图利奥·列维-齐维塔Tullio Levi-Civita

图利奥·列维-齐维塔的父亲是Giacomo Levi-Civita，是一名律师。事实上，Giacomo后来于1908年被任命为意大利参议员。列维-齐维塔出生于一个犹太家庭，在帕多瓦上中学，在那里显露出杰出的才能。随后，他于1890年进入帕多瓦大学数学系攻读学位。他的两位老师是朱塞佩·韦罗内塞和格雷戈里奥·里奇-库尔巴斯托罗，列维-齐维塔后来与后者合作。他写了一篇由格雷戈里奥·里奇-库尔巴斯托罗指导的关于绝对不变量的学位论文，但这也标志着他开始使用tensor calculus[30]：-

通过将格雷戈里奥·里奇-库尔巴斯托罗的算法与索菲斯·李变换群理论的一些结果结合起来，列维-齐维塔将绝对不变量理论推广到了比格雷戈里奥·里奇-库尔巴斯托罗所考虑的更为一般的情形。

他于1892年毕业，其学位论文在作了一些小修改后于次年发表。他于1894年获得教师资格证书，次年被任命到帕维亚理学院附属的师范学院。

列维-齐维塔于1898年被任命为帕多瓦大学理性力学讲席，他担任此职达20年。然而在这二十年中，曾多次有人试图让他迁往罗马。特别是在1909年，Castelnuovo竭力劝他迁往罗马，但列维-齐维塔乐于留在帕多瓦。列维-齐维塔是一位和平主义者，抱有坚定的社会主义思想，很可能他觉得当时帕多瓦比罗马更适合他的个性。当然，他是一位杰出的数学家，享有令人瞩目的国际声誉，因此罗马大学试图吸引他是很自然的。在帕多瓦任教期间，Libera Trevisani是他的学生之一，他们于1914年结婚。

第一次世界大战结束后，罗马大学大力加强教学与研究，许多顶尖科学家被吸引到那里。列维-齐维塔的视野一向非常国际化，而罗马能够吸引来自国外的顶尖学生，这必定是他此时想要迁往那里的原因之一。1918年，他被任命为罗马大学高等分析讲席，两年后又被任命为那里的力学讲席。

第一次世界大战后的几年对希望与所有国家的科学家在平等基础上合作的科学家来说是艰难的。美国总统伍德罗·威尔逊于1917年1月8日提出了结束第一次世界大战的十四点原则。这些原则并未得到协约国的同意。1918年10月4日，德国政府接触威尔逊，希望开始和平谈判，威尔逊向他们提出了十四点原则。经过近三周的谈判，在其他协约国未参与的情况下，德国于10月23日接受了十四点原则。英国和法国当然对十四点原则中的一些内容感到不满，随后是一段艰难的时期。在这一切之中，威尔逊向英国和法国提出了另一个想法，即应建立一个架构来重建科学领域的国际合作。他的提议是成立一个国际研究理事会，该理事会将围绕各个科学学科的国际联合会来组织。这些国际联合会将通过十一个协约国国家的国家委员会运作，每个国家委员会由其国家科学院和国家研究理事会支持。国际联合会有权邀请中立国加入，但不得邀请协约国曾与之交战的国家。威尔逊的提议被接受，1919年国际研究理事会成立。根据国际研究理事会的规定，德国、奥地利、匈牙利和保加利亚不能成为成员。列维-齐维塔反对这样的想法，正如他在1920年写给阿诺·索末菲的一封信中所明确表示的：-

我一直是一个坚定的国际主义者，不仅在科学方面……我们在一个关键点上意见一致——我对此感到高兴——即来自不同国家的科学家之间的科学关系和个人关系不应受到国家或政府分歧的偶然事件或记忆的干扰。

