格雷戈里奥·里奇-库尔巴斯托罗Gregorio Ricci-Curbastro

格雷戈里奥·里奇-库尔巴斯托罗 的父亲是 Antonio Ricci-Curbastro，母亲是 Livia Vecchi。这是一个在整个拉韦纳省都知名的高门第家庭。Antonio Ricci-Curbastro 虽然肯定从未取得接近他儿子 里奇-库尔巴斯托罗 所获得的那种名声，但他本人作为一名工程师也很有名。里奇-库尔巴斯托罗 和他的兄弟 Domenico 都没有上过学。他们在进入大学之前所受的全部教育都是在家中完成的，父母在那里为他们聘请了私人教师。

1869年，里奇-库尔巴斯托罗 进入罗马大学，打算学习数学和哲学。当时他只有十六岁，虽然他没有上过学，但在学业上准备得很好。然而，政治事件凑在一起，使罗马成了一个多少有些不幸的选择，尽管考虑到他的出生地，这又是一个非常自然的选择。当 里奇-库尔巴斯托罗 在罗马开始他的学业时，虽然意大利王国几年前已经建立，但罗马并不属于这个王国，而是属于教皇国，Ricci 就是在教皇国出生和长大的。罗马曾在1867年遭到意大利军队进攻，但法国保卫了这座城市，并动用其军队抵御进攻。然而在1870年，意大利军队攻占了罗马，它成为意大利王国的首都。里奇-库尔巴斯托罗 从1869年到1870年在罗马学习了一年，然后回到父母家中，在那里待了两年，之后才开始第二段大学生涯。

这一次他去的地方不是罗马，而是博洛尼亚大学。他在1872至1873年间在那里学习，随后转到比萨，进入比萨高等师范学校，在恩里科·贝蒂的领导下，该校正成为意大利数学研究和数学教育的领先中心。除了在比萨听恩里科·贝蒂的课，里奇-库尔巴斯托罗还听了乌利塞·迪尼的课。1875年，里奇-库尔巴斯托罗因其学位论文On Fuchs's research concerning linear 微分方程获得博士学位。他留在比萨撰写一篇论文，并于次年提交以满足任教所需的条件。这篇论文是On a generalisation of Riemann's problem concerning 超几何函数。无论是这篇论文还是他的博士论文，都未曾发表。

细心的读者会注意到，里奇-库尔巴斯托罗的这两部早期著作都基于德国数学家而非意大利数学家的著作。他接下来进行的工作同样不是基于意大利数学家的思想。首先是关于詹姆斯·克拉克·麦克斯韦的电动力学理论和鲁道夫·克劳修斯工作的一系列文章，这是恩里科·贝蒂请他写的。其中三篇于1877年发表在Nuovo Cimento上，同年，乌利塞·迪尼请他撰写的关于约瑟夫·拉格朗日的线性微分方程组问题的文章发表在Giornale di matematiche di Battaglini上。

里奇-库尔巴斯托罗这时在竞争一项奖学金，他赢得了一项，这使他得以在1877—78年间出国。他选择去德国并不令人意外，事实上他选择在慕尼黑的技术高等学校学习，菲利克斯·克莱因两年前已被任命为该校讲席。除了菲利克斯·克莱因，Brill也在慕尼黑的技术高等学校工作，里奇-库尔巴斯托罗听了这两位著名数学家的课。正如Speziali在[1]中所写：-

Ricci非常钦佩菲利克斯·克莱因，而他的敬意很快也得到了回报；尽管如此，Ricci似乎并未决定性地受到Klein教学的影响。真正激发他未来研究的，毋宁说是波恩哈德·黎曼、埃尔温·布鲁诺·克里斯托费尔和鲁道夫·利普希茨。事实上，他们对他的影响甚至超过了他的意大利老师们。

1879年回到比萨后，里奇-库尔巴斯托罗成为乌利塞·迪尼的助手。然后，从1880年到1925年去世，他一直是帕多瓦大学的数学物理教授。然而，他不仅教授数学物理，因为从1891年起他还在帕多瓦教授高等代数课程。直到他被任命为帕多瓦的讲席后，他才有了结婚的保障，并于1884年与Bianca 路易吉·比安基 Azzarani结婚。他们有三个孩子；两个儿子和一个女儿。

里奇-库尔巴斯托罗的早期工作是在数学物理领域，特别是关于电路定律和微分方程。他稍微改变了领域，从事微分几何的研究，并在1884年至1894年间发明了绝对微分学。最初的贡献是由卡尔·弗里德里希·高斯做出的，然后这些思想在波恩哈德·黎曼1854年的ProbevorlesungⓉ（试讲）以及他为巴黎Académie des Sciences的奖金竞赛所写的一篇1861年论文中得到了发展。然而，是埃尔温·布鲁诺·克里斯托费尔的一篇论文，于1868年发表在Crelle's Journal上，对里奇-库尔巴斯托罗在1884年开始研究二次微分形式产生了主要影响。他于1888年在一篇为博洛尼亚大学800周年校庆所写的论文中首次系统地提出了这些重要想法。Speziali在[1]中写道：-

