Erich Kahler (1906-2000)
Erich Kähler
Mathematische Werke Mathematical Works Edited by Rolf Berndt and Oswald Riemenschneider
W DE G Walter de Gruyter · Berlin · New York
Editors Rolf Berndt Mathematisches Seminar Universität Hamburg Bundesstraße 55 20146 Hamburg Germany
[email protected] Oswald Riemenschneider Mathematisches Seminar Universität Hamburg Bundesstraße 55 20146 Hamburg Germany
[email protected] © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data Kähler, Erich, b. 1906 Mathematische Werke = Mathematical works / Erich Kähler; edited by Rolf Berndt and Oswald Riemenschneider, p. cm. English and German. Includes bibliographical references. ISBN 3-11-017118-X (alk. paper) 1. Mathematics. I. Title: Mathematical works. II. Berndt, Rolf. III. Riemenschneider, Ο. (Oswald). IV. Title. QA3. K34 2003 510-dc22 2003062669
ISBN 3-11-017118-X Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
© Copyright 2003 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Werner Hildebrand, Berlin Binding: Lüderitz & Bauer GmbH, Berlin
Preface
For most mathematicians and many mathematical physicists, Erich Kähler's name is strongly tied to important geometric notions such as Kähler metrics, Kähler manifolds and Kähler groups. These ideas go back to a paper of 14 pages that appeared in 1932. However, this is just a small part of Kähler's many outstanding achievements, which only specialists may be acquainted with. We hope that this volume will disseminate his ideas to a broader audience. After a short description of Kähler's life we continue with a summary of the mathematical part of his scientific work that consists of several papers of moderate length that are somewhat inaccessible, two small books (on systems of partial differential equations and on algebra and infinitesimal arithmetic), his opus magnum Geometria Aritmetica (a huge article of 399 pages which fills a complete volume of the Annali di Matematica), and some carefully worked out texts of various university courses. A bibliography can be found at the end of this book. For lack of space, we reproduce in this volume only the short introduction to the Geometria Aritmetica. Also, we only report on the important book on differential equations in a Survey of Kühlers Mathematical Work and Some Comments (pages 11-40 in this volume). The ideas laid down in Kähler's other book Algebra und Differentialrechnung (pages 282-387 in this volume) and in the long Italian article are summed up in two essays (on Kähler differentials and Kähler's Zeta function), which are part of a special section in our volume that assembles commentaries on the main topics of Kähler's mathematical research and describes some recent developments in these fields. Kähler's interests went far beyond the realm of mathematics and mathematical physics. He wrote several essays that intertwined standard notions from mathematics with central notions in philosophy, biology and even theology, and gave courses and lectures on these topics. Whether this part of Kähler's work merits deeper investigation may be left to the judgement and taste of the reader and a future evaluation. But, any picture of Kähler's personality would be incomplete without touching on this, perhaps controversial, facet of his work. In order to demonstrate how universal and far reaching his interests were, we reproduce three of those articles in an appendix. It is our agreeable duty to thank several persons who helped us to put this volume together. First of all, our thanks go to Erich Kähler's wife, Charlotte Kähler, and his son, Helmuth Kähler, who supported this project from the bottom of their hearts. We would also like to thank those who participated by submitting articles which partly were presented on the occasion of a colloquium, In memoriam Erich Kähler, held in Hamburg in January 2001: A. Böhm, J. P. Bourguignon, J.-B. Bost, S.S. Chern, A. Deitmar, I. Ekeland, A. Krieg, Ε. Kunz, Κ. Maurin, W. Neumann, and
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Preface
Η. Nicolai. We are indebted to W. Müller-Schauenburg and M. Gaudefroy-Bergmann for the translation of Maurin's text and to those who assisted us by giving mathematical and editorial advice. Here we would like to name first of all P. Slodowy who - despite his fatal illness - even helped with linguistic problems, and again J. P. Bourguignon, U. Jannsen, E. Kunz, Η. J. Nastold, H. Schumann and D. Zagier, and finally H. Knorr and our colleagues from the Mathematisches Seminar, in particular, H. Brückner, J. Michaliöek, G. Mülich and U. Semmelmann. Last but not least, our thanks are due to Mrs. E. Dänhardt who typed most of our editorial texts and helped in every possible way with the organization. Finally we thank Dr. Manfred Karbe from the Verlag Walter de Gruyter for his patience and everlasting encouragement and his team for professional support. Hamburg, September 2003 Oswald
Rolf Berndt Riemenschneider
Contents
Preface
ν
A Tribute to Herrn Erich Kahler by S. S. Chern
1
Life of Erich Kahler by R. Berndt and A. Böhm
3
Survey of Kahler's Mathematical Work and Some Comments by R. Berndt and O. Riemenschneider
11
Erich Kahler's Mathematical Articles Transformation der Differentialgleichungen des Dreikörperproblems [1]
43
Die Reduktion des Dreikörperproblems in geometrischer Form dargestellt [2] . . . 59 Über ein geometrisches Kennzeichen der analytischen Abbildungen im Gebiete zweier Veränderlichen [3]
64