当西奥多·冯·卡门于1922年接触列维-齐维塔，建议就流体动力学举行一次科学会议时，他知道如果德国和意大利科学家都参与，这样的会议不能是正式的大会，因此他提议举行一次非正式会议。列维-齐维塔热情支持，但当会议于当年9月在因斯布鲁克举行时，来自协约国的唯一参与科学家是列维-齐维塔及其研究小组成员。然而，这标志着国际应用力学大会的开始，因斯布鲁克会议决定涵盖应用力学的所有领域，第一次完整的大会于1924年在代尔夫特举行。列维-齐维塔的作用在Battimelli的[10]中有详细描述。他写道：-

列维-齐维塔是第一次世界大战后几年中创建国际应用力学大会的主要人物之一，并且直到生命的尽头一直是大会委员会的活跃成员。……列维-齐维塔对大会的生命做出了[重大]贡献，从1922年因斯布鲁克会议的早期到三十年代后期，[其中]他在这个新生机构所创建的国际网络中发挥了作用……

给列维-齐维塔带来问题的不仅是国际局势，还有极权主义和反犹太主义对科学和大学生活的影响。他发现意大利国内随着法西斯主义的兴起，局势日益艰难。1931年，所有意大利教授都被要求签署效忠法西斯主义的誓言。维多·沃尔泰拉拒绝宣誓并被解职。尽管他深深反对这样的想法，列维-齐维塔觉得为了家庭和他在罗马的研究学派，尽管有强烈的道德异议，他还是不得不签署。他于1933年在美国讲学，1935年在莫斯科和基辅讲学。1936年他回到美国，在哈佛、普林斯顿和莱斯研究所讲学。在休斯顿期间，他接受了一次采访，该采访被视为对意大利的批评，意大利领事要求澄清。他被召回意大利，但由于他的国际领先地位，意大利政府觉得不应反应过于强烈。1936年晚些时候，国际数学大会在奥斯陆举行，但列维-齐维塔和所有其他意大利数学家被他们的政府禁止参加。尽管如此，列维-齐维塔被任命为颁发菲尔兹奖的委员会成员。

1938年9月5日，种族法获得通过，将所有有犹太背景的人排除在大学、学校、科学院和其他机构之外。列维-齐维塔被解除教授职务，被迫离开Zentralblatt für Mathematik的编辑委员会，并被阻止参加在美国举行的第五届国际应用力学大会。他于1939年5月写信给一位以前的学生（例如见[30]）：-

我过着退休生活，不外出；不过，如果我的个人条件允许，夏天除外。你可能知道，犹太人已被完全逐出意大利的文化生活；特别是，我不会参加“Volta Congress”，也不会在九月去罗马。

[30]的作者写道：-

在他生命的最后几年，尽管精神和身体上都处于抑郁状态，列维-齐维塔仍然忠于科学国际主义的理想，并帮助了反犹太主义的受害同事和学生；多亏了他，他们中的许多人在南美或美国找到了职位。

列维-齐维塔对纯数学有很强的掌握，尤其具有极强的几何直觉，他将其应用于应用数学的各种问题。他在1895年的一篇论文改进了波恩哈德·黎曼关于给定区间内primes数量的围道积分公式。然而，他最著名的是他在绝对微分计算及其在相对论中的应用方面的工作。1886年，他发表了一篇著名论文，在埃尔温·布鲁诺·克里斯托费尔工作的基础上发展了张量计算，包括协变微分。1900年，他与格雷戈里奥·里奇-库尔巴斯托罗联合发表了Méthodes de calcul differential absolu et leures applicationsⓉ中的张量理论（绝对微分计算方法及其应用），其形式被阿尔伯特·爱因斯坦在15年后使用。这篇论文是菲利克斯·克莱因在1899年在帕多瓦遇到列维-齐维塔时要求的，并按照菲利克斯·克莱因的意愿，发表在Mathematische Annalen上。

赫尔曼·外尔将采纳列维-齐维塔的思想，并将其发展为引力和电磁力的统一理论。列维-齐维塔的工作在相对论中极为重要，他发表了一系列论文，优雅地处理了静态引力场问题。这个主题在列维-齐维塔和阿尔伯特·爱因斯坦之间的通信中进行了讨论。论文[15]探讨了：-

