By putting together Ricci-Curbastro's algorithm with some results from Lie's theory of transformation groups, Levi-Civita extended the theory of absolute invariants to more general cases than those considered by Ricci-Curbastro.He graduated in 1892 and his dissertation was published in the following year after he had made some minor changes to it. He was awarded his teaching diploma in 1894 and in the following year he was appointed to the teacher's college which was attached to the Faculty of Science at Parvia.
Levi-Civita was appointed to the Chair of Rational Mechanics at Padua in 1898, a post which he was to hold for 20 years. However several times during these twenty years attempts had been made to have him move to Rome. In particular in 1909 Castelnuovo tried hard to persuade him to move, but Levi-Civita was happy to remain in Padua. Levi-Civita was a pacifist with firm socialist ideas and it may well have been that he felt Padua suited his personality better than Rome at the time. Of course he was an outstanding mathematician with an impressive international reputation so it was natural for the University of Rome to try to attract him. While teaching at Padua, Libera Trevisani was one of his pupils and they married in 1914.
After World War I ended, the University of Rome made strenuous efforts to strengthen both its teaching and research and many leading scientists were attracted there. Levi-Civita was always very international in his outlook and the ability of Rome to attract top quality students from abroad must have figured in his reasons to now want to make the move there. In 1918 he was appointed to the Chair of Higher Analysis at Rome, and two years later he was appointed to the Chair of Mechanics there.
The years following World War I were difficult for scientists who wanted to collaborate with those in all countries on an equal footing. The President of the United States, Woodrow Wilson, drew up Fourteen Points on 8 January 1917 on which to end World War I. These had not been agreed by the allies. On 4 October 1918 the German government approached Wilson, looking to start peace negotiations and Wilson presented them with the Fourteen Points. After nearly three weeks of negotiations, without the other allies being involved, Germany accepted the Fourteen Points on 23 October. The British and French were certainly unhappy with some of the Fourteen Points and a difficult period followed. In the middle of all of this, Wilson proposed another idea to Britain and France, namely that a structure should be put in place to re-establish international cooperation in science. His proposal was for an International Research Council which would be organised round International Unions for each of the various scientific subjects. These International Unions would operate through National Committees in the countries of the eleven Allied Powers, with these National Committees each supported by its National Academy of Science and National Research Council. The International Unions would have the power to invite neutral countries to join, but not those countries against whom the Allied Powers had fought. Wilson's proposal was accepted and in 1919 the International Research Council was founded. Germany, Austria, Hungary and Bulgaria could not be members under the terms of the International Research Council. Levi-Civita was opposed to such ideas as he made clear in a letter he wrote to Sommerfeld in 1920:-
I have always been, and not only in science, a convinced internationalist ... we agree on an essential point - and I am pleased about it - that scientific relationships and personal relationships between scientists coming from different countries should not be perturbed by contingencies or memories of national or state disagreements.When Von Kármán approached Levi-Civita in 1922 suggesting a scientific meeting on fluid dynamics he knew that such a meeting could not be an official congress if German and Italian scientists were both involved so he proposed an informal one. Levi-Civita was enthusiastic but when the meeting took place in Innsbruck in September of that year the only scientists from the Allied Powers to participate were Levi-Civita and members of his research group. This, however, marked the start of the International Congresses of Applied Mechanics with the decision taken at the Innsbruck meeting to include all areas of applied mechanics and the first full congress took place in Delft in 1924. Levi-Civita's role is described in detail by Battimelli in . He writes:-
Tullio Levi-Civita was one of the leading figures in the creation, in the years following World War I, of the International Congresses of Applied Mechanics, and remained an active member of the Congress committee to the end of his life. ... Levi-Civita [made a major] contribution to the life of the Congresses, from the early days of the 1922 Innsbruck conference to the late thirties [with] his role in the international network created by the newborn institution ...It was not just the international situation which gave Levi-Civita problems but also the effect of totalitarianism and anti-Semitism on scientific and university life. He found the national situation in Italy with the rise of Fascism increasingly difficult. In 1931 all Italian professors were required to sign an oath to Fascism. Volterra refused to take the oath and was dismissed. Although he was deeply opposed to such ideas, Levi-Civita felt that for the sake of his family and his research school in Rome he had to sign despite his strong moral objections. He lectured in the United States in 1933 and in Moscow and Kiev in 1935. In 1936 he returned to the United States, lecturing at Harvard, Princeton and the Rice Institute. While in Houston he gave an interview which was seen as critical of Italy and the Italian consul asked for clarification. He was recalled to Italy but because of his leading international status the Italian government felt that it should not react too strongly. Later in 1936 the International Mathematical Congress was held in Oslo but Levi-Civita, and all other Italian mathematicians, were forbidden to attend by their government. Despite this Levi-Civita was appointed as a member of the Commission for awarding Fields Medals.