Über die Existenz von Gleichgewichtsfiguren rotierender Flüssigkeiten, die sich aus gewissen Lösungen des n-Körperproblems ableiten (Dissertation) [4]
69
Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle [5]
87
Zur Theorie der algebraischen Funktionen zweier Veränderlichen. I [6]
104
Über den topologi^chen Sinn der Periodenrelationen bei vierfach-periodischen Funktionen [7]
116
Über die Integrale algebraischer Differentialgleichungen (Habilitationsschrift) [8]
123
Zur Invariantentheorie von Differentialoperatoren [9]
154
Sui periodi degli integrali multipli sopra una varietä algebrica (with an appendix by F. Severi: Osservazioni a proposito della nota di Erich Kähler: "Sui periodi degl'integrali multipli sopra una varietä algebrica,,) [10]
162
Forme differenziali e funzioni algebriche [11]
175
Über eine bemerkenswerte Hermitesche Metrik [12]
190
Bemerkungen über die Maxwellschen Gleichungen [14]
204
Über eine Verallgemeinerung der Theorie der Pfaffschen Systeme [15]
232
Über rein algebraische Körper [18]
236
Sur la theorie des corps purement algebriques [19]
260
Zahlentheorie und Physik [21]
274
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Algebra und Differentialrechnung [22]
282
Osservazioni a proposito della dinamica [23]
388
a
Tensori razionali di \ specie sopra una varietä algebrica [25]
402
Über die Beziehungen der Mathematik zu Astronomie und Physik [27]
406
Geometria aritmetica, Introduzione [28]
419
Innerer und äußerer Differentialkalkül [29]
421
Die Dirac-Gleichung [30]
449
Der innere Differentialkalkül [31]
483
Der innere Differentialkalkül [33]
497
Infinitesimal-Arithmetik [34]
596
Die Poincare-Gruppe [42]
621
The Poincare group [43]
653
Raum-Zeit-Individuum [46]
661
Comments to the Mathematical Work of Erich Kahler Topology of Hypersurface Singularities by W. D. Neumann
727
The Unabated Vitality of Kählerian Geometry by J.-P. Bourguignon
737
Some Applications of the Cartan-Kähler Theorem to Economic Theory by I. Ekeland
767
Kähler Differentials and Some Applications in Arithmetic Geometry by R. Berndt
777
Why 'Kähler' Differentials? by E. Kunz
848
A Neglected Aspect of Kähler's Work on Arithmetic Geometry: Birational Invariants of Algebraic Varieties over Number Fields by J.-B. Bost
854
Kähler's Zeta Function by R. Berndt
870
Panorama of Zeta Functions by A. Deitmar
880
Eisenstein Series on Kähler's Poincare Group by A. Krieg
891
Supersymmetry, Kähler Geometry and Beyond by H. Nicolai
907
Appendix Selecta of Erich Kähler's Philosophical Articles Wesen und Erscheinung als mathematische Prinzipien der Philosophie [36]
919
Contents
ix
U regno delle idee [38]
932
Saggio di una dinamica della vita [39]
939
Comments to the Philosophical Work of Erich Kahler Erich Kähler's Vision of Mathematics as a Universal Language by R. Berndt...
955
An Approach to the Philosophy of Erich Kahler by K. Maurin
960
Addresses of the Authors
965
Acknowledgements
967
Bibliography
969
A Tribute to Herrn Erich Kähler Shiing-Shen Chern
In 1931 Professor Wilhelm Blaschke visited China. He gave a series of lectures on the Topologische Fragen der Differentialgeometrie (nowadays known as web geometry) in Peiping. I was a graduate student at Tsing Hua University and attended these lectures. I was fascinated. So, in the summer of 1934, when I finished my graduate studies and was awarded a fellowship to study abroad, I requested permission to go to Hamburg instead of going to an American university, as usually the case. Fortunately this was approved. I arrived at Hamburg in the summer of 1934. The University began in November and I attended, among other classes, Herrn Kähler's seminar on exterior differential systems. He had just published his booklet entitled Einführung in die Theorie der Systeme von Differentialgleichungen, which gives a treatment of the theory developed by Professor Elie Cartan. I devoted much time to the subject and in 1935 wrote a thesis on the applications of exterior differential systems to web geometry. I received my Doktor der Wissenschaften in February 1936 and the thesis was published in the Hamburger Abhandlungen in the same year. Clearly it received much advice from Herrn Kähler, from whom I learned the subject of exterior differential calculus and what is now known as the Cartan-Kähler theory. We had frequent lunches together at the restaurant Curio Haus near the Seminar. He told me many things, mathematical or otherwise. My gratitude to him cannot be overstated. At that time Herr Kähler had just published his article Über eine bemerkenswerte Hermitesche Metrik in the Hamburger Abhandlungen which is the starting-point of Kähler geometry. In this paper he showed a clear understanding of the complex structure and proved in particular that a Kähler metric (later name) has a potential relative to the operator 33. The paper affects me in my concept of the complex structure, which later becomes an important part in my work on the Gauss-Bonnet formula and characteristic classes. The exterior differential calculus is a fundamental tool in manifolds, making the infinitesimal calculus available in high-dimensional geometry. After its discovery by Elie Cartan and the topological work of George de Rham, Herr Kähler was the one who clearly saw its importance and made use of it. Herr Kähler also did important work on algebraic geometry. Unfortunately it overlapped with the great development in algebraic geometry in the sixties and had not been sufficiently recognized.
