**Hanno Rund**was born in Schwerin, a town in northern Germany. His parents were Victor Rund and Martha von Neumann. Victor was an academic with doctorates in law, theology, and philosophy. Victor and Martha had two children, the eldest being a daughter Armgard (1921-2014) who was nearly five years older than her brother Hanno, the subject of this biography. The reason that Victor and Martha chose the names Armgard and Hanno for their children was that they were fans of the 1897 novel

*Buddenbrooks*by Thomas Mann about a family from northern Germany and these names were those of two characters in the novel.

On 30 January 1933 Hitler came to power in Germany and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course removing those of Jewish descent from other roles. This made the Rund family decide to emigrate and, later in 1933, they travelled to Eldoret in Kenya. After a short while in Eldoret, the family moved again, this time to South Africa. Victor Rund earned a living as a teacher of German, teaching in schools in Natal and the Cape of Good Hope. Hanno, at the age of ten, attended a primary school in Cape Town.

The Groote Schuur Primary School, in Rondebosch, a southern suburb of Cape Town, was founded in January 1936 and Hanno Rund attended the school from the time it opened in temporary accommodation. For his secondary education he attended the Rondebosch Boys High School, founded February 1897 in Rondebosch, which by the time Rund entered had achieved a high reputation. He graduated from the school in 1942. Although he was interested in science, and in mathematics in particular, he won a King's Entrance Scholarship to the University of Cape Town for his performance in Latin and English. At the University of Cape Town, Rund took courses in Pure Mathematics, Applied Mathematics and Physics and graduated with a B.Sc. in 1946.

Rund was appointed to the University of Cape Town and began teaching there in 1946. In 1948 he moved to the University of Natal in Pietermaritzburg and taught there for one year before returning to the University of Cape Town in 1949. He undertook research for his Ph.D. advised by Christian Yvon Pauc (1911-1981). Pauc was a Frenchman who had studied at the École Normale Supérieure in Paris, in Vienna with Karl Menger, and in Rome. Then he was awarded a doctorate from the Sorbonne for his thesis *Les methodes directes en calcul des variations et en geometrie differentielle* Ⓣ having been advised by Maurice Fréchet. After teaching in Germany and France Pauc was appointed to the University of Cape Town in 1948. Rund submitted his thesis *The Geometry of Finsler Spaces Considered as Generalized Minkowskian Spaces* to the University of Cape Town in August 1950. He dedicated his thesis to "Alexander Brown, Professor of Applied Mathematics, University of Cape Town (1904-1947)." In the thesis Rund writes:-

Laurence Chisholm Young had been appointed as Professor of Mathematics at the University of Cape Town in 1938 where he was the first head of the Department of Mathematics. He was very successful in building the Department at Cape Town during the ten years he spent there.My best thanks are due to Prof L C Young who first introduced me to the Calculus of Variations, and to Dr C Y Pauc for his encouragement and many very valuable suggestions.

Wilhelm Süss was appointed as the external examiner of Rund's thesis. Süss, as well as being professor of mathematics at the University of Freiburg, Germany, was also the director of the Oberwolfach Mathematical Research Institute in the Black Forest in southwest Germany. Rund went to Germany in 1950 spending some time at Oberwolfach before being appointed as a docent at the University of Freiburg in 1951. He sailed from Durban, South Africa, on the *Winchester Castle*, arriving in Southampton, England, on 11 August 1950. From England he travelled to Germany.

Rund began publishing papers, the first to appear being *Über die Parallelverschiebung in Finslerschen Räumen* Ⓣ (1951). In this paper he gives his address as Cape Town. The paper was written in German, as were a number of his early papers, but from 1957 onwards all his papers were written in English. He attended the Colloque de Topologie de Strasbourg in 1951 and gave a lecture entitled *A theory of curvature in Finsler spaces*. The published version is a 12-page paper reviewed by Shiing-shen Chern who writes:-

His next papers wereThis is a summary of a theory of curvatures in Finsler spaces. Most of the results have since been published in detail in other papers[by Hanno Rund]. The main idea is a notion of parallelism different from the classical ones of Berwald and Cartan. It has the advantage of depending only on the point and not on the direction, but the disadvantage that the length of a vector is not preserved. In a sense the resulting curvature measures the difference of a Finsler metric from the tangent Minkowskian metric. The author shows how various notions in the differential geometry of Finsler spaces can be established on the basis of this parallelism.