他用来证明[二次形式的不变性]的方法引导他走向了绝对微分学的技巧，他在1888年至1892年间撰写的四篇出版物中完整地讨论了这一技巧。

里奇-库尔巴斯托罗在1900年之后的大部分工作是与他的学生图利奥·列维-齐维塔共同完成的。在那年的一篇基础性联合论文Méthodes de calcul différentiel absolu et leurs applicationsⓉ（绝对微分学方法及其应用）中，他（唯一一次）使用了Ricci这个名字，而不是他的全名。这篇论文是五年前应菲利克斯·克莱因的要求而写的。作者在他们重要的七十七页论文的序言中陈述了他们的目标：-

绝对微分学的算法，即这些方法的物质工具……可以在埃尔温·布鲁诺·克里斯托费尔的一条注记中找到完整的表述。但方法本身及其所提供的优势，其存在理由和来源在于它们与$n$维簇概念的密切联系，这一概念我们要归功于卡尔·弗里德里希·高斯和波恩哈德·黎曼的卓越思想。……由于以本质的方式与$V_{n}$相关联，它是所有那些研究的自然工具，这些研究以这样一个簇为主题，或者在其中人们会遇到一个由$n$个变量的微分或其导数构成的正定二次形式作为特征元素。

在论文中，里奇-库尔巴斯托罗和图利奥·列维-齐维塔给出了对微分的二次型分类的应用，还有其他分析应用；他们给出了几何应用，包括曲面论和运动的群；以及力学应用，包括动力学和约瑟夫·拉格朗日方程的解。这篇论文的主要思想在[1]中进行了讨论。里奇-库尔巴斯托罗的绝对微分学成为tensor analysis的基础，并被阿尔伯特·爱因斯坦用于他的广义相对论中。

论文[7]由里奇-库尔巴斯托罗的学生图利奥·列维-齐维塔撰写，列出了他的六十一种出版物。然而，他抽出时间也为地方政府做出贡献，正如他那个时代的许多意大利数学家一样。他曾担任家乡卢戈的议员，并以此身份参与了许多与供水和沼泽排水有关的项目（这是许多意大利数学家在几个世纪中参与的活动）。后来，他担任帕多瓦的议员，在那里他的兴趣包括学校教育和财政。然而，当被提供帕多瓦市长的职位时，他拒绝了。

里奇-库尔巴斯托罗因其杰出贡献而获得许多荣誉，尽管人们不得不说，在他做出这些工作时，其重要性并未被充分理解，而是在一段时间后才被认识到。他被授予多个科学院的院士称号，如威尼斯学院，他于1892年被接纳，并于1916-18年担任院长。他还从1899年起成为Accademia dei Lincei的成员，从1905年起成为帕多瓦学院的成员，从1918年起成为都灵科学院的成员，从1921年起成为Società dei Quaranta的成员，从1922年起成为博洛尼亚皇家学院的成员，从1925年起成为教皇学院的成员。

Gregorio Ricci-Curbastro's father was Antonio Ricci-Curbastro and his mother was Livia Vecchi. It was a family of high status known throughout the province of Ravenna. Antonio Ricci-Curbastro, although certainly never achieving anything close to the fame achieved by his son Gregorio, nevertheless was himself well known as an engineer. Neither Gregorio nor his brother Domenico attended school. All their education prior to entering university was carried out at home where their parents employed private tutors.

In 1869 Ricci-Curbastro entered the University of Rome with the intention of studying mathematics and philosophy. He was only sixteen years old at the time and, although he had not attended school, he was well prepared academically. Political events, however, conspired to make Rome a somewhat unfortunate choice, although a very natural one given his place of birth. When Ricci-Curbastro began his studies in Rome, although the Kingdom of Italy had been created a few years earlier, Rome was not part of that Kingdom being part of the Papal States in which Ricci was born and brought up. Rome had been attacked by Italian troops in 1867 but France had defended the city and employed its troops against the attack. In 1870, however, Italian troops captured Rome and it became the capital of the Kingdom of Italy. Ricci-Curbastro studied at Rome for one year from 1869 to 1870 and then returned to his parents home where he remained for two years before beginning a second university career.

This time he went, not to Rome but to the University of Bologna. He studied there during the years 1872-73, then moved to Pisa where he attended the Scuola Normale Superiore which, under Betti's leadership, was becoming the leading Italian centre for mathematical research and mathematical education. As well as attending lectures by Betti in Pisa, Ricci-Curbastro also attended lectures by Dini. In 1875 Ricci-Curbastro was awarded a doctorate for his thesis On Fuchs's research concerning linear differential equations. He remained at Pisa working on a paper which he presented the following year to fulfil the requirements necessary to teach. The paper was On a generalisation of Riemann's problem concerning hypergeometric functions. Neither this paper, nor his doctoral thesis, have been published.