1
Life of Erich Kähler RolfBerndt and Arno Böhm
On May 31,2000, Erich Kähler died at the age of 94 in his home in Wedel. Until 1974 he was director of the Institute for Pure Mathematics of the University of Hamburg (Mathematisches Seminar) and professor of the Technical University in Berlin. Since 1976 he was an honorary member of the Hamburg Mathematical Society (Mathematische Gesellschaft in Hamburg), and since 1949 member of the Academy of Sciences of Saxony (Sächsische Akademie der Wissenschaften), since 1955 a member of the Academy of Sciences in Berlin (Berliner Akademie der Wissenschaften), since 1957, a member of the German Academy of Scientists Leopoldina (Deutsche Akademie der Naturforscher Leopoldina) and the Italian Accademia Nazionale dei Lincei and since 1992, a member of the Accademia di Scienze e Lettere Milano. His contributions to mathematical physics, as well as algebraic and arithmetic geometry assure him a permanent place in the history of mathematics. Mathematics was for him allencompassing world-view. The unity of mathematics and its progress was his foremost concern. His passion to reach beyond the traditional boundaries of mathematics can be understood from his biography. He had a happy childhood and adolescence, but his life was later influenced by unusually tragic events. Erich Kähler was born on January 16, 1906 in Leipzig. He attended school there throughout his high school and college years. His father, who had been an oboist in the Navy for 12 years, worked as a telegraph inspector. From Kähler's autobiographical essays, which he dictated to his wife in 1998, one learns that during his youth he showed interest in geography and ethnology. Inspired by the books of Sven Hedin, he wanted to become a world explorer and visit those parts of the earth that he had read about in the books provided by his otherwise rather frugal mother. From seafaring his interests expanded to astronomy and finally, at the age of 12, he became interested in mathematics. His teacher, Dr. Wiese, suggested that he pursue mathematics. In the final year of high school, he was excused from the mathematics classes and his principal Mr. Donat, a student of Weierstraß, gave him his university lecture notes of Weierstraß' courses. In this way Kähler became familiar with the works of Gauß and the theory of elliptic and abelian functions at an early age. When he was 17 he had the idea to study fractional differentiation. He wrote a 50 pages treatise and submitted it to Otto Holder, hoping that he could receive a PhD for this work. But he and his father were told that in order to receive a PhD he needed at least six semesters of classes,
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Rolf Bemdt and Arno Böhm
and a little later he himself noticed that the subject of his treatise had already been studied by Liouville though in a different way. The 17 year old Erich Kahler then started a six semester study of mathematics as protege of Lichtenstein. In the first semester he took Galois theory and read the papers of Gauß, Abel, Weierstraß, Riemann, Lagrange and he soon became acquainted with Emil Artin. In 1928 he received his PhD under Lichtenstein with the dissertation entitled "Über die Existenz von Gleichgewichtsfiguren, die sich aus gewissen Lösungen des η-Körperproblems ableiten" ("The existence of equilibrium figures that are derived from certain solutions of the η-body problem"), a topic which he had chosen himself. Subsequent to his graduation, Kahler received a fellowship of the "Notgemeinschaft der Deutschen Wissenschaften" which enabled him to continue his work in mathematics. He chose to study the quadruple periodic functions. In the summer of 1929, on the way to a vacation in Laboe near Kiel, he had a short meeting with Artin in Hamburg, which set the direction for his future. Erich Kähler became a scientific assistant to Blaschke after he had spent the summer of 1929 at an assistant position at the University in Königsberg. In 1930 he completed his habilitation with work on the integrals of algebraic differential equations. The years 1931-1932 he spent in Rome as a Rockefeller Fellow. He studied with the representatives of the Italian school of algebraic geometry, in particular with Enriques, Castelnuovo, Levi-Civita, Severi and B. Segre. In Rome he also met Andre Weil with whom he had a lifelong friendship and scientific exchange. Already in 1931, Blaschke had recommended Erich Kähler for a professorship in mathematics at the University in Rostock, but Kähler preferred to stay in Hamburg to participate in the inspiring atmosphere at the Mathematical Institute. One result of his stay in Hamburg was the paper "Über eine bemerkenswerte Hermitesche Metrik" ("On a remarkable Hermitian metric"). This metric was later