# Johannes Haantjes

### Born: 18 September 1909 in Itens, Friesland, Netherlands

Died: 8 February 1956 in Leiden, Netherlands

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**Johannes Haantjes**was the second son of Hantje Johans Haantjes (1880-1939) and Reinskje Spijksma (1883-1943). Hantje and Reinskje married in 1907. Johannes had one older brother Johan Haantjes (1908-1978) who studied physics and became director of the Philips Physics Laboratory in Eindhoven, and one younger brother Jetze Haantjes (1914-1975). He also had four younger sisters Pietje Haantjes (1911-2002), Janneke Haantjes (1913-1994), Hinke Haantjes (1916-1992), and Grietje Haantjes (1922-2014). Hantje Haantjes was a teacher and, at the time his son Johannes was born, he was the headmaster of the school in the village of Itens. Reinskje had also been a teacher before she married. Johannes attended his father's school in Itens for his primary education, then moved to the Hogere Burgerschool in Leeuwarden for his secondary education. Itens is only a small village and it had no secondary school. Leeuwarden, about 14 km northeast of Itens, is the closest large town. When Johannes was in the middle of his secondary education, his father was appointed as headmaster of the school in Valkenburg (South Holland) so he completed his schooling in Leiden, a major city a few kilometres to the east of Valkenburg (which is now part of Katwijk).

After graduating from the High School in Leiden in 1927, Haantjes entered the University of Leiden where his chosen major subjects were mathematic and physics. His lecturers included Jan Cornelis Kluyver (1860-1932) and Willem van der Woude (1876-1974). Van der Woude had studied for his doctorate at Groningen advised by Pieter Hendrik Schoute and, from 1916, was professor of mathematics and mechanics at the University of Leiden. Haantjes was also taught by Jan Arnoldus Schouten who was the professor of mathematics at Delft University but taught a course on differential geometry at Leiden as a guest lecturer. Another of Haantjes' lecturers was Johannes Droste (1886-1963). Droste had received a Ph.D. in 1916 for his thesis *The gravitational field of one or more bodies according to the theory of Einstein* (Dutch) which he worked on at Leiden advised by H A Lorentz. After teaching at the Gymnasium in Gorkum, he was appointed as Paul Ehrenfest's assistant at Leiden before being promoted to professor in 1930.

Haantjes continued his studies at the University of Leiden, undertaking research for his doctorate advised by van der Woude. Although Jan Schouten was not on the faculty of the University of Leiden, he did advise Haantjes in an unofficial capacity. On 19 May 1933 Haantjes was awarded a Ph.D. for his 102-page thesis *Het beweeglijk assenstelsel in de affiene ruimte* Ⓣ. In [1] van der Woude gives the background to the thesis:-

A few months after Haantjes' thesis was accepted, his first paper, a joint publication with his thesis advisor van der Woude appeared. This paperLet me first remind you that at the beginning of this century[20th century], certainly also in the Netherlands, there was interest in multi-dimensional and non-Euclidean geometry. Yet differential magnetism was still predominantly three-dimensional. It had found a certain conclusion through rather extensive textbooks such as the "Lezioni" by Luigi Bianchi, but especially through the encyclopaedic four-part "Théorie des Surfaces" by Gaston Darboux. But then comes an almost stormy interest in many more general forms of differential magnetism. Why the sudden reversal? Undoubtedly under the influence of the then physical theories. Think, for example, Einstein; he needs a space that is neither three-dimensional nor Euclidean. Thus suddenly multi-dimensional, flat or curved, affine, projective, conformal geometries arise.

*Über das bewegende Achsensystem im affinen Raum: Ableitung der Grundformeln der Flächentheorie*Ⓣ was written in German and was published in the

*Proceeding*of the Koninklijke Nederlandse Akademie van Wetenschappen. The authors begin their introduction as follows:-

By the time this paper was published, Haantjes was already working as an assistant to Jan Schouten at Delft University. Willem van der Woude writes [1]:-It is well known that the use of a moving axis system for the handling of many questions from differential geometry makes a considerable simplification. In the present work an axis system is considered whose movement depends on two parameters. By allowing the origin of this system to describe an area in three-dimensional(affine)space, the basic formulas(relations between the two fundamental-value tensors)are derived simultaneously with the formulas, which are of great importance for the applications ...

Schouten and Haantjes published two joint papers in 1934, namelyIt did not take him much time to familiarize himself with the new calculation method and to immerse himself completely in the geometric world of thought of his boss. He was precise, critical and - a characteristic that repeatedly emerges - very keen on completing an investigation once it had begun.