A perceptive reader will have noticed that both of these first two works by Ricci-Curbastro were based on works by German, rather than by Italian, mathematicians. The next pieces of work which he undertook were, likewise, not based on ideas by Italian mathematicians. The first was a series of articles on Maxwell's theory of electrodynamics and the work of Clausius which Betti asked him to write. Three of these articles appeared in Nuovo Cimento in 1877 and, in the same year, an article appeared in Giornale di matematiche di Battaglini which Dini had asked him to write on Lagrange's problem on a system of linear differential equations.

Ricci-Curbastro now competed for a scholarship and he won one which allowed him to spend the year 1877-78 abroad. That he chose to go to Germany should be no surprise and in fact he chose to study at the Technische Hochschule in Munich where Klein had been appointed to the chair two years earlier. As well as Klein, Brill worked at the Technische Hochschule in Munich and Ricci-Curbastro attended lectures by both these famous mathematicians. As Speziali writes in [1]:-

Ricci greatly admired Klein, and his esteem was soon reciprocated; nevertheless, Ricci does not seem to have been decisively influenced by Klein's teaching. It was, rather, Riemann, Christoffel, and Lipschitz who inspired his future research. Indeed, their influence on him was even greater than that of his Italian teachers.

Returning to Pisa in 1879, Ricci-Curbastro became Dini's assistant. Then, from 1880 until his death in 1925 he was professor of mathematical physics at the University of Padua. He did not only teach mathematical physics, however, for from 1891 he also taught courses on advanced algebra at Padua. It was only after he was appointed to the chair at Padua that he had the security that would allow him to marry and, in 1884, he married Bianca Bianchi Azzarani. They had three children; two sons and a daughter.

Ricci-Curbastro's early work was in mathematical physics, particularly on the laws of electric circuits and differential equations. He changed area somewhat to undertake research in differential geometry and was the inventor of the absolute differential calculus between 1884 and 1894. The initial contributions had been made by Gauss, then the ideas had been developed in Riemann's 1854 Probevorlesung Ⓣ and in an 1861 paper which he wrote for a prize contest of the Paris Académie des Sciences. However, it was a paper of Christoffel, published in Crelle's Journal in 1868, which was the main influence on Ricci-Curbastro to begin his investigations in 1884 on quadratic differential forms. He first systematically presented the important ideas in 1888 in a paper written for the 800th anniversary of the University of Bologna. Speziali writes in [1]:-

The method he used to demonstrate [the invariance of the quadratics] led him to the technique of absolute differential calculus, which he discussed in its entirety in four publications written between 1888 and 1892.

Much of Ricci-Curbastro's work after 1900 was done jointly with his student Levi-Civita. In a fundamental joint paper that year Méthodes de calcul différentiel absolu et leurs applications Ⓣ he used (for the only time) the name Ricci instead of his full name. This paper had been requested five years earlier by Klein. The authors state their aims in the preface to their important seventy-seven page paper:-

The algorithm of absolute differential calculus, the instrument matériel of the methods ... can be found complete in a remark due to Christoffel. But the methods themselves and the advantages they offer have their raison d'être and their source in the intimate relationships that join them to the notion of an $n$-dimensional variety, which we owe to the brilliant minds of Gauss and Riemann. ... Being thus associated in an essential way with $V_{n}$, it is the natural instrument of all those studies that have as their subject, such a variety, or in which one encounters as a characteristic element a positive quadratic form of the differentials of $n$ variables or of their derivatives.

In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations. The main ideas of this paper are discussed in [1]. Ricci-Curbastro's absolute differential calculus became the foundation of tensor analysis and was used by Einstein in his theory of general relativity.

The paper [7], written by Ricci-Curbastro's student Levi-Civita, lists sixty-one of his publications. However, he found time to also contribute to local government as did many of the Italian mathematicians of his time. He served as a councillor for his home town of Lugo and in this capacity was involved in many projects relating to the supply of water and to swamp drainage (an activity which many Italian mathematicians became involved with over several centuries). Later he served as a councillor for Padua and there his interests included school education and finance. Offered the position of mayor of Padua, however, he declined.

Ricci-Curbastro received many honours for his outstanding contributions, although one would have to say that the importance of his work was not fully understood at the time when he produced it, but rather it was realised some time later. He was honoured with membership of several academies such as the Istituto Veneto which he was admitted to in 1892 and which he served as president in 1916-18. He also was a member of the Accademia dei Lincei from 1899, the Accademia di Padua from 1905, the Academy of Sciences of Turin from 1918, the Società dei Quaranta from 1921, the Reale Accademia di Bologna from 1922 and the Accademia Pontifica from 1925.