called "Kähler metric". It led to the Kähler manifolds which is the foundation of Kähler geometry and plays an important role in string theory which dominates present day theoretical physics. Α Kähler manifold whose Ricci tensor vanishes is called a Calabi-Yau manifold. A Calabi-Yau manifold has a one-dimensional canonical bundle. This fact plays a very important role in string theory compactifications and such Kähler manifolds lead to compactifications with space-time supersymmetry. In Hamburg Kähler also met Chern. This meeting led to a lifelong relationship of mutual respect and admiration. In 1934 Blaschke took Kähler along to Moscow to give lectures on his results on systems of differential equations. From the lecture notes emerged the book [13]*. This journey to Moscow also brought him in contact with Elie Cartan. There he also met Alexandrov. In 1935 Kähler went to Königsberg where he became full professor in the following year. During his time in Königsberg, he did research in mathematical physics including his new formulation of Maxwell's equations using differential forms. To be noted is his lecture "Über die Beziehungen der Mathematik zu Astronomie und Physik" on the *References containing only numbers refer to Erich Kähler's Bibliography at the end of this volume.
4
Erich Kahler around 1934
Life of Erich Kahler
7
relations between mathematics and astronomy and physics which was delivered during the Kant-Copernicus week in 1939 at the Königsberg University. The quotation at the beginning of this essay is from these lectures and can be seen as the formulation of the general program of Erich Kahler. In 1938 Erich Kähler married the physician Luise Günther, and in 1939 their first son Helmuth was born. Then World War II interrupted his scientific work. Kähler had never been a member of the Nazi party. But throughout his life he was enamored by Nietzsche's teaching, which curiously can be traced back to a romantic relationship during the summer vacation of 1928. Attracted since childhood to the sea and influenced by Nietzsche's ideas he had earlier volunteered to serve in the Navy. Now, two weeks after the birth of his son, he was called to active duty. In his memoirs Kähler wrote that the military service freed him from having to participate in Nazi activities. He had too many Jewish friends to be able to voice his support for any activities that expressed antisemitism. At the end of the war Kähler was captured at the fortress St. Nazaire in Brittany from where he was sent to the French prisoner of war camp on the lie de Re and later to the one in Mulsanne near Le Mans. He describes this time at the prisoner of war camp as a paradise for scientific endeavors. Since he had been an officer he was not assigned to manual labor. Because of his connections with French mathematicians in the years before the war he was provided with an exemption signed by Joliot-Curie, director of the French CNRS, that allowed him to purchase books. Elie Cartan, Chern and many others sent him preprints and mathematical literature in response to his requests. Andre Weil who left for Chicago in 1947 tried to bring him to Sao Paulo. But after release from the prisoner of war camp in 1947, Kähler went as a lecturer to Hamburg and then, in 1948, he became the successor of Koebe in Leipzig. After he left the prisoner of war camp, he again lived with his family. In 1942 his daughter Gisela was born and, at the end of the war, his wife succeeded to catch one of the last ships that left East Prussia and found refuge in Seedorf in Schleswig-Holstein. In 1948 his son Reinhard was born. He was handicapped and thus the more beloved by parents and siblings. About the years in Leipzig there exists a fine report from Horst Schumann [Schu]^. In retrospect the time in Leipzig was the most fruitful scientific period of Erich Kähler's life. His influence in Leipzig is still felt today and is documented by the support of his work by Zeidler, director of the Max Planck Institute in Leipzig (who as student had taken classes from Kähler). In Leipzig Kähler gave a 5 semester course with sometimes more than 10 hours of classes per week, in which he expostulated on algebra, algebraic geometry, function theory and arithmetic. The results presented in this course were then published in 1958 in a 399 pages volume of Annali di Matematica under the title "Geometria Aritmetica" [28]. Kähler was called "Master" by a circle of students which included H. Schumann, Lustig, Mehner, Eisenreich, Häuslein, Grosche and Uhlmann. Together with his students he wrote several drafts and the contributions of these students can be seen throughout the "Geometria Aritmetica". Further ^See the reference in Survey ofKähler's Mathematical Work and Some Comments, p. 40 in this volume.
