Coordinates [Reprint 2010 ed.] 9783110866353, 9783110148527

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Wolfram Neutsch Coordinates

Wolfram Neutsch

Coordinates

w DE

Walter de Gruyter · Berlin · New York 1996

Author Priv.-Doz. Dr. Wolfram Neutsch Institut für Astrophysik der Universität Bonn Auf dem Hügel 71 53121 Bonn

Figure in cover illustration by kind permission of Oxford University Press, from Ray d'Inverno: Introducing Einstein's Relativity, Clarendon Press, Oxford, 1992

Library of Congress Cataloging-in-Publication Data Neutsch, W. Coordinates / Wolfram Neutsch. p. cm. Includes bibliographical references (p. ISBN 3-11-014852-8 (alk. paper) 1. Coordinates. I. Title. QA556.N48 1996 516'.16-dc20

—

) and index.

96-24425 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Neutsch, Wolfram: Coordinates / Wolfram Neutsch. — Berlin ; New York : de Gruyter, 1996 ISBN 3-11-014852-8

© Copyright 1996 by Walter de Gruyter & Co., D-10785 Berlin 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: WB-Druck, Rieden. — Binding: Heinz Stein, Berlin. Cover design: Hansbernd Lindemann, Berlin.

Preface Some time ago, and other

I began to collect systematically transformation

important

formulas

information about certain coordinate frames which I

found useful when dealing with mathematical and physical problems. Unfortunately, nomenclature,

in the available literature there is no universally

accepted

and no comprehensive account is readily at hand. This caused

me to prepare a number of tables which initially had been intended f o r my personal usage in a more systematical way (and with uniform terminology) in order to publish them. At the same time, I extended the somewhat scattered data, thus attempting to get a more complete presentation. It soon turned out that a simple compilation of the basic equations would not

be s u f f i c i e n t

for

the

practitioner;

a theoretical

introduction

is

in-

dispensable. For the same reason I also included a number of novel developments which are only loosely connected with the main topic of these books, among them a great many quadrature methods, some of which are new. Over the years, the short list of reference frames grew into the two heavy volumes of the present monograph.

In spite of

its size,

it is believed to

be welcomed as a handy companion f o r all who have to apply coordinates. It is my pleasant duty to thank all those who contributed to this

project

by countless critical remarks, a lot of new thoughts and insights, and r e f erences to books and research papers which I had overlooked. The number of single pieces of advice is much too large to mention them separately.

Without wanting to b e l i t t l e the others,

I would like to name only

Horst Fichtner and Klaus Scherer to whom I am most g r a t e f u l f o r their untiring willingness to discuss various aspects of the present book. I f e e l under an obligation to Reinhold Schaaf who drew my attention to an e r r o r in an e a r l i e r publication. My special thanks go to Günter Lay f o r his invaluable help (not only) concerning questions of

computing.

By f a r the greatest support, however, I owe to my mother, Ingeborg Neutsch, who during the preparation of the manuscript patiently bore my whims w i t h out complaining. I furthermore want to apologize to her f o r the chaos which I temporarily

produced by paper

lying around in her

f o r g i v e me, and thanks f o r everything!

living

room.

Please

Volume 1: Theory 0. Introduction

1

Α. Foundations 1. Historical development of the coordinate concept

1.1. Geography

9

9

1.2. Affine and projective geometry

15

1.3. Differential geometry

20

1.4. Geodesy and cartography

24

1.5. Special coordinates

34

2. Notation and conventions

2.1. Sets and topological spaces

38

38

2.2. Groups

42

2.3. Matrices

48

2.4. Multilinear forms sind tensors

56

2.5. Polynomials

65

B. Geometry 3. Manifolds

3.1. Continuous coordinate systems

70

70

3.2. Smooth manifolds

73

3.3. Curves and tangent spaces

76

3.4. Vector bundles

80

3.5. Differential and exterior derivation

85

3.6. Partition of unity

94

3.7. Oriented manifolds

102

3.8. Integration of differential forms

106

3.9. Stokes' theorem

110

4. Riemannian spaces

113

4.1. The metric tensor

113

4.2. Christoffel symbols and covariant derivation

117

4.3. Normal coordinates

123

4.4. Curvature

128

4.5. Volume

136

VIII

4.6. Duality 4.7. Classical vector analysis 5. Applications to physics

145 149 160

5.1. Mechanics

160

5.2. Hydrodynamics

166

5.3. Relativity

172

5.4. Electromagnet ism

175

5.5. Optics

182

5.6. Quantum mechanics

187

6. Complex analysis

190

6.1. Elementary properties of complex numbers

190

6.2. Convergence of function series

193

6.3. Power series

198

6.4. Analytical continuation

205

6.5. Holomorphic functions

211

6.6. Angle-preserving transformations

225

6.7. Lie groups

233

6.8. Lie algebras

237

7. Projective Geometry

246

7.1. Affine and projective coordinates

246

7.2. Projectivities

255

7.3. The cross ratio

260

7.4. B6zout' s theorem

268

7. 5. Planar algebraic curves

279

7.6. Stereographic projection

287

7.7. Higher dimensional spaces

293

C. Rotations 8. Orthogonal groups

295

8.1. Isometries and Euclidean transformations

295

8.2. The exponential mapping

301

8.3. Rational parametrization

311

9. Linear transformations of complex spaces

318

9.1. Pauli matrices

318

9.2. Cayley-Klein parameters

322

9.3. The angular momentum algebra

327

9.4. Gaussian mutations of space

336

IX

9.5. Euler angles 9.6. Elementary geometry of the Riemann number sphere 10. Quaternions 10.1. The skew field of quaternions