On 5 September 1938 the Racial Laws were passed which excluded all those of Jewish background from universities, schools, academies and other institutions. Levi-Civita was dismissed from his professorship, forced to leave the editorial board of Zentralblatt für Mathematik, and prevented from attending the Fifth International Congress of Applied Mechanics in the United States. He wrote to a former student in May 1939 (see for example ):-
I live as a retired person and I do not move; except in summer, however, if my personal conditions allow me to move. As you maybe know, Jews have been completely expelled from Italian cultural life; in particular, I will not participate in the "Volta Congress" and will not be in Rome in September.The authors of  write:-
In the last years of his life, in spite of his moral and physical depression, Levi-Civita remained faithful to the ideal of scientific internationalism and helped colleagues and students who were victims of anti-Semitism; thanks to him, many of them found positions in South America or in the USA.Levi-Civita had very great command of pure mathematics, with particularly strong geometric intuition which he applied to a variety of problems of applied mathematics. One of his papers in 1895 improved on Riemann's contour integral formula for the number of primes in a given interval. He is best known, however, for his work on the absolute differential calculus and with its applications to the theory of relativity. In 1886 he published a famous paper in which he developed the calculus of tensors, following on the work of Christoffel, including covariant differentiation. In 1900 he published, jointly with Ricci-Curbastro, the theory of tensors in Méthodes de calcul differential absolu et leures applications Ⓣ in a form which was used by Einstein 15 years later. The paper was requested by Klein when he met Levi-Civita in Padua in 1899 and, following Klein's wishes, it appeared in Mathematische Annalen.
Weyl was to take up Levi-Civita's ideas and make them into a unified theory of gravitation and electromagnetism. Levi-Civita's work was of extreme importance in the theory of relativity, and he produced a series of papers elegantly treating the problem of a static gravitational field. This topic was discussed in a correspondence between Levi-Civita and Einstein. The paper  looks at:-
... the main mathematical and physical questions discussed by Einstein and Levi-Civita in their 1915 - 1917 correspondence: the variational formulation of the gravitational field equations and their covariance properties, and the definition of the gravitational energy and the existence of gravitational waves.Analytic dynamics was another topic studied by Levi-Civita, many of his papers examining special cases of the three-body problem. He began publishing papers on the subject in 1903, with another important paper appearing in 1906 which strengthened his earlier results. In 1920 he published a compendium on the three-body problem in Acta Mathematica. Then near the end of his career he became interested in the n-body problem. In 1950 (nine years after his death) a book by Levi-Civita entitled Le problème des n corps en relativité générale Ⓣ was published. H P Robertson writes in a review:-
This excellent monograph on the n-body problem in the general theory of relativity was prepared about ten years ago, but its appearance now is none the less timely for those who have worried themselves with one or another aspect of the problem. Its major achievements are two: a derivation of the equations of motion of n point masses, free from the subtle errors besetting most of the standard treatments; and a careful discussion of the possible contributions, in the Einsteinian approximation, of the finite size and internal constitution of the bodies involved.He also wrote on the theory of systems of ordinary and partial differential equations. In  the authors argue that Levi-Civita was interested in the theory of stability and qualitative analysis of ordinary differential equations for three reasons: his interest in geometry and geometric models; his interest in classical mechanics and celestial mechanics, in particular, the three-body problem; and his interest in stability of movement in the domain of analytic mechanics. He added to the theory of Cauchy and Kovalevskaya and wrote up this work in an excellent book written in 1931.
Levi-Civita's interest in hydrodynamics began early in his career with his paper Note on the resistance of fluids appearing in 1901. He worked later on waves in a canal and his proof of the existence of irrotational waves was a major contribution to a long standing open question. In  Levi-Civita's work with his student L S Da Rios on three-dimensional vortex filament dynamics is discussed in detail. Ricca writes:-
Their results include the conception of the localized induction approximation for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations, and the stability analysis of circular vortex filaments. In the light of modern developments in nonlinear fluid mechanics, their work strikes for modernity and depth of results. ... Levi-Civita's work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory and capacity theory.In 1933 Levi-Civita contributed to Dirac's equations of quantum theory.
The Royal Society conferred the Sylvester medal on Levi-Civita in 1922, while in 1930 he was elected a foreign member. He was also an honorary member of the London Mathematical Society, the Royal Society of Edinburgh, and the Edinburgh Mathematical Society. He attended the 1930 Colloquium of the Edinburgh Mathematical Society in St Andrews.
After he was dismissed from his post the blow soon told on his health and he developed severe heart problems. He died of a stroke. Nastasi and Tazzioli write :-
He was one of the most eminent professors in Italy for 40 years and attracted students coming from all countries, whom he encouraged with patience and nobility. Kindness and modesty were manifestations of his soul. Many people benefited from his kindness and retained an ineffaceable memory of his extraordinary personality.
Article by: J J O'Connor and E F Robertson