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travels to Moscow and his academy membership gave Erich Kähler an independent position during the period of increased political tension in East Germany. However, as a consequence of his support for the release of the imprisoned Student Chaplain Georg-Siegfried Schmutzler, the tension between Erich Kähler and the East German system became so overwhelming that he decided to leave East Germany. Fortunately he got an offer of the Technical University in West Berlin which he accepted. At the Technical University he was heralded as the greatest living mathematician and his lectures overflowed with 600 students majoring in engineering and sciences. In addition he gave advanced special topics classes, among them a class on the exterior and interior differential calculus in which also the second of the two authors of this essay participated. Kähler also discussed the application of the interior differential calculus to the Dirac equation of the electron in this class. He also gave a course on Algebra I - I V in which he revisited parts of the Geometria Aritmetica. In 1959, van der Waerden tried to bring Kähler to Zürich. But he did not want to relocate immediately again. Instead he tried to bring Hirzebruch to Berlin, in which however, he did not succeed. In 1964 Kähler accepted an offer to succeed E. Artin in Hamburg. Together with Sperner, Hasse, Witt, Collatz, he became one of the directors of the Mathematical Institute. For some time, he still continued to live in Berlin where he also gave lectures as honorary professor. In Berlin, he lived in one of the pleasant villas in Lankwitz, filled and decorated with books of all kind. The wonderful atmosphere which the students enjoyed in the house of Erich Kähler, will always remain in the memory of the authors of this article. Tragedy struck the family again with the drowning death on August 19, 1966 of his son Reinhard, 18 years old, in a rafting accident on Lake Wannsee. In 1970, his wife Luise Kähler died of leukemia and in 1988, his daughter Gisela died from the same illness. The illness of his wife and the tragic events of those years had a large effect on Kähler's life and influenced his scientific thinking. Already in his "Geometria Aritmetica" and in his "Algebra und Differentialrechnung" from 1953, he had tried to find connections between mathematics and philosophy for which he found its mathematical expression in Leibniz' Monadology. Now in view of his personal suffering, his need became ever stronger to find a connection of mathematics to chemistry, biology and even to theology and the ingenious mathematician became a mathematical dreamer who thought he could solve all problems of this world by mathematical methods. This is expressed in his papers [38] and [39]. Happiness returned again to his life in 1972 when he married the pharmacist Charlotte Kähler, widow of his brother since the last months of the war. Until Kähler's death, they were always together. She accompanied him on his weekly trips between Berlin and Hamburg before the family moved from Berlin to Wedel along the Elbe, rekindling the dreams of Kähler's childhood and the sea. The sea also played a prominent role in his son Helmuth's life; first he studied naval architecture, but then changed to physics and is now an astrophysicist at the Bergedorf Observatory, devoting most of his research to the development of star models. Kähler became more and more
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Life of Erich Kahler
9
interested in the idea of using mathematics for the creation of a "Weltbild". In particular, he wanted to use modular forms on Siegel's half plane and later quaternion half-spaces. For this purpose he studied the philosophy of Plato, Schopenhauer, Nietzsche and, most of all, Leibniz; he attended courses in theology and in addition read about far Eastern philosophy. Already in Leipzig he had started to study Russian, Chi : nese and Sanskrit. He continued this in Berlin when during the recitation session of his assistants, Mertin, Lehmann and Schulze, he sat in the back row copying Chinese characters. In Hamburg he gave a series of courses, Mathematics I-VIII, the lecture notes of which have been catalogued in the library of Hamburg. In addition, in 1973, he gave a course called "Über die Mathematik als Sprache und Schrift" ("The Mathematics as Language and Script") and in 1981 a course on "Nietzsches Philosophie als höchstes Stadium des Idealismus" ("Nietzsche's philosophy as the highest state of idealism"). It must be said that in Hamburg he did not always find as positive a response as he would have wished. This was probably caused by the presence in Hamburg of such great mathematicians and theoretical physicists as Helmut Hasse, Ernst Witt, Emanuel Sperner, Hei Braun, Pascual Jordan, Harry Lehmann, Hans Joos, Rudolf Haag and Carl Friedrich von Weizsäcker. Another reason may have been that Kahler isolated himself in order to develop his ideas without exterior influence. After his formal retirement, Kähler still participated in the scientific life of the Mathematics Institute. He invited his colleagues to his home in Wedel after seminars and special festivities. His wife created an atmosphere in which discussions flourished and which remained in the memory of all participants. Subjects discussed were of course mathematics, but not only mathematics. One example of the questions posed is whether mathematics was the creation of the human mind. In Kähler's lectures and talks, many listeners always retained the impression that mathematics is created, at least this was the impression which both authors of this essay garnered in his class. In contrast to the impression that Kähler conveyed in his lectures, Kähler's opinion was (cf. [38], II Regno delle Idee) that mathematical structures exist in the realm of the ideas and that the mathematician discovers these structures. Other subjects discussed in the meetings at his home in Wedel were music, philosophy, theology, sciences and current events. With increasing age Kähler believed that all these subjects needed to be treated with the help of mathematics. Until his late age Kähler travelled often and participated in conferences and events honoring colleagues or himself. In 1986 he gave a talk at a physics workshop at Canterbury and participated at the XV International Conference on Differential Geometric Methods in Physics in Clausthal. In 1991, one day after his 85th birthday and again on the 18th of April, he gave lectures at the Academy of Sciences in Berlin. On several occasions he lectured in Leipzig and he went to the conference on string theory in Potsdam to meet Witten and Hawking. In Hamburg Kähler participated in the Colloquium organized on the occasion of his 80th birthday with talks by Mackey: "Weyl's Program and the Symbiosis between Quantum Mechanics and the Theory of Group Representations" and by H. Grauert: "Komplexe und meromorphe Äquivalenzrelationen". He also gave a talk at a Col-
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loquium organized to celebrate his 90th birthday with the title "Vom Relativen zum Absoluten", borrowed from Max Planck's talk of 1924 on a related subject. There he advocated the idea of an absolute space endowed with a hyperbolic metric and fixed by a 10-parameter group of quaternions which Kahler called the "new Poincare group". And until shortly before his death Kahler pursued the idea of a possible arithmetization of physics originally mentioned in the talk by Max Planck.