*Generelle Feldtheorie. VIII. Autogeodätische Linien und Weltlinien*Ⓣ and

*Über die konforminvariante Gestalt der Maxwellschen Gleichungen und der elektromagnetischen Impuls-Energiegleichungen. (Konforme Feldtheorie I)*Ⓣ. After a year at Delft, in 1934, he was appointed as a docent at the University of Leiden but also continued to hold the assistantship at Delft until 1938. At Leiden he taught geometry and mechanics to students of mathematics and theoretical physics. He interrupted his work in Delft and Leiden with a short stay at the University of Edinburgh, Scotland, where he gave differential geometry lectures.

In 1936 Haantjes and Jan Schouten were co-authors of the article *On the theory of the geometric object* which was published in the *Proceedings* of the Edinburgh Mathematical Society. The authors begin the article by reviewing the history of the idea of a "geometric object."

We give a version of the start of the Introduction at THIS LINK.

In 1938 he became professor of mathematics, mechanics and the theory of relativity at the Vrije Universiteit Amsterdam (Free University of Amsterdam).

On 4 April 1939 Haantjes married Marie Antoinette Mulder in Leiden. Marie was the daughter of Jean Pierre Mulder (1879-1959), a book printer, and Anna Marie van der Most van Spijk (1880-1960). At the time of the marriage, Haantjes was 29 years old and his wife Marie was 24. They had four children, the eldest being Hanneke Haantjes (1940-2007). Marie Antoinette Haantjes died in 2001.

Following World War II, in 1945 he was made a full professor of mathematics and mechanics at the Vrije Universiteit Amsterdam. He delivered his inaugural lecture *De zekerheid der meetkunde* Ⓣ on 4 October 1945 (it was published in 1946). For a version of this speech, see THIS LINK.

He remained in Amsterdam until 1948 when he was appointed as full professor of geometry at the University of Leiden. This position had become vacant since his former Ph.D. supervisor van der Woude had retired. Haantjes gave his inaugural lecture *Over enige grondbegrippen uit de meetkunde* Ⓣ in Leiden on 28 May 1948.

Haantjes published over 50 papers in his short career and we will now look at the titles and give a short extract from a review, of a few of these. We give English versions of the titles of the Dutch papers.

**Conformal differential geometry. Curves in conformal euclidean spaces (1941)**: The object of this paper is to obtain intrinsic equations for a curve in Euclidean n-space which determine it up to conformal point transformations. The method is that of proportional Euclidean metrics rather than polyspherical coordinates and therefore ordinary tensor analysis is used.

**(with C Smits) The differential geometry of Möbius in the plane (1943)**: In this paper the differential geometry of plane curves is developed with respect to the group of conformal transformations, which transform circles into circles (the so-called Möbius-group), by studying the group of linear fractional transformations in a complex variable*z*=*x*+*iy*.

**Conformal differential geometry. V. Special surfaces (1943)**: This paper uses the methods developed by the author in four previous papers. Some special surfaces in three-space are studied. Among the results are the following three theorems. (1) If a system of lines of curvature of a surface consists of Darboux curves, then it consists of circles. (2) A necessary and sufficient condition that a line of curvature is spherical is that the angle between the conformal osculating plane and the tangent plane is constant along the curve. (3) If one system of lines of curvature consists of conformal geodesics the other system consists of circles. There are also conditions that the normal circles of a surface all pass through a fixed point and studies of surfaces of which the normal circles form a normal congruence.

**The certainty of geometry (1946)**: Lecture at the Free University of Amsterdam.

**Symmetrization in the hyperbolic plane (1947)**: According to Steiner, a closed curve is "symmetrized" by the process of shifting every section perpendicular to a given line till its midpoint lies on the line. This process diminishes the circumference*L*of any curve not symmetrical about the given line, while leaving the area*O*unchanged. Hence any closed curve other than a circle yields a shorter closed curve for some direction of the line. The author considers, in the hyperbolic plane, a closed curve whose geodesic curvature is everywhere greater than 1 and shows that this property is retained after symmetrization.

**Equilateral point-sets in elliptic two- and three-dimensional spaces (1948)**: A point set in a metric space is called equilateral if its pairs of different points have the same distance. A method is given for deriving all equilateral n-tuples in elliptic space. In particular, in the elliptic plane there are equilateral sextuples, but no septuples. In three-dimensional space there are six kinds of (noncongruent) equilateral quintuples, two kinds of sextuples and no septuples.

