Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics Lizhen Ji Athanase Papadopoulos Editors Editors: Lizhen Ji Athanase Papadopoulos Department of Mathematics Institut de Recherche Mathématique Avancée University of Michigan CNRS et Université de Strasbourg 530 Church Street 7 Rue René Descartes Ann Arbor, MI 48109-1043 67084 Strasbourg Cedex USA France 2010 Mathematics Subject Classification: 01-00, 01-02, 01A05, 01A55, 01A70, 22-00, 22-02, 22-03, 51N15, 51P05, 53A20, 53A35, 53B50, 54H15, 58E40 Key words: Sophus Lie, Felix Klein, the Erlangen program, group action, Lie group action, symmetry, projective geometry, non-Euclidean geometry, spherical geometry, hyperbolic geometry, transitional geometry, discrete geometry, transformation group, rigidity, Galois theory, symmetries of partial differential equations, mathematical physics ISBN 978-3-03719-148-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2015 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface The Erlangen program provides a fundamental point of view on the place of trans- formation groups in mathematics and physics. Felix Klein wrote the program, but Sophus Lie also contributed to its formulation, and his writings are probably the best example of how this program is used in mathematics. The present book gives the first modern historical and comprehensive treatment of the scope, applications and impact of the Erlangen program in geometry and physics and the roles played by Lie and Klein in its formulation and development. The book is also intended as an introduc- tion to the works and visions of these two mathematicians. It addresses the question of what is geometry, how are its various facets connected with each other, and how are geometry and group theory involved in physics. Besides Lie and Klein, the names of Bernhard Riemann, Henri Poincar´e, Hermann Weyl, Elie´ Cartan, Emmy Noether and other major mathematicians appear at several places in this volume. A conference was held at the University of Strasbourg in September 2012, as the 90th meeting of the periodic Encounter between Mathematicians and Theoretical Physicists, whose subject was the same as the title of this book. The book does not faithfully reflect the talks given at the conference, which were generally more specialized. Indeed, our plan was to have a book interesting for a wide audience and we asked the potential authors to provide surveys and not technical reports. We would like to thank Manfred Karbe for his encouragement and advice, and Hubert Goenner and Catherine Meusburger for valuable comments. We also thank Goenner, Meusburger and Arnfinn Laudal for sending photographs that we use in this book. This work was supported in part by the French program ANR Finsler, by the GEAR network of the National Science Foundation (GEometric structures And Rep- resentation varieties) and by a stay of the two editors at the Erwin Schr¨odinger Insti- tute for Mathematical Physics (Vienna). Lizhen Ji and Athanase Papadopoulos Ann Arbor and Strasbourg, March 2015 Contents Preface v Introduction xi 1 Sophus Lie, a giant in mathematics ..................... 1 Lizhen Ji 2 Felix Klein: his life and mathematics .................... 27 Lizhen Ji 3 Klein and the Erlangen Programme .................... 59 Jeremy J. Gray 4 Klein’s “Erlanger Programm”: do traces of it exist in physical theories? 77 Hubert Goenner 5 On Klein’s So-called Non-Euclidean geometry ............... 91 Norbert A’Campo, Athanase Papadopoulos 6 What are symmetries of PDEs and what are PDEs themselves? .....137 Alexandre Vinogradov 7 Transformation groups in non-Riemannian geometry ..........191 Charles Frances 8 Transitional geometry ............................217 Norbert A’Campo, Athanase Papadopoulos 9 On the projective geometry of constant curvature spaces ........237 Athanase Papadopoulos, Sumio Yamada 10 The Erlangen program and discrete differential geometry ........247 Yuri B. Suris 11 Three-dimensional gravity – an application of Felix Klein’s ideas in physics ...................................283 Catherine Meusburger 12 Invariances in physics and group theory ..................307 Jean-Bernard Zuber viii Contents List of Contributors ...............................325 Index .......................................327 Sophus Lie. Felix Klein. Introduction The Erlangen program is a perspective on geometry through invariants of the auto- morphism group of a space. The original reference to this program is a paper by Felix Klein which is usually presented as the exclusive historical document in this matter. Even though Klein’s viewpoint was generally accepted by the mathematical commu- nity, its re-interpretation in the light of modern geometries, and especially of modern theories of physics, is central today. There are no books on the modern developments of this program. Our book is one modest step towards this goal. The history of the Erlangen program is intricate. Klein wrote this program, but Sophus Lie made a very substantial contribution, in promoting and popularizing the ideas it contains. The work of Lie on group actions and his emphasis on their impor- tance were certainly more decisive than Klein’s contribution. This is why Lie’s name comes first in the title of the present volume. Another major figure in this story is Poincar´e, and his role in highlighting the importance of group actions is also critical. Thus, groups and group actions are at the center of our discussion. But their importance in mathematics had already been crucial before the Erlangen program was formulated. From its early beginning in questions related to solutions of algebraic equations, group theory is merged with geometry and topology. In fact, group actions existed and were important before mathematicians gave them a name, even though the for- malization of the notion of a group and its systematic use in the language of geometry took place in the 19th century. If we consider group theory and transformation groups as an abstraction of the notion of symmetry, then we can say that the presence and importance of this notion in the sciences and in the arts was realized in ancient times. Today, the notion of group is omnipresent in mathematics and, in fact, if we want to name one single concept which runs through the broad field of mathematics, it is the notion of group. Among groups, Lie groups play a central role. Besides their mathematical beauty, Lie groups have many applications both inside and out- side mathematics. They are a combination of algebra, geometry and topology. Besides groups, our subject includes geometry. Unlike the word “group” which, in mathematics has a definite significance, the word “geometry” is not frozen. It has several meanings, and all of them (even the most recent ones) can be encompassed by the modern interpretation of Klein’s idea. In the first version of Klein’s Erlangen program, the main geometries that are em- phasized are projective geometry and the three constant curvature geometries (Eu- clidean, hyperbolic and spherical), which are considered there, like affine geometry, as part of projective geometry. This is due to the fact that the transformation groups of all these geometries can be viewed as restrictions to subgroups of the transfor- mation group of projective geometry. After these first examples of group actions in geometry, the stress shifted to Lie transformation groups, and it gradually included many new notions, like Riemannian manifolds, and more generally spaces equipped xii Introduction with affine connections. There is a wealth of geometries which can be described by transformation groups in the spirit of the Erlangen program. Several of these ge- ometries were studied by Klein and Lie; among them we can mention Minkowski geometry, complex geometry, contact geometry and symplectic geometry. In modern geometry, besides the transformations of classical geometry which take the form of motions, isometries, etc., new notions of transformations and maps between spaces arose. Today, there is a wealth of new geometries that can be described by trans- formation groups in the spirit of the Erlangen program, including modern algebraic geometry where, according to Grothendieck’s approach, the notion of morphism is more important than the notion of space.1 As a concrete example of this fact, one can compare the Grothendieck–Riemann–Roch theorem with the Hirzebruch–Riemann– Roch. The former, which concerns morphisms, is much stronger than the latter, which concerns spaces. Besides Lie and Klein, several other mathematicians must be mentioned in this venture. Lie created Lie theory,