10
Survey of Kähler's Mathematical Work and Some Comments Rolf Berndt and Oswald Riemenschneider
Erich Kähler's research covers an unusually wide area. From celestial mechanics he went into complex function theory, differential equations, analytic and complex geometry with differential forms, and then into his main topic, combining all this with arithmetic, i.e. arithmetic geometry. His principal interest was in finding the unity in the variety of mathematical themes and establishing thus mathematics as a universal language for physics, chemistry, biology and even philosophy and theology. As usual, it is difficult to put good mathematics into baskets labeled for instance differential geometry, number theory or whatever. And as said above, it is particularly difficult in Kähler's case. But to arrange for some structure, we will enumerate labels in five blocks following roughly the chronological order and the stations of his scientific life in Leipzig, Hamburg, Rome, Hamburg, Königsberg, and - after the war - Leipzig, Berlin and again Hamburg.
1 On the η-Body Problem Influenced by having read Gauß, Weierstraß and Lagrange already at school, Kähler's first publications starting in 1926 treated classical differential equations. The papers [1] 1 and [2] treat the three-body problem and [4] the application of certain principles of the rc-body problem to the determination of equilibrium configurations of rotating liquids: The aim of [1], Transformation der Differentialgleichungen des Dreikörperproblems (1926,16 pages), is to rearrange the differential equations of the three-body problem in such a way that Lagrange's well-known solutions appear in a simple fashion. This succeeds by the introduction of special coordinates (in particular the anomalies) φι (i = 1, 2, 3). After writing the differential equations in these adapted coordinates, the question whether it is possible that the three area velocities are constant leads immediately to the two Lagrangian solutions: - the three bodies form an equilateral triangle during their whole movement, i.e. they move in a plane along conics with focus in the center of gravity, and 1
References containing only numbers refer to Erich Kähler's Bibliography at the end of this volume.
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Rolf Berndt and Oswald Riemenschneider
- the three bodies are arranged in one line and move again in a plane. The note [2], Reduktion des Dreikörperproblems in geometrischer Form dargestellt (1926, 6 pages), shows a geometrical way how to reduce the differential equations to a seventh-order system. The paper [4], Über die Existenz von Gleichgewichtsfiguren rotierender Flüssigkeiten, die sich aus gewissen Lösungen des n-Körperproblems ableiten (1928, 17 pages), is Kähler's thesis. It generalizes fundamental research by Lichtenstein and leads to a system of integro-differential equations whose solution depends on certain linear integral equations. The starting point is here that the plane η-body problem has special solutions, where the η mass points describe circles around the joint center of gravity. In general, these solutions produce equilibrium configurations of a rotating liquid by replacing the mass points by particles of the liquid. In the paper the exact conditions are formulated for this procedure to work.