339 342 353 353

10.2. Left and right multiplication

359

10.3. Rotations of the quaternion algebra

362

10.4. Representation by complex matrices

367

10.5. Finite groups of quaternions

371

11. Octaves

379

11.1. Doubling method of Cayley and Dickson

379

11.2. Alternative division algebras

389

11.3. The theorem of Hurwitz

404

11.4. Quadratic algebras

411

11.5. Parallelizability and regular vielbeine

425

12. Hopf mappings

429

12.1. Homotopy groups of spheres

429

12.2. Homotopy invariants

433

12.3. Duplication of angles and classical Hopf fibration

438

12.4. Generalizations

444

12.5. Geometric peculiarities 13. Spinors

448 452

13.1. Schur extensions of the symmetric groups

452

13.2. Spin groups

466

13.3. Graßmann algebras

470

13.4. Clifford algebras

476

13.5. Dirac matrices

486

14. Lorentz transformations

492

14.1. The Poincar6 group

492

14.2. Boosts

499

14.3. Complex Lorentz matrices

508

14.4. Description by quaternions

S16

14.5. Representation theory of the Lorentz group

522

D. Reflections 15. Coxeter groups

528

15.1. Discrete symmetries

528

15.2. Invariant theory; Molien's theorem

532

χ

15.3. Polynomial Invariant rings

538

15.4. Reflection groups

550

15.5. Fundamental systems

558

15.6. The Euclidean Coxeter groups

566

16. Invariant rings of finite Weyl groups

574

16.1. Root systems

574

16.2. Weyl groups

589

16.3. Elementary symmetric functions

599

17. Basic invariants

605

17.1. Generic Weyl groups (A,IB,ID)

605

17.2. Weyl groups of type IE

607

17.3. Weyl groups of types F,(B,IH, D 17.4. Basic degrees

623 635

Volume 2: Applications E. Lattices 18. Elliptic functions and modular forms

637

18.1. Doubly periodic functions

637

18.2. Weierstraß functions

643

18.3. The modular group

654

18.4. Modular forms

661

18.5. Cusp forms

672

18.6. Modular functions; the J-invariant

678

19. Euclidean lattices

688

19.1. Foundations of lattice theory

688

19.2. Theta functions

698

19.3. The Gösset lattice

712

19.4. Niemeier lattices

717

20. Linear codes

726

20.1. Encoding of information

726

20. 2. The Hexacode

730

20.3. The binary Golay code

738

20.4. Miracle octad generator

751

20.5. Alternative constructions of the Golay code

757

20.6. Steiner systems and Mathieu groups

767

XI

21. The Leech lattice

781

21.1. Uniqueness of the Leech lattice

781

21.2. Explicit representations

789

21.3. Aut omorphi sms

792

21.4. Holes in the Leech lattice

799

21.5. Invariants of the Conway group

806

21.6. Lorentz lattices and sporadic groups

819

F. Spheres 22. Harmonic functions

837

22.1. The theorems of Green

837

22.2. Dirichlet's principle

844

22.3. The Poisson integral

854

22.4. Potential functions

862

23. Spherical surface functions

866

23.1. Spherical harmonics

866

23.2. Legendre polynomials

870

23.3. Orthogonal functions on the 2-sphere

881

23.4. C1 ebsch-Gordan coefficients

897

24. Lattice integration

913

24.1. Integration points and weights

913

24.2. Numerical integration after Gauß

920

24.3. Simple examples

928

24.4. Integration by means of the Gösset lattice

935

24.5. Transfer via Hopf mappings

941

25. Spherical designs

945

25.1. Integration methods with equal weights

945

25.2. Tight designs

951

25.3. Optimal integration on the 2-sphere

956

25. 4. The hypersphere

966

25.5. Very high precision numerical integration

976

25.6. Four-dimensional root systems and quaternions

981

G. Coordinate systems 26. Linear and reducible coordinates 26.1. Cartesian and oblique linear frames

985 985

XII

26.2. Polar coordinates

991

26.3. Classical cylindrical coordinates

995

26.4. Separable isothermal systems 27. Three-dimensional Stäckel coordinates

997 1006

27.1. Rotational symmetry

1006

27.2. Stäckel spaces

1015

27.3. Classification of Stäckel-type metrics

1020

27.4. Bipolar coordinates

1029

28. Confocal Coordinates

1037

28. 1. Orthogonal families of quadrics

1037

28.2. Metric and differential operators

1045

28.3. Separation of the potential equation

1053

28.4. Lam£ functions

1058

28.5. Geodesies on the ellipsoid

1064

29. Gauß-Krüger Coordinates

1068

29.1. Soldner's parameterization of the spheroid

1068

29.2. Series expansions

1077

29.3. Transversal Mercator projection

1083

29.4. Besse 1's figure of the Earth

1095