**Ptolemy's theorem (1952)**: If*p*_{1},*p*_{2},*p*_{3},*p*_{4}are any four points of euclidean three-space, the (extended) theorem of Ptolemy asserts that the three products

*p*_{1}*p*_{2}×*p*_{3}*p*_{4},*p*_{1}*p*_{3}×*p*_{2}*p*_{4},*p*_{1}*p*_{4}×*p*_{2}*p*_{3}

of opposite distances satisfy the triangle inequality, and the four points are concyclic if and only if the sum of two of these products equals the third. This note contains a purely metric proof of the theorem ...

*Inleiding tot de Differentiaalmeetkunde*Ⓣ (1954). Dirk Struik writes in a review:-

Today Haantjes is remembered by those working in metric geometry for the Haantjes-Finsler curve. It is one of the many extensions of the standard concept of curvature, including applications to wavelets. The Haantjes-Finsler curve is a metric notion of curvature suggested by Paul Finsler and developed by Haantjes in his paperThis attractive book is an introduction to the classical differential geometry of curves and surfaces in euclidean three-dimensional space with the aid of the vector and the tensor calculus. In its seven chapters we find not only a clear and comprehensive exposition of such subjects as the Frenet formulas, the Gauss-Codazzi equations and the Gauss-Bonnet theorem, but also a short but adequate treatment of the elements of the analysis of vectors and tensors. These methods are introduced only at the place where they are needed for the geometrical demonstrations. Some special features are a sketch of Bonnet's existence theorem for surfaces with given first and second fundamental tensors and a derivation of the Gauss-Bonnet theorem based on the consideration of triangles of which two sides are geodesic. There are more than a hundred problems, with indications for their solution at the end, and thirty-five good illustrations.

*Distance geometry. Curvature in abstract metric spaces*(1947).

Willem van der Woude writes that at the University of Leiden, Haantjes [1]:-

Haantjes was a dedicated Christian who had many social interests. When in Leiden he was a student elder in the Reformed Church. After he moved to Amsterdam he held the office of elder of the Reformed Church. He was the secretary of the "Free Courses" of the Vrije Universiteit Amsterdam until 1953. He was also much involved with school mathematics examinations. He often acted as an examiner for the "Lager Onderwijs", the primary certificate, and for the "Middelbaar Onderwijs", the Hogere Burgerschool examination. He regularly attended international conferences, for example the Colloque de Géométrie Différentielle, in Louvain in 1951, publishing... was an excellent teacher ... He was greatly appreciated in the Faculty of Mathematics and Physics and in the Senate, also in the positions in which he performed outside of a university context. No wonder, he was kind and kind-hearted, knew what he wanted, but easily adapted to others. He had ecclesiastical and social interest, was a student elder of the Reformed Church and a member of the supervisory committee for secondary education in Leiden. He was known as a geometric expert of very general knowledge and as one of the best experts in differential geometry at all levels.

*Sur la géométrie infinitésimale des espaces métriques*Ⓣ in the

*Proceedings*, and the Convegno Internazionale di Geometria Differenziale, Italia, 1953, publishing

*On the notion of geometric object*in the

*Proceedings*.

Among the honours he received, we mention his election to the Royal Netherlands Academy of Arts and Sciences (Koninklijke Nederlandse Akademie van Wetenschappen) in 1952. He was also a member of the Dutch Mathematical Society (the Koninklijke Wiskundig Genootschap) and served as its treasurer in the years 1939-1955. In October 1950 the Board of the Koninklijke Wiskundig Genootschap appointed a small committee with the task of drawing up a report on the structure and regulations for the International Congress of Mathematicians to be held in Amsterdam in 1954. H D Kloosterman was chair of this committee with Haantjes as its secretary. In 1955 he was elected as president of the Dutch Mathematical Society but due to ill health he was soon forced to resign.

In 1955 he received a request from the Clarendon Press to undertake the writing of a treatise on the foundations of differential geometry for publication in their Oxford Tracts series. This shows the strong international reputation that Haantjes had acquired. Sadly illness prevented him from writing this book. Around the middle of 1955 he was rather tired when the summer vacation began but after that his health began to rapidly deteriorate. His last papers*On*

*X*

_{m}

*-forming sets of eigenvectors*and (with R Nottrot)

*Distance geometry. Directions in metric spaces. Torsion*were published in 1955. Despite receiving the best medical help and the most loving nursing he died in February 1956 and was buried on 11 February at the Rhijnhof Cemetery in Oegstgeest.

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

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