2 On the Theory of Complex Functions (in Two Variables) The papers [3] and [5-7] treat problems in complex function theory of several variables. Each case is restricted to the treatment of two variables where already the typical problems show up which arise by leaving Riemann's one variable theory. In note [3], Über ein geometrisches Kennzeichen der analytischen Abbildungen im Gebiete zweier Veränderlichen (1928, 5 pages), it is shown in which sense analytical maps in two variables can be called conformal. The paper [5], Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle (1929, 17 pages), uses algebroid functions on the base of former work by Brauner. This paper has had a large influence on the theory of knots (as to be seen in the book by Epple [Epp]) and of singularities. Here we have in this volume the report Topology of Hypersurface Singularities by W. D. Neumann ([Ne]) giving details on Kähler's work and a survey of developments since then, including a brief discussion of the topology of isolated hypersurface singularities in higher dimension. The paper [6], Zur Theorie der algebraischen Funktionen zweier Veränderlichen I (1929,12 pages), pursues the research on the problem to capture properties of functions in two complex variables by topological notions. To study an algebraic function ζ = z(x, y) with defining equation f ( x , y, z) = 0, Kähler introduces a four-dimensional Riemannian manifold R with Betti numbers Pi and gives an explicit formula for Pi ~ 2Λ (using different topological characteristics of R). As an example Kähler treats the equation z3 -3xz
+ 2y = 0
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Survey of Kähler's Mathematical Work and Some Comments
13
where he gets P2 —2P\ = 2. He indicates (p. 262) that the determination of Pi and P2 separately seems to be a much harder problem. The note [7], Über den topologiscken Sinn der Periodenrelationen bei vier-fach periodischen Funktionen (1929,6 pages), has obviously impressed Artin so much that he influenced Blaschke to provide for Kähler a position as an assistant in Hamburg. Here we look at a univalent analytic function φ(μ, ν) which has four periods (Μ,· , υ,·), i = 1 , . . . , 4. A necessary and sufficient condition for the existence of such a function is given by the period relations Σ C i k ( " < V k ~ u kVi) = 0 i(F(x,y)),
where φ is an analytic function. This stimulates the question to ask for those differential equations for which integrals F exist such that all substitutions φ are linear. If one presumes that F(x,y) has only isolated singular curves by 1., one concludes that F can be represented as a quotient of two independent solutions of a simultaneous system of second order dz
dz
dx
dy
_
3 2z dxz
3ζ dx
with coefficients univalent on R. The differential equations coming up this way get a particular interest by the fact that they are valid for Picard's hyperabelian functions φ{χ\,χΐ) with an invariance for the substitutions Xi ι—• x[ = a ' X l + f ' , YiXi + for exterior differentiation of a form ω replacing the older ω'). And Kähler denoted by the letter ο a differential ideal in the ring Ω*(Μ) of exterior differential forms on a manifold M, i.e. a graded ideal with the additional property that άω is an element of α for each ω e a.
17
18
Rolf Berndt and Oswald Riemenschneider
1. Let Μ be a real-analytic manifold of dimension m and I a differential ideal in the ring Ω* (M) of exterior differentials on M. The most important special case is when I is generated by Pfaffian forms (i.e. of degree one). Sometimes it is useful to assume that I contains no forms of degree zero, i.e. functions. Then, we have the following associated concepts (as everything will be real-analytic, we will not restate it each time). i) A submanifold —> Μ
i\X
of Μ is called an integral manifold of I if ϊ*(φ) = 0
for all φ e I .
ii) A p-dimensional linear subspace Ε = E p c TXM of the tangent space to Μ in χ e Μ is called an integral element of I if one has φ Ε ('•= restriction of φ\χ to E) = 0 The set of all /^-dimensional integral elements E
for all φ e X.
p
of I is denoted by
vp(i). Its intersection with the Grassmann space G P ( T X M ) , V P ( I ) Π G P ( T X M ) is an algebraic subvariety of G P { T X M ) . For an η-form Ω e Ω"(Μ), we define G
n
( T M , Ω) : = { Ε e G
n
(TM), Ω
Ε
Φ 0}
and ν
η
(Ι,Ω) : = ν
η
( Ι ) η ΰ
η
( Τ Μ , Ω).
iii) For each ρ with 1 < ρ < m, we introduce the polar space of an integral element Ε = E p C TXM with basis e\,... ,ep H ( E ) := {υ e T X M , φ(ν, e i , . . . , ep) = 0 for all φ G l
p + l
}
and the function r = rp defined by r: V , ( X ) — Ε
ι—> r ( E ) : = dim H ( E ) - (p + 1).
We have r ( E ) > —1 and r ( E ) measures the possibilities to extend a pdimensional integral element E p to a (p + 1)-dimensional one. iv) An integral element Ε = E n e V n ( I ) is called Kähler-ordinary if there is an η η-form Ω e Ω ( Μ ) with Ώε Φ 0 and Ε is an ordinary zero (i.e. a smooth point) of the set of functions
Fv{l)=
18
{ C W1 xRxW
xRns
(za) &W, p = (pf) e Kni, an open domain,
19
20
Rolf Berndt and Oswald Riemenschneider G: o —> Μ5 a real-analytic mapping so that the Γ / : = {(*, yoi /(·*)> Df(x))
f-graph
I Χ € £)Q)} C D for some constant yo,
where Df {x) € M.ns,theJacobianoff, is described by the condition that pf(Df(x)) a df (x)/dxi. Then there is an open neighborhood D\ C 0 x Μ o/5)o x {yo} rea/ analytic mapping F: £)\ W which satisfies the P.D.E. ^ with initial
= G(x, y, F,
= α
dF/dx)
condition F(x, yo) = f ( x ) for all χ e £>o.