29.5. Recent models

1100

30. Coordinates for special applications

1104

30.1. Clairaut coordinates

1104

30.2. Roche coordinates

1109

30.3. Planar Weyl coordinates

1113

30.4. Weyl coordinates in three-dimensional space

1118

30.5. Higher dimensional Weyl coordinates

1126

H. Tables Calculation and organization of the tables 2

Coordinates in IR

1132 1136

Cartesian coordinates

1136

Polar coordinates

1139

Parabolic coordinates

1142

Elliptic coordinates

1145

Confocal coordinates

1148

Bipolar coordinates

1152

Digonal coordinates

1156

XIII

Trigonal coordinates

1159

Tetragonal coordinates

1162

Pentagonal coordinates

1165

Hexagonal coordinates 3

1168

Coordinates in IR

1171

Cartesian coordinates

1171

Cylindrical coordinates

1174

Polar coordinates

1178

Geographic coordinates

1182

Coordinates of the parabolic cylinder

1186

Coordinates of the elliptic cylinder

1190

Confocal cylindrical coordinates...

1194

Coordinates of the circular paraboloid

1198

Coordinates of the elliptic paraboloid

1202

Ellipsoidal coordinates (prolate)

1207

Ellipsoidal coordinates (oblate)

1211

Spheroidal coordinates (prolate)

1215

Spheroidal coordinates (oblate)

1219

Conical coordinates

1223

Confocal coordinates

1228

Bicylindrical coordinates

1234

Bispherical coordinates

1238

Torus coordinates

1243

Tetrahedral coordinates

1248

Octahedral coordinates

1253

Icosahedral coordinates

1259

Coordinates in IR4

1265

Cartesian coordinates

1265

Cylindrical coordinates of type (1,3)

1268

Cylindrical coordinates of type (2,2)

1272

Polar coordinates

1276

Geographic coordinates

1280

Double polar coordinates

1284

Confocal coordinates

1288

Appendix References

1297

Index

1322

Der N i e d e r r h e i n e r w e i ß nichts,

kann aber a l l e s

erklären. Hanns D i e t e r

HUsch

0. Introduction

C o o r d i n a t e s a r e ubiquitous in science and t e c h n o l o g y .

T h e r e is v i r t u a l l y

question

use.

of

solvability tion.

importance of

which

a particular

does not

problem

require

depends

strongly

A c l e v e r t r i c k may lead to new insights,

thematical

description

complicates

the

their

on the

the

parametriza-

w h i l e an i n a p p r o p r i a t e

investigation

or

no

As a r u l e ,

even

makes

ma-

it

im-

possible. Therefore,

i t is e s s e n t i a l

to be w e l l

i n f o r m e d about the a v a i l a b l e

Quite o f t e n ideas which at a f i r s t glance look a bit " e x o t i c " a r e helpful,

at

least

if

they

are

flexible

enough

to

be

adapted

frames. extremely

to

changing

conditions. The v a r i e t y

of

all

possible

the p a s t ) is immense;

reference

frames

(and even those

in

e v e r y d i f f e o m o r p h i s m of an a r b i t r a r y space o n t o some

r e g i o n in IRn or Cn d e f i n e s an n-dimensional mapping, nate

published

i. e.

a smooth

coordi-

system.

F o r t h i s r e a s o n a comprehensive r e p r e s e n t a t i o n of all imaginable zations

is t o t a l l y

out of

question,

parametri-

unless we a r e content w i t h g e n e r a l explicit.

finitions

and equations which a r e not v e r y

however,

these a r e of no use; he or she demands f o r m u l a s which a r e r e a d i l y

at hand and may be employed w i t h o u t long p r i o r

For

the

de-

calculations.

practitioner,

Thus,

a

re-

s t r i c t i o n t o the most i m p o r t a n t special systems is unavoidable. Universal

rules

appropriate

for

allowing

us to decide,

a given

purpose,

which

coordinate

do not e x i s t .

Rather,

the s k i l l f u l n e s s w i t h which one a t t a c k s a p r o b l e m . charm of

our f i e l d l i e s just in this " a r t i s t i c

The p r o p e r use of alternatives tively

which w e want to choose.

d i f f i c u l t t o d e r i v e f r o m the l i t e r a t u r e

are

On the o t h e r hand,

of

sufficiently

Unfortunately,

reasonably

it

complete

is

but may be i m p o r t a n t on special o p p o r t u n i t i e s .

can be g i v e n .

of the

many rela-

informa-

t i o n c o n c e r n i n g t y p e s of c o o r d i n a t e s which a r e employed c o m p a r a t i v e l y ly,

most

is a m a t t e r

freedom".

c o o r d i n a t e s r e q u i r e s the knowledge

between

systems it

S e v e r a l reasons f o r

rarethis

0.

