转化群与几何物理：艾朗根纲领及其影响

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"The Erlangen Program and Its Impact in Mathematics and Physics" 是一本深入探讨埃朗根计划及其在几何学与物理学中影响的书籍。该计划由费利克斯·克莱因提出，并得到索非斯·李的贡献，它提供了一个关于变换群在数学和物理中的基础视角。这本书不仅详述了埃朗根计划的范围、应用和影响，还介绍了李和克莱因在形成和发展该计划中的作用，同时作为这两位数学家工作的入门读物。书中涉及的主要人物还包括伯恩哈德·黎曼、亨利·庞加莱、赫尔曼·外尔、埃利·嘉当、艾米·诺特等数学巨匠。埃朗根计划的核心是将几何学看作是变换群的不变性质的研究，即不同的几何学可以被理解为不同变换群的不变性质。这一观点对现代数学和理论物理学产生了深远影响。例如，通过李群理论，我们可以研究连续对称性，这在量子力学和相对论中至关重要。克莱因和李的工作揭示了几何与代数之间的紧密联系，为后来的数学家如黎曼、庞加莱和嘉当等人开辟了新的研究领域。黎曼的几何思想，特别是在非欧几何方面的贡献，为理解广义相对论中的时空弯曲提供了基础。而庞加莱的拓扑学工作，特别是他的不动点定理，对理解物理系统的稳定性有着深远的影响。外尔的工作则将群论引入到量子力学中，发展了规范场论，这是现代粒子物理学的基础之一。嘉当的工作进一步深化了李群和微分几何的结合，他的联络理论在现代物理学中扮演着关键角色，特别是在规范场论和弦理论中。艾米·诺特定理则揭示了守恒定律与对称性的关系，成为量子场论中的一条基本原理。此外，该书还可能探讨了这些数学家如何通过他们的理论影响了20世纪的物理学发展，包括量子力学、相对论和宇宙学等领域。它不仅是对数学历史的回顾，也是对数学与物理交叉领域的深度探索，旨在使广大数学家、物理学家和学生能够理解和欣赏这些理论的深远意义。 "Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics" 提供了一个全面的历史视角，展示了数学中的几何和群论如何深刻地塑造了我们对自然界的理解，并且继续在现代科学中发挥着重要作用。

Introduction xv

papers cited above, gives models of the three constant-curvature geometries (hyper-

bolic, Euclidean and spherical) in the setting of projective geometry. He deﬁnes the

distance functions in each of these g eometries by ﬁxing a conic (the “conic at inﬁn-

ity”) and taking a constant multiple of the logarithm of the cross ratio of four points:

the given two points and the two intersection points of the line joining them with the

conic at inﬁnity. The hyperbolic and spherical geometries are obtained by using real

and complex conics respectively, and Euclidean geometry by using a degenerate one.

The authors in Chapter 5 comment on these two important papers of Klein and they

display relations with works of other mathematicians, including Cayley, Beltrami,

Poincar´e and the founders of projective geometry.

Klein’s interaction with Lie in their formative years partly motivated Lie to de-

velop Lie’s version of Galois theory of differential equations and hence of Lie trans-

formation theory.

In fact, a major motivation for Lie for the introduction of Lie

groups was to understand differential equations. This subject is treated in Chapter 6

of this volume. The author, Alexandre Vinogradov, starts by observing that Lie initi-

ated his work by transporting the Galois theory of the solvability of algebraic equa-

tions to the setting of differential equations. He explains that the major contribution

of Lie in this setting is the idea that symmetries of differential equations are the basic

elements in the search for their solutions. One may recall here that Galois approached

the problem of solvability of polynomial equations through a study of the symmetries

of their roots. This is based on the simple observation that the coefﬁcients of a poly-

nomial may be expressed in terms of the symmetric functions of their roots, and that

a permutation of the roots does not change the coefﬁcients of the polynomial. In the

case of differential equations, one can naively deﬁne the symmetry group to be the

group of diffeomorphisms which preserve the space of solutions, but it is not clear

how such a notion can be used. There is a differential Galois theory which is parallel

to the Galois theory of polynomial equations. In the differential theory, the question

“what are the symmetries of a (linear or nonlinear, partial or ordinary) differential

equation?” is considered as the central question. Chapter 6 also contains reviews of

the notions of jets and jet spaces and o ther constructions to explain the right setup for

formulating the question of symmetry, with the goal of providing a uniform frame-

work for the study of nonlinear partial differential equations. The author is critical

of the widely held view that each nonlinear partial differential equation arising from

geometry or physics is special and often requires its own development. He believes

that the general approach based on symmetry is the right one.