*Zur Begründung der Differentialgeometrie der Minkowskischen Räume*Ⓣ (1952), with the address Cape Town, and

*Die Hamiltonsche Funktion bei allgemeinen dynamischen Systemen*(1952) which gives Freiburg im Breisgau as his address. This last mentioned paper has the acknowledgement:-

In 1952 Rund habilitated in Freiburg, sponsored by Wilhelm Süss and Gerrit Bol (1906-1989). He was then appointed as a Diätendozent at the University of Bonn. A Diätendozent is a university teacher who has habilitated but is still self-employed as they were when a docent. In July 1954 he was appointed as an Assistant Professor of Applied Mathematics at the University of Toronto in Canada but before taking up the appointment he returned to South Africa. TheI would like to take this opportunity to thank the "Royal Commissioners for the Exhibition of1851" for their generous support.

*Rondebosch Magazine*[6] has an item in the Old Boys' Notes:-

About eighteen months after arriving in Canada, Rund met the girl who would become his wife a few months later [3]:-Hanno Rund, after three years at Bonn University, during part of which time he was senior lecturer, has been appointed professor of mathematics at Toronto University. He spent five weeks with his parents at Rondebosch before proceeding to Canada.

Marian Biddlecomb, daughter of R W Biddlecomb, was English, being born in London on 19 March 1934. She had sailed from Southampton to New York in May 1955 on route to Canada. She gave her occupation as 'hairdresser'. Hanno and Marian were married on 27 April 1956 then, three weeks later, they sailed on theOn Christmas day in1955he met Marian Iris(Frances)Biddlecomb in Toronto, and they were married four months later.

*Homeric*from Quebec, Canada, to Southampton, England, on their way to South Africa. Rund had been appointed Professor and Head of the Department of Mathematics at the University of Natal. On 21 April 1958 Hanno and Marian's daughter Ingrid was born. He served as Dean of the Faculty of Science at Natal in 1958 while he was working on writing his first monograph. In 1959 he published

*The differential geometry of Finsler spaces*. A Kawaguchi writes in a review:-

Rund moved from Natal to the University of South Africa where he was a Research Professor from 1962 to 1966. He published a second major book in 1966, namelyThis is the first well-arranged book furnishing a reasonably comprehensive account of the new field, theory of Finsler spaces, developed by a large number of geometers very rapidly in the last30years. The method of tensor calculus is used, mainly, and it is intended that this book may serve also as an introduction to a branch of differential geometry which is closely related to various topics in theoretical physics, notably analytical dynamics and geometrical optics.

*Hamilton-Jacobi Theory in the Calculus of Variations: Its Role in Mathematics and Physics*. The publisher's description is as follows:-

Leopold Pars writes in the review [7]:-This rigorous self-contained account demonstrates the role of the calculus of variations in unifying some of the most fundamental branches of pure mathematics and theoretical physics. The importance of the Hamilton-Jacobi theory is stressed from the start, and so the pure mathematician gains immediate access to the theory of first-order partial differential equations, to that of some second-order partial differential equations, and to metric geometries. The theoretical physicist is shown how the theory of non-homogeneous single integral problems give rise to relativistic particle mechanics, in which the special invariant Hamiltonian function permits a particularly simple method of quantization, from which the relativistic wave equations(Dirac, Kemmer, etc.)may be obtained directly. A very substantial part is devoted to multiple integral problems, with special reference to modern field theories and areal spaces. The same fundamental methods are used to discuss rigorously the problem of Lagrange, and they therefore lead to an analysis of fundamental dynamical laws for both holonomic and non-holonomic systems. The fundamental pure mathematics is derived from Carathéodory's approach to the calculus of variations, of which no account exists in English, although his methods may be said to have revolutionized the most basic aspects of the subjects. Many of the methods and some of the results now presented are original, and their is little overlap with existing literature, particularly on multiple integral theories.

In 1966 Rund became Professor and Head of the Department of Mathematics at the University of the Witwatersrand in Johannesburg. He continued in this role until 1969 when he moved to Canada. After the year 1970-71 as Professor and Chairman of the Department of Applied Mathematics at the University of Waterloo in Ontario, Canada, he moved to Tucson, Arizona in the United States where he would remain for the rest of his career living at 3150 N Bear Canyon Road. Appointed to the University of Arizona in 1971 he was Chairman of the Department of Mathematics from that time until 1978 when he became a Professor in the Committee on Applied Mathematics. Richard Pierce writes that Rund [8]:-This book is concerned with problems in the Calculus of Variations, with particular emphasis on the use of Hamilton's equations and of the Hamilton-Jacobi theorem. The approach is not that of the familiar derivation of Hamilton's equations from the Euler-Lagrange equations, but is based on Carathéodory's notion of the "complete figure". ... The design is highly ambitious, and the author is to be congratulated on a courageous attempt to present a comprehensive account, including much recent work, of these difficult topics.