2

In the f i r s t place,

Introduction

only a few systems are documented well enough;

formularies and t e x t books r e s t r i c t to the most common ones,

most

while more

" e x o t i c " ideas are discussed only occasionally and rudimentarily,

e. g.

by

doing no more than j u s t offering the definitions. Before we can apply those coordinates,

we have to f a c e more or less involved auxiliary calculations.

This is annoying and acts as a deterrent. In addition, each author has his or her own scheme of notation. This makes comparisons between d i f f e r e n t sources laborious. The above-mentioned reasons led me already a couple of years ago to compile the basic information on those coordinates which I found advantageous f o r my mathematical,

physical, and astronomical studies.

In the beginning,

I collected data a r b i t r a r i l y ,

but l a t e r a more and more

systematic treatment emerged. I always kept an eye on c l a r i t y and uniformity of notation.

After a while I developed a scheme which in my opinion

comes quite close to these ideals. Several colleagues prompted me to extend the mass of data and to publish them in concise form. The question

arose which degree of explicitness

is adequate.

Initially

I

tended to edit no more than a simple formulary, amended by a few comments and a t h e o r e t i c a l introduction. The bulk of the work was planned to consist of t a b l e s of the most important f a c t s like definitions, tial operators,

metrics,

differen-

etc. f o r as many systems as possible.

My own experience and, even more, numerous talks with colleagues from d i f f e r e n t branches, however, soon taught me that this kind of presentation is inadequate f o r several reasons: (1) A simple listing is of no value without some advice which coordinates should be applied under given circumstances. The most important c r i t e rion is the agreement of the symmetry properties of the problem to be solved and its required parametrization.

Accordingly,

I put the empha-

sis j u s t on this aspect. (2) Many mappings elude the s t r i c t schematization of a pure collection tables.

Therefore

a detailed evaluation

of more theoretical

ideas

of is

required. (3) Although most of the tabulated coordinates r e f e r to Euclidean spaces 2 3 4 (usually IR , R and IR ), charts of other domains are also of great int e r e s t . For instance, a discussion of our present topic would be incomplete without a thorough investigation of rotations or, to speak somewhat more generally,

isometries of the spaces in question.

Representa-

0.

Introduction

3

t i o n t h e o r y and i t s p r a c t i c a l c o n s e q u e n c e s t h e r e f o r e p l a y a m a j o r in t h i s m o n o g r a p h ;

This l e a d s us t o an overview of t h e c o n t e n t s . into eight p a r t s ,

role

they may even be c o n s i d e r e d a s a c e n t r a l t o p i c . The w o r k a s a w h o l e

splits

t h e f i r s t seven of which (A t h r o u g h G) c o n s t i t u t e a t e x t

book w h i l e t h e t a b l e s f o r m t h e l a s t one (H). P a r t A is d e d i c a t e d t o r i c a l and t e c h n i c a l p r e p a r a t i o n s , its various shades,

histo-

p a r t s Β t o Ε develop t h e t h e o r y p r o p e r in

p a r t F c o n t a i n s some i m p o r t a n t a p p l i c a t i o n s

concerning

t h e t r e a t m e n t of s p h e r i c a l h a r m o n i c s which a r e of wide use in s c i e n c e .

Fi-

n a l l y , p a r t s G and Η f o r m an i n t e g r a t e d whole. In t h e f o r m e r , t h e n e c e s s a r y e x p l i c a t i o n s t o t h e t a b l e s a r e t o be f o u n d . Both p a r t s , nates.

G and H, a r e r e s t r i c t e d t o t h e d i s c u s s i o n of c o n c r e t e

All f r a m e s l i s t e d

in p a r t

Η are described

coordi-

in m o r e o r l e s s

The i n t e r r e l a t i o n b e t w e e n p a r t s G and Η is n o t 1: 1, h o w e v e r .

detail.

The t h e o r e t i -

cal e v a l u a t i o n c a r r i e s much m o r e i n f o r m a t i o n t h a n t h e t a b l e s .

As i n d i c a t e d

above, a n u m b e r of s p e c i a l c o o r d i n a t e s could not be a d a p t e d t o t h e c o n c e p t of p a r t H; t h u s t h e y a r e m e n t i o n e d only in p a r t G. On t h e o t h e r h a n d , I i n cluded e x a m p l e s w h e r e v e r

possible.