The author mentions developments of these ideas that were originally formulated

by Lie and Klein in works of E. Noether, B¨acklund,

E. Cartan, Ehresmann and others.

A lot of questions in this domain remain open, and this chapter will certainly give

the reader a new perspective on the geometric theory of nonlinear partial differential

equations.

In Chapter 7, Charles Frances surveys the modern developments of geometric

structures on manifolds in the lineage of Klein and Lie. The guiding idea in this

Lie has had a course at Oslo by Sylow on Galois and Abel theory before he meets Klein, but it is clear that

Klein also brought some of his knowledge to Lie.

xvi Introduction

chapter is the following question: When is the automorphism group of a geometric

structure a Lie group, and what can we say about the structure of such a Lie group?

The author considers the concept of Klein geometry, that is, a homogeneous space

acted upon by a Lie group, and a generalization of this notion, leading to the con-

cept of a Cartan geometry. (Cartan used the expression espace g

eralis

e.) Besides

the classical geometries, like constant curvature spaces (Euclidean, Lobachevsky and

spherical) as well as projective geometry which uniﬁes them, the notion of Cartan ge-

ometry includes several differential-geometric structures. These notions are deﬁned

using ﬁber bundles and connections. They describe spaces of variable curvature and

they also lead to pseudo-Riemannian manifolds, conformal structures of type .p; q/,

afﬁne connections, CR structures, and the so-called parabolic geometries. The author

presents a series of important results on this subject, starting with the theorem of My-

ers and Steenrod (1939) saying that the isometry group of any Riemannian manifold

is a Lie group, giving a bound on its dimension, and furthermore, it says that this

group is compact if the manifold is compact. This result gave rise to an abundance

of developments and generalizations. The author also explains in what sense pseudo-

Riemannian manifolds, afﬁne connections and conformal structures in dimensions

 3 are rigid, symplectic manifolds are not rigid, and complex manifolds are of an

intermediate type.

Thus, two general important questions are addressed in this survey:

 What are the possible continuous groups that are the automorphism groups of

a geometry on a compact manifold?

 What is the inﬂuence of the automorphism group of a structure on the topology

or the diffeomorphism type of the underlying manifold?

Several examples and recent results are given concerning Cartan geometries and in

particular pseudo-Riemannian conformal structures.

Chapter 8, by Norbert A’Campo and Athanase Papadopoulos, concern transitional

geometry. This is a family of geometries which makes a continuous transition be-

tween hyperbolic and spherical geometry, passing through Euclidean geometry. The

space of transitional geometry is a ﬁber space over the interval Œ1; 1 where the ﬁber

above each point t is a space of constant curvature t

if t>0and of constant curva-

ture t

if t>0. The ﬁbers are examples of Klein geometries in the sense deﬁned

in Chapter 7. The elements of each geometry are deﬁned group-theoretically, in the

spirit of Klein’s Erlangen program. Points, lines, triangles, trigonometric formulae

and other geometric properties transit continuously between the various geometries.

In Chapter 9, by Athanase Papadopoulos and Sumio Yamada, the authors intro-

duce a notion of cross ratio which is proper to each of the three geometries: Euclidean,

spherical and hyperbolic. This highlights the relation between projective geometry

and these geometries. This is in the spirit of Klein’s view of the three constant cur-

vature geometries as part of projective geometry, which is the subject of Chapter 5 of

the present volume.

Chapter 10, by Yuri Suris, concerns the Erlangen program in the setting of dis-

crete differential geometry. This is a subject which recently emerged, whose aim is

to develop a theory which is the discrete analogue of classical differential geometry.

Introduction xvii

It includes discrete versions of the d ifferential geometry of curves and surfaces but

also higher-dimensional analogues. There are discrete notions of line, curve, plane,

volume, curvature, contact elements, etc. There is a unifying transformation group

approach in discrete differential geometry, where the discrete analogues of the clas-

sical objects of geometry become invariants of the respective transformation groups.

Several classical geometries survive in the discrete setting, and the author shows that

there is a discrete analogue of the fact shown by Klein that the transformation groups

of several geometries are subgroups of the projective transformation group, namely,

the subgroup preserving a quadric.