...阿尔伯特·爱因斯坦和列维-齐维塔在1915-1917年通信中讨论的主要数学和物理问题：引力场方程的变分表述及其协变性质，以及引力能量的定义和引力波的存在。

分析动力学是列维-齐维塔研究的另一个主题，他的许多论文考察了三体问题的特殊情形。他于1903年开始发表关于该主题的论文，1906年又发表了一篇重要论文，加强了他先前的结果。1920年，他出版了关于Acta Mathematica中三体问题的纲要。然后在他职业生涯接近尾声时，他对$n$体问题产生了兴趣。1950年（他去世九年后），列维-齐维塔的一本题为Le problème des n corps en relativité générale Ⓣ（广义相对论中的n体问题）的书出版了。H P Robertson在一篇评论中写道：-

这本关于广义相对论中$n$体问题的优秀专著大约在十年前就已准备好，但它的问世对于曾为这一问题某一或另一侧面而苦恼的人来说，仍然同样及时。它的主要成就有两项：推导出$n$个质点的运动方程，摆脱了困扰大多数标准处理的微妙错误；以及仔细讨论了在爱因斯坦近似下，所涉及物体的有限大小和内部构造可能作出的贡献。

他还撰写了关于常微分方程和偏微分方程方程组的理论的著作。在[18]中，作者们认为列维-齐维塔对常微分方程的稳定性理论和定性分析感兴趣有三个原因：他对几何和几何模型的兴趣；他对经典力学和天体力学，特别是三体问题的兴趣；以及他对分析力学领域中运动稳定性的兴趣。他对奥古斯丁·路易·柯西和柯瓦列夫斯卡娅的理论有所贡献，并在1931年写成的一本优秀著作中记述了这项工作。

列维-齐维塔对流体力学的兴趣始于他职业生涯早期，其论文Note on the resistance of fluids于1901年发表。他后来研究运河中的波，他对无旋波存在的证明是对一个长期未决问题的重要贡献。在[33]中，列维-齐维塔与其学生L S Da Rios关于三维涡丝动力学的工作得到了详细讨论。Ricca写道：-

他们的成果包括：针对细涡丝诱导速度的局部诱导近似的构想，本征运动方程的推导，应用于涡管的渐近势论，螺旋涡和环生成涡构型形式的定常解的推导，以及圆形涡丝的稳定性分析。鉴于非线性流体力学中的现代发展，他们的工作以其现代性和结果的深度而引人注目。……列维-齐维塔关于细长管渐近势的工作处于势论和容量论数学表述的核心。

1933年，列维-齐维塔对保罗·狄拉克的quantum theory方程作出了贡献。

Royal Society于1922年向列维-齐维塔授予了詹姆斯·约瑟夫·西尔维斯特奖章，而1930年他当选为外籍成员。他还是伦敦数学会、Royal Society of Edinburgh和爱丁堡数学会的名誉成员。他参加了1930年在圣安德鲁斯举行的爱丁堡数学会学术讨论会。

他被免职后，这一打击很快影响了他的健康，他出现了严重的心脏问题。他死于中风。Nastasi 和 Tazzioli 写道[30]：-

四十年来，他是意大利最杰出的教授之一，吸引了来自各国的学生，他以耐心和高尚鼓励他们。善良和谦逊是他灵魂的表现。许多人受益于他的善良，并对他非凡的个性留下了不可磨灭的记忆。

Tullio Levi-Civita's father was Giacomo Levi-Civita who was a lawyer. In fact Giacomo was later, in 1908, appointed as an Italian senator. Tullio, born into a Jewish family, attended secondary school in Padua where he showed his outstanding abilities. He then studied for his degree in the Faculty of Mathematics of the University of Padua where he enrolled in 1890. Two of his teachers were Giuseppe Veronese and Ricci-Curbastro and Levi-Civita later collaborated with the latter. He wrote a dissertation, which was supervised by Ricci-Curbastro, on absolute invariants but this also marks the beginning of his use of the tensor calculus [30]:-

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 Pavia.

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 [10]. 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 [30]):-

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 [30] 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 [15] 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 [18] 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 [33] 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 [30]:-

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.