F o r i n s t a n c e , t h e t h o r o u g h t r e a t m e n t in c h a p t e r 29 of t h e G a u ß - K r i i g e r map ( a l s o known a s t r a n s v e r s a l M e r c a t o r p r o j e c t i o n ) which is of w i d e s p r e a d use in geodesy and s u r v e y i n g h a s been e n r i c h e d by many n u m e r i c a l s e r i e s d e v e l o p m e n t s . To a s s i s t t h e r e a d e r , t h e c o n c r e t e n u m b e r s d e r i v e d f r o m t h e m o s t common s p h e r o i d a l m o d e l s of t h e E a r t h (among t h e m t h o s e of Bessel, H a y f o r d , and K r a s s o w s k i ) a r e s t a t e d a l s o . F u r t h e r m o r e a f e w o t h e r e x c e p t i o n a l p a r a m e t r i z a t i o n s like Roche and C l a i raut

c o o r d i n a t e s a r e described as well,

though their relatively

domain of a p p l i c a t i o n does not j u s t i f y a m o r e e x t e n s i v e In c l e a r c o n t r a s t t o t h i s ,

restricted

presentation.

t h e Weyl or C o x e t e r c o o r d i n a t e s which have been

d i s c o v e r e d only r e c e n t l y ( d u r i n g t h e l a s t d e c a d e ) will r e c e i v e g r o w i n g t e n t i o n in t h e n e a r f u t u r e .

They a r e a l w a y s s u i t a b l e if we a r e

w i t h a g e o m e t r y which p o s s e s s e s a n o n t r i v i a l f i n i t e g r o u p of

at-

concerned

symmetries.

This o f t e n o c c u r s in s o l i d s t a t e p h y s i c s ( c r y s t a l l o g r a p h y ) and e l s e w h e r e . Up t o now no c o m p r e h e n s i v e a c c o u n t of Weyl c o o r d i n a t e s h a s been p u b l i s h e d . All a v a i l a b l e

i n f o r m a t i o n is contained

correspondingly d i f f i c u l t to obtain.

in a f e w m a t h e m a t i c a l

papers

and

In t h e p r e s e n t book we a t t e m p t f o r t h e

f i r s t t i m e t o e s t a b l i s h a sound t h e o r y of t h i s c a t e g o r y of c h a r t s .

Except

f o r c e r t a i n known f a c t s , some new r e s u l t s have been f o u n d which open a l t e r n a t i v e p a t h s and a r e a d v a n t a g e o u s in c o m p a r i s o n t o t h e m o r e methods.

conventional

4

0.

Introduction

Anyway, t h e a s s e r t i o n of many a u t h o r s t h a t a couple of c o o r d i n a t e

systems

a r e by t h e m s e l v e s much m o r e i m p o r t a n t t h a n t h e o t h e r s b e c a u s e t h e y may be applied

more

universally,

very f e w i n s t a n c e s ,

e.g.

has to

be q u a l i f i e d .

Cartesian

or

It

is c e r t a i n l y

true

(with some r e s t r i c t i o n s )

in

a

spherical

coordinates. Beyond t h i s ,

in my opinion t h i s s t a t e m e n t

seems t o be an i n t e r c h a n g e

c a u s e and e f f e c t . If one is c o n f r o n t e d w i t h a c o n c r e t e m a t h e m a t i c a l lem,

it is q u i t e n a t u r a l t o a p p l y t h e b e s t known p r o c e d u r e s i n s t e a d of

ing t o new t e r r i t o r y . ches a r e p r e f e r r e d ,

Thus t h e m o r e e x p l i c i t l y d e s c r i b e d "simple" even if o t h e r p o s s i b i l i t i e s might be b e t t e r .

of

probgo-

approaIt is one

of t h e g o a l s of t h e s e t w o volumes t o achieve a s o m e w h a t m o r e b a l a n c e d p r e sentation. Coming back t o t h e t h e o r e t i c a l p a r t s A t o F, some r e m a r k s a r e n e c e s s a r y : At t h e h e a r t of t h e i n t r o d u c t o r y p a r t A l i e s c h a p t e r 2,

where I collected

t h o s e f u n d a m e n t a l d e f i n i t i o n s and t h e o r e m s which a r e p r o b a b l y known t o many readers,

b u t may be a l i t t l e b i t u n f a m i l i a r t o o t h e r s .

The aim of t h i s i n s e r t i o n is t w o f o l d . F i r s t , tion,

we p r e s e n t t h e b a s i c i n f o r m a -

which m a i n l y c o n s i s t s of w h a t is c a l l e d m a t h e m a t i c a l

"folklore",

c o m p r e s s e d f o r m t o avoid i n t e r r u p t i o n s of t h e t e x t p a s s a g e s t o f o l l o w .

in La-

t e r - o n t h e s e s e c t i o n s of t h e t e x t a r e c i t e d w i t h o u t f u r t h e r e x p l a n a t i o n . Beside t h i s , t h e n o t a t i o n is d e f i n e d and e x p l a i n e d . The n o m e n c l a t u r e e s s e n tially f i t s into e a r l i e r systems derived f r o m the relevant l i t e r a t u r e .