In 1975 Rund, together with David Lovelock, published the book... energetically pursued the development program ... adding more than a dozen new faculty members. One of the important occurrences in the history of mathematics at the University of Arizona was the introduction around1977of an interdisciplinary group in Applied Mathematics. Mathematics is used and taught in many of the university programs. The faculty members who view themselves as mathematicians are by no means confined to the Mathematics Department. The formal recognition of a common bond of interest that links a large group of people on campus came about with the formation of the Committee on Applied Mathematics. This committee consists of several dozen faculty members from various areas, including the Mathematics Department of course. Some of the most interesting and exciting mathematical developments in Arizona during the last eight years have occurred under the stimulus of the Applied Mathematics Committee.

*Tensor, differential forms, and variational principles*. The authors write in the Preface:-

Abraham Haskel Taub, in the review [9], describes the book as "well-written" and its "purpose is well accomplished."The objective of this book is two-fold. Firstly, it is our aim to present a self-contained, reasonably modern account of tensor analysis and the calculus of exterior differential forms, adapted to the needs of physicists, engineers, and applied mathematicians in general. Secondly, however, it is anticipated that a substantial part of the material included in the later chapters is of interest also to those who have some previous knowledge of tensors and differential forms: we refer in particular to the remarkable interaction between the concept of invariance and the calculus of variations, which has profound implications in almost all physical field theories.

Let us say a little about Rund's co-author David Lovelock since his career is closely connected with Rund's. Lovelock was born in 1938 in Bromley, Kent, England. His family emigrated to Durban, South Africa, in 1947 and he studied for his B.Sc. at the University of Natal from 1956 to 1959. Here he was taught by Rund and, after graduating, he studied for a Ph.D. advised by Rund. He was a professor of Applied Mathematics at the University of Waterloo, Canada, during the time that Rund was there, then moved to the University of Arizona in 1974 where again he became Rund's colleague. Both remained at the University of Arizona for the rest of their careers.

We note that although Rund held his position at the University of Arizona for the rest of his career, he did spend the year 1983-84 back in South Africa at the University of Cape Town. We quote now from [3]:-

This last statement is significant since Rund had a major impact on African mathematics particularly through the many Ph.D. students he advised while at various South African universities. For example, James Ray Vanstone (1959), John Wainwright (1965), Johannes Christiaan Du Plessis (1965), Duncan H Martin (1965), John Henry Swart (1969), Stanley Paul Lipshitz (1970), and Sarel Venter (1975) were all students of Rund and are listed inThroughout his academic career Hanno Rund lectured widely and often throughout the United States, Canada, Western Europe , and South Africa, on topics ranging from differential geometry and Lie theory to the calculus of variations to theoretical physics and gauge field theory. At age34he was the youngest person ever to be awarded the Havenga Prize by the South African Academy of Arts and Sciences, and in1983he was the first recipient of the South African Mathematical Society Award. He was awarded honorary doctorates by the Universities of Natal(1982)and Waterloo(1984). He served on the editorial boards of several journals, including 'Algebras, Groups and Geometries', 'Aequationes Mathematicae', 'Journal of Mathematical and Physical Sciences', 'Quaestiones Mathematicae', 'Resultate der Mathematik', 'Tensor', and 'Utilitas Mathematica'. He was an unusually conscientious contributor to reviewing journals such as 'Mathematical Reviews' and 'Zentralblatt fur Mathematik'; he once estimated that he had reviewed more than2500articles. Ever very popular as a teacher and adviser, Rund continually supervised Ph.D. candidates throughout his teaching career.

**African men with a doctorate in mathematics 1**at THIS LINK.

In addition to Rund's books that we mentioned above, we also note that he published *Invariant theory of variational problems on subspaces of a Riemannian manifold *(1971), (with his Ph.D. student Jonathan H Beare) *Variational properties of direction-dependent metric fields *(1972), and *Generalized connections and gauge fields on fibre bundles* (1981).

In addition to the Havenga Prize mentioned above, he was honoured with the award of a D.Sc. (honoris causa) from the University of Natal in 1982 and a Dr Math (honoris causa) from the University of Waterloo in 1984. He received the South African Mathematical Society Award in 1983.

Rund was a keen mountaineer and was a member of the Mountain Club of South Africa. His other hobbies include listening to classical music.

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