Moreover, F is unique in the sense that any other real analytic solution agrees with F on some neighborhood of DQ χ {yo}· The strong form of the Cartan-Kähler theorem is given in [BCGGG], Theorem 2.2 as follows. Let be I C Ω*(Μ) a real-analytic differential ideal, Ρ C Μ a connected, ρ-dimensional, manifold of I , r = r{P) € No,
real-analytic,
Kahler-regular
R C Μ a real analytic submanifold of codimension r with Ρ C R C and H(TXR) transverse in TXM for each χ e P. Then there is a real-analytic ρ + l-dimensional I with Ρ C X C R. This X is unique locally.
integral
M,TXR
connected integral manifold X of
(The introduction of R converts the underdetermined Cauchy problem of extending Ρ to a (p + 1)-dimensional integral manifold to a determined problem). Often the Cartan-Kähler theorem is understood as the following weak form given as Corollary 2.3 in [BCGGG]. Let I be a real-analytic differential ideal of Μ and Ε C TXM an ordinary integral element of I. Then there is an integral manifold of I which passes through χ and whose tangent space at χ is Ε. The question of generality of the solution is resolved by using the Cartan characters sk = Oc — Ck-1 introduced above in 1. vi). Here, we assume that
20
Survey of Kähler's Mathematical Work and Some Comments
21
I is a real-analytic differential system on Μ containing no forms of degree zero, (0)z c Ei c · · · C En c TZM is an ordinary integral flag, s = cn = dim M„, χ = (xl),u = (ua) a ζ-centered coordinate system so that vectors {9/3*-' } 1 < ; · < λ and that for/: < η
is spanned by the
H(Ek) = {u e TZM I dua(υ) = 0 for all a < ck}, Ω = dx1 a • • • A dxn, Ek:={v
dxj(v)
eE\
= 0} for each Ε e Vn(I, Ω).
Then we have a (connected) neighborhood U of En in Vn ( I , U) with the properties: Ek is Kähler-regular for all k < η and all Ε e U and and {dua}a>Ck
[dxi]i<j 0 and < 0 if Τ has negative discriminant discr Τ — ac — bb = —p2 < 0. In particular, for
we have T(q) = qj — jq = 2t = 0 as the equation of the wall dividing both cells. In [46], 15, Kähler proves Proposition. The following statements are equivalent: 1.) Μ or Μ
j
0 )
€
GK. ds2.
2.) q ι-»· Μ(q) is an isometryfor
3.) q — ι >· M(q) is a symmetry of the original cell, i.e. for Τ hermitian with discr Τ < 0 maps the cell with T(q) ^ 0 either onto itself or onto the cell with T(q) § 0. iv) As in the two other papers, a large part is devoted to the study of the Dirac equation δu = a v u,
in particular a = λ = const.,
for the metric ds^. Using the volume differential τ
=
dx A dy Λ dz A dt ?
the inner calculus allows for a decomposition U =
V+
ν
e+ +
v
ν e ,
e± :=
1 ±τ 2
where ν± are space differentials in the sense that they are outer polynomials which do not contain dt. These space differentials ν = a + bidx + bidy + b^dz + c\dy A dz + C2dz A dx + c^dx a dy + hdx Ady Adz determine two scalars a and h and two vector fields (bi, b2, h) '•= 6
and
(ci, C2, C3) := c.
Putting c + b := iZ,
c-b
:= S)
t2h + t~2a =: ρ,
(i :=
v^T)
t2h - t~2a =: ia,
30
Survey of Kähler's Mathematical Work and Some Comments
31
the homogeneous Dirac equation Su = 0 leads to Maxwell-type equations ([46], 28): . , 1 3.fj rotmodule W. The Lie bracket on pi χ pi is given by Π. Any such extended Poincare algebra ρ(Π) admits a derivation D with eigenspace decomposition p ( n ) = o(V) + V + W and corresponding eigenvalues (0, 1, 1/2). ρ(Π) can be extended by D to a new Z2-graded Lie algebra 0 = 0(Π) : = MD + po + Ρι· The adjoint representation of 0 is faithful and hence defines on g the structure of a linear Lie algebra. LetG = G(FI) c Aut g denote the corresponding connected linear group. Moreover, Κ = Κ (Ε) c G is the connected linear Lie group with Lie algebra t = o(E) 0 oiE^-), where Ε c V is a three-dimensional Euclidean subspace. Cortes has as his (first) principal result ([Co], Theorem A, p. 4, resp. Theorem 8, p. 24): 1.) Μ = G/K has G-equivariant quaternionic structures. 2.) If Π is nondegenerate (i.e. if W 3 s ι-» Π (5·i;·) € W*