Only

o c c a s i o n a l l y d e v i a t i o n s or a m e n d m e n t s t u r n e d out t o be n e c e s s a r y . They a r e all s t a t e d

explicitly.

Whenever a l t e r n a t i v e a s s i g n m e n t s of s y m b o l s a r e in u s e , I t o o k one of t h e m according to two d i f f e r e n t c r i t e r i a . was p r e f e r r e d . suggested

Sometimes,

however,

Normally the more "abundant" notation f o r r e a s o n s of e f f i c a c y o t h e r

choices

themselves.

In any c a s e I c a r e f u l l y o b s e r v e d t h e d e m a n d s of f a i t h f u l n e s s and

retainabi-

lity. R e a d e r s who a r e f a m i l i a r w i t h t h e r e l e v a n t i d e a s may skip c h a p t e r 2 w i t h o u t serious loss,

but should go back if they a r e in doubt a b o u t t h e m e a n i n g of

a p a r t i c u l a r concept.

Those who a r e n o t so w e l l - v e r s e d in a c e r t a i n m a t h e -

m a t i c a l f i e l d a r e a d v i s e d t o s t u d y t h e s e c t i o n s r e g a r d i n g it c a r e f u l l y . t e r w a r d s t h e y w i l l be p r e p a r e d t o f o l l o w t h e d i s c u s s i o n w i t h o u t (and w i t h o u t r e f e r r i n g t o o t h e r

Af-

difficulty

sources).

The p r i m e t o p i c s of p a r t Β a r e t o d e f i n e t h e t e r m " c o o r d i n a t e s " and t o p r o vide a couple of

elementary geometric tools.

Here I t o o k a v e r y

general

0.

Introduction

5

p o i n t of view in o r d e r t o leave s p a c e f o r a l l i t s v a r i o u s f a c e t s ,

even if

f o r m o s t a p p l i c a t i o n s a n a r r o w e r f o r m u l a t i o n would have s u f f i c e d . A r e s t r i c t i o n t o , e. g. , r e a l or c o m p l e x c o o r d i n a t e s would mean t h e l o s s of several extremely interesting parametrizations f o r c e r t a i n s c i e n t i f i c or technical

which a r e a l s o v e r y u s e f u l

problems.

C o n c e r n i n g t h e c o n t e n t s of p a r t B, we may be b r i e f . (differential)

geometric,

i.e.

analytical,

It l a r g e l y g i v e s

foundations

without

the

neglecting

a f f i n e and p r o j e c t i v e g e o m e t r y . Here and in o t h e r p a s s a g e s of t h e book we go as f a r a s j u s t i f i a b l e t h e l i m i t a t i o n s of s p a c e . Where it w a s p o s s i b l e ,

under

I f u r t h e r m o r e included

f o r m a t i o n which is s o m e w h a t o u t s i d e t h e r e a l m of p r a c t i c a l n e c e s s i t y . is i n t e n d e d t o make f u t u r e d e v e l o p m e n t s e a s i e r and t o widen t h e

This

horizons.

It w a s ( r a r e l y ! ) n e c e s s a r y t o r o u n d o f f t h e p r e s e n t a t i o n by s k e t c h i n g c i a l r e s u l t s of r e c e n t r e s e a r c h ,

cru-

though the p r o o f s a r e too d i f f i c u l t or

volved t o p r e s e n t t h e m in t h i s book.

I believe t h a t such o c c a s i o n a l

e x c u r s i o n s a r e i n d i s p e n s a b l e f o r a deeper

in-

short

understanding.

In p a r t C we c o n s i d e r p a r a m e t r i z a t i o n s of r o t a t i o n s , i n t r o d u c t i o n of c o o r d i n a t e s on o r t h o g o n a l g r o u p s . discussed,

in-

and in p a r t i c u l a r

the

A number of m e t h o d s

is

t h e bulk of which a r e of a l g e b r a i c o r i g i n .

By t h e way, d u r i n g t h e w r i t i n g of t h e m a n u s c r i p t t h e w e i g h t i n g s h i f t e d f r o m a mainly analytical to a more algebraic i n t e r p r e t a t i o n . p e c t is c a u s e d by t h e i n s i g h t t h a t

the most

T h i s c h a n g e of

"elegant" mathematical

asideas

a r e a l w a y s t h o s e which p o s s e s s s p e c i a l s y m m e t r i e s . This does n o t mean a n y t h i n g e l s e t h a n t h e e x i s t e n c e of a " l a r g e " g r o u p of i n v a r i a n c e

operations.

