe Țițeica in 1939. In 1948 Vrănceanu was appointed Head of Geometry and Topology at Bucharest University. He retired from his chair in 1970, continuing to take an active interest in mathematics at the university.
During his time at Cernauti University Vrănceanu became known as one of the leading geometers in the world. In 1928 at the International Congress of Mathematics in Bologna, the notion of a non-holonomic space which he had discovered was studied by Schouten and Cartan. Meanwhile Vrănceanu made new discoveries in global geometry.
At Bucharest University Vrănceanu began to organise the mathematics library in a similar way to the one in Iasi. He formed his own group of young geometers and together they wrote teaching texts, as well as the 4 volumes of a differential geometry text, later translated in German and French. Besides his major contributions to science, Vrănceanu took an active interest in politics. In 1944 he was one of the founders of a movement which tried to prevent Romania from fighting against Russia.
Vrănceanu organised the Mathematical Institute of the Romanian Academy, a very important step for theoretical and applied researches in his country. Until his death, he was an editor of the Mathematical Studies and Researches and the Revue Roumaine de Mathématiques Pures et Appliquées. He tried to make known all the Romanian discoveries in mathematics to the international mathematical community. He also organised many scientific conferences, both inside and outside Romania, the last one being held in September 1978 in Craiova. He was much in demend as a lecturer, being invited to lecture at over 30 institutions world-wide, for example he lectured at universities in Paris, Rome, Princeton, Moscow, Peking, Berlin, London, Salamanca, Geneva and many others.
During his career, Vrănceanu published over 300 articles in journals throughout the world. They cover all the branches of modern geometry, from the classical theory of surfaces to the notion of non-holonomic spaces which he discovered, creating efficient methods and solving fundamental problems. Other topics he studied include the absolute differential calculus of congruences, analytical mechanics, partial differential equations of the second order, non-holonomic unitary theory, conformal connection spaces, metrics in spherical and projective spaces, Lie groups, global differential geometry, discrete groups of affine connection spaces, locally Euclidean connection spaces, Riemannian spaces of constant connection, differentiable varieties, embedding of lens spaces into Euclidean space, tangent vectors of spheres and exotic spheres, the equivalence method, non-linear connection spaces, and the geometry of mechanical systems.
Vrănceanu won many honours, both in his own country and elsewhere. He was elected to the Romanian Academy as a corresponding member in 1946, then as a full member in 1955. From 1964 he was President of the Mathematics Section of the Romanian Academy. He won a Government Award (1952) and other medals and awards for excellence. He was awarded honorary degrees from Bologna University (1967) and Iasi University (1970). He was also elected to the Peloritana dei Pericolanti University in Messina (1968) and the Royal Flamand Academy of Brussels (1970). He was elected a member of the Royal Society of Liège in 1972.
Vrănceanu served as a member of the International Committee of the International Mathematical Union for many years and, in that capacity, he was involved in publishing the complete works of Élie Cartan. In 1975 Vrănceanu was elected Vice-president of the International Mathematical Union.
Article by: J J O'Connor and E F Robertson