Examples of discrete differential geometric geometries reviewed in this chapter

include discrete line geometry a nd discrete line congruence, quadrics, Pl¨ucker line

geometry, Lie sphere geometry, Laguerre geometry and M¨obius geometry. Important

notions such as curvature line parametrized surfaces, principal contact element nets,

discrete Ribeaucour transformations, circular nets and conical nets are discussed. The

general underlying idea is that the notion of transformation group survives in the d is-

cretization process. Like in the continuous case, the transformation group approach is

at the same time a unifying approach, and it is also related to the question of “multi-

dimensional consistency” of the geometry, which says roughly that a 4D consistency

implies consistency in all higher dimensions. The two principles – the transforma-

tion group principle and consistency principle – are the two guiding principles in this

chapter.

Chapter 11 by Catherine Meusburger is an illustration of the application of Klein’s

ideas in physics, and the main example studied is that of three-dimensional gravity,

that is, Einstein’s general relativity theory

with one time and two space variables.

In three-dimensions, Einstein’s general relativity can be described in terms of cer-

tain domains of dependence in thee-dimensional Minkowski, de Sitter and anti de

Sitter space, which are homogeneous spaces. After a summary of the geometry of

spacetimes and a description of the gauge invariant phase spaces of these theories,

the author discusses the question of quantization of gravity and its relation to Klein’s

ideas of characterizing geometry by groups.

Besides presenting the geometrical and group-theoretical aspects of three-dimen-

sional gravity, the author mentions other facets of symmetry in physics, some of them

related to moduli spaces of ﬂat connections and to quantum groups.

Chapter 12, by Jean-Bernard Zuber, is also on groups that appear in physics, as

group invariants associated to a geometry. Several physical ﬁelds are mentioned,

including crystallography, piezzoelectricity, general relativity, Yang–Mills theory,

quantum ﬁeld theories, particle physics, the physics of strong interactions, electro-

magnetism, sigma-models, integrable systems, superalgebras and inﬁnite-dimensional

algebras. We see again the work of Emmy Noether on g roup invariance principles

in variational problems. Representation theory entered into physics through quan-

tum mechanics, and the modern theory of quantum group is a by-product. The au-

thor comments on Noether’s celebrated paper which she presented at the occasion of

Klein’s academic Jubilee. It contains two of her theorems on conservation laws.

We recall by the way that Galileo’s relativity theory is at the origin of many of the twentieth century theories.

Chapter 1

Sophus Lie, a giant in mathematics

Lizhen Ji

Contents

1 Introduction ........................................ 1

2 Some general comments on Lie and his impact ...................... 2

3 A glimpse of Lie’s early academic life .......................... 4

4 A mature Lie and his collaboration with Engel ...................... 6

5 Lie’s breakdown and a ﬁnal major result ......................... 10

6 An overview of Lie’s major works ............................ 11

7 Three fundamental theorems of Lie in the Lie theory .................. 13

8 Relation with Klein I: the fruitful cooperation ...................... 15

9 Relation with Klein II: conﬂicts and the famous preface ................. 16

10 Relations with others ................................... 21

11 Collected works of Lie: editing, commentaries and publication ............. 23

Bibliography .......................................... 25

1 Introduction

There are very few mathematicians and physicists who h ave not heard of Lie groups

or Lie algebras and made use of them in some way or another. If we treat discrete

or ﬁnite groups as special (or degenerate, zero-dimensional) Lie groups, then almost

every subject in mathematics uses Lie groups. As H. Poincar´etoldLie[25] in October

1882, “all of mathematics is a matter of groups.” It is clear that the importance of

groups comes from their actions. For a list of topics of group actions, see [17].

Lie theory was the creation of Sophus Lie, and Lie is most famous for it. But Lie’s

work is broader than this. What else did Lie achieve besides his work in Lie theory?

This might not be so well known. The differential geometer S. S. Chern wrote in 1992

that “Lie was a great mathematician even without Lie g roups” [7]. What did and can

Chern mean? We w ill attempt to give a summary of some major contributions of Lie

in 6.

One purpose of this chapter is to give a glimpse of Lie’s mathematical life by

recording several things which I have read about Lie and his work. Therefore, it is

short and emphasizes only a few things about his mathematics and life. For a fairly de-

tailed account of his life (but not his mathematics), see the full length biography [27].

We also provide some details about the unfortunate conﬂict b etween Lie and Klein

and the famous quote from Lie’s preface to the third volume of his books on trans-

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