Beside a f e w d i f f e r e n t d e s c r i p t i o n s of t h e r o t a t i o n g r o u p s in IRn f o r

arbi-

t r a r y ( f i n i t e ) n, we a l s o d i s c u s s some m e t h o d s which a r e i m p o r t a n t t o t h e practitioner,

b u t a p p l i c a b l e only t o t h e most o f t e n needed low

dimensions

2, 3, and 4. They a r e n o t e x t e n s i b l e t o h i g h e r d i m e n s i o n a l s p a c e s since t h e y depend on g r o u p t h e o r e t i c a l p e c u l i a r i t i e s of t h e complex n u m b e r s , q u a t e r n i o n s o r C a y ley n u m b e r s ( o c t a v e s ) . Here I added a couple of r e m a r k s on f u r t h e r i n v e s t i g a t i o n s and d e e p e r r e s u l t s in t o p o l o g y and t h e n e i g h b o u r i n g b r a n c h e s . The l a s t t h r e e c h a p t e r s of p a r t C c o n t a i n s e v e r a l c o n c e p t s which a r e very u s e f u l b u t n o t widely known o u t s i d e

mathematics.

In c h a p t e r 12 we c o n s i d e r t h e s o - c a l l e d Hopf m a p p i n g s . Up t o now, t h e i r r e al m e a n i n g s e e m s t o have been acknowledged only by t h e ( a l g e b r a i c ) gists.

(Later,

topolo-

in c h a p t e r 24, we s h a l l e x p l o i t t h e m t o c o n s t r u c t high

cision numerical i n t e g r a t i o n

methods).

pre-

0.

6

In the sequel,

a thorough investigation of the nontrivial coverings of

thogonal and related groups f o l l o w s . spin groups.

Introduction

or-

These are known to the physicist

as

We c l a s s i f y the real and complex C l i f f o r d algebras and apply

them to spinors.

As a by-result

we get the representation theory of

the

three-dimensional rotation group. This is not merely fundamental f o r quantum mechanics; there are applications in many other f i e l d s , a f e w of which are mentioned in the text. The remainder of part C is a chapter about Lorentz transformations.

Here,

except f o r the very well known real forms we also give somewhat less common descriptions by means of complex matrices and quaternions. Among the isometries of Euclidean spaces, tions,

not only the rotations,

i.e.

the orthogonal

transforma-

but also the r e f l e c t i o n s are of utmost

inte-

rest. T h e r e f o r e the leitmotif of part D is a combination of representation theory of Coxeter groups (which are generated by r e f l e c t i o n s ) and invariant theory (Molien's

theorem).

with the outcome;

Mathematicians I nevertheless

or

at

least

algebraists

are

familiar

attempted to give a self-contained

and

comprehensive account although this goes a bit beyond the limitations kept elsewhere in this monograph. To be precise, I not only included a c l a s s i f i cation of the Weyl representations of all f i n i t e Coxeter groups; compiled a complete list of the polynomial invariants,

I moreover

sometimes in several

equivalent ways. These occasionally longish formulas are the basis of several later

investi-

gations.

integra-

We need them to construct extremely e f f i c i e n t numerical

tion methods on spheres (chapters 24 and 25, see below). The explanations

in this part

are a l i t t l e

more involved than

but I tried to present them in an easily understandable f o r m .

elsewhere, This should

allow readers who are not so well-versed mathematically or more inclined to practical applications to find their way to this fascinating subject. Lattice theory as developed largely by elementary

means in part Ε is

of

fundamental importance f o r the chapters to f o l l o w . Unfortunately here again a f u l l discussion has to be ruled out because of only the most significant ideas can be dealt with.

its enormous extent,

so

Among them is a short

account of the complex-analytical properties of theta functions and a concise treatment of some important lattices,

particularly those discovered by

Gösset and Leech. A f e w remarks on recent progress in sporadic group r e search f o l l o w (without p r o o f s ) . Virtually all terms discussed previously are applied in part F to the prob-

0.

Introduction

7

lem of a p p r o x i m a t i n g s p h e r i c a l f u n c t i o n s by s t a n d a r d t y p e s ( s p h e r i c a l monics)

and

to

mechanical

integration.

The

cross

relations

to

har-

potential

t h e o r y (key w o r d : h a r m o n i c f u n c t i o n s ) a r e shown and d e s c r i b e d in much d e tail. We a l s o d i s c u s s t h e s t r u c t u r e c o n s t a n t s ,

known f r o m q u a n t u m

mechanics,

which a r i s e d u r i n g t h e r e d u c t i o n of t e n s o r r e p r e s e n t a t i o n s of t h e

angular

momentum a l g e b r a ( C l e b s c h - G o r d a n c o e f f i c i e n t s and t h e i r r e l a t i v e s , t h e Wigner symbols, cf. section 23.4). The c u l m i n a t i o n p o i n t of t h e s e c o n s i d e r a t i o n s a r e t h e two f i n a l c h a p t e r s of p a r t F, which a r e c o n c e r n e d w i t h t w o a l t e r n a t i v e n u m e r i c a l i n t e g r a t i o n m e t h o d s on s p h e r e s (of v a r i o u s d i m e n s i o n s ) . The f i r s t ( c h a p t e r 23) is f o u n d e d on g e o m e t r i c and a l g e b r a i c p r o p e r t i e s of special lattices; last few years.

m e t h o d s of t h i s c l a s s have been developed only w i t h i n t h e They t u r n out t o be very u s e f u l .

Another a p p r o a c h ,

s t e m m i n g f r o m c o m b i n a t o r i c s (= f i n i t e g e o m e t r y ) ,

the recently discovered so-called spherical designs.

By t h i s one m e a n s

t e g r a t i o n p r o c e d u r e s w i t h equal w e i g h t s f o r a l l p o i n t s . configurations, exhibiting complete symmetry,

is via

A s u b s e t of

is p a r t i c u l a r l y

in-

these

valuable.

Many of t h e c o n s t r u c t i o n s in c h a p t e r 25 have been a v a i l a b l e u n t i l t o d a y o n ly in m a t h e m a t i c a l

journals,

but not in t e x t books.

O t h e r s have been

in-

v e n t e d newly f o r t h e p r e s e n t m o n o g r a p h . Some of t h e m p o s s e s s d e g r e e s of a p p r o x i m a t i o n which s u r p a s s t h o s e of t h e o l d e r ones, t h u s d e m o n s t r a t i n g t h e i r high q u a l i t y . As i n d i c a t e d above,

it is i n e v i t a b l e t h a t t h e v a r i o u s c o n t r i b u t i o n s have t o

be h a n d l e d on d i f f e r e n t levels.

In c o n n e c t i o n

with

s i m p l e and d i f f i c u l t q u e s t i o n s t o be c o n s i d e r e d .

coordinates

there

This need not d e t e r

dy; even t h e m o r e a d v a n c e d p a s s a g e s should be a c c e s s i b l e t o wide

are

anybosections

of t h e r e a d e r s h i p . One may skip o c c a s i o n a l deeper a r g u m e n t s or g l a n c e t h r o u g h t h e m

without

loss. Both v o l u m e s of t h i s m o n o g r a p h a r e d i r e c t e d a t s t u d e n t s of science

or

especially

technical at

disciplines

practitioners

as well

(engineers)

They a r e w r i t t e n f o r p r i v a t e s t u d y ,

as l e c t u r e r s

of

mathematics,

these

in d i f f e r e n t b r a n c h e s

of

fields

and

industry.

but a r e a t l e a s t p a r t i a l l y a p t a l s o t o

support appropriate university courses. Above a l l t h e p a r t s c o n c e r n e d w i t h s p e c i a l s y s t e m s and t h e f o r m u l a r y may a l s o s e r v e a s a g u i d a n c e t o decide which c o o r d i n a t e s should be used f o r a particular

purpose.

I tried

hard

to present

all

information

as

explicitly

δ

0.

Introduction

a s p o s s i b l e in a c c o r d a n c e w i t h t h e l i m i t e d s p a c e . T h e r e f o r e in any c a s e a f t e r t h e d e f i n i n g t r a n s f o r m a t i o n s ( f r o m which e v e r y t h i n g e l s e w a s d e r i v e d ) t h e c o n t r a - and c o v a r i a n t m e t r i c t e n s o r s , f e r e n t i a l s of t h e c o o r d i n a t e s ,

the d i f -

and t h e volume e l e m e n t a r e given.

They a r e

f o l l o w e d by t h e C h r i s t o f f e l s y m b o l s ( a f f i n e c o n n e c t i o n ) and t h e m o s t i m p o r tant differential operators, (div) and L a p l a c i a n omitted,

curl (rot),

divergence

s i n c e it would r e q u i r e very much r o o m w h i l e being of

low i n t e r e s t . the

namely g r a d i e n t ( g r a d ) ,

(Δ). Only f o r t h e f o u r - d i m e n s i o n a l f r a m e s t h e c u r l

If n e c e s s a r y ,

Christoffel

is

relatively

it can be deduced w i t h a l m o s t no e f f o r t f r o m

coefficients;

the

rules

required

for

obtaining

it

are

in-

cluded. T h e r e is no need t o e m p h a s i z e t h a t I took g r e a t p a i n s t o e l i m i n a t e t o t h e b e s t of my a b i l i t y .

errors

All c a l c u l a t i o n s have been c a r r i e d out w i t h

the

h e l p of c o m p u t e r a l g e b r a p r o g r a m s (Reduce) w h e n e v e r t h i s w a s f e a s i b l e . The r e s u l t s were t r a n s f e r r e d directly into the t e x t f i l e s . Only t h e l a y o u t h a s been improved by hand,

and I s t r i c t l y avoided t o

s c r i b e f o r m u l a s o r l a r g e n u m b e r s and copied them Clearly,

automatically

tran-

instead.

many b l u n d e r s will have s u r v i v e d , t h i s is i n e v i t a b l e .

Of c o u r s e ,

a l l c a l c u l a t i o n s w e r e checked i n d e p e n d e n t l y if p o s s i b l e ,

for

ex-

a m p l e by c o m p a r i s o n w i t h t h e l i t e r a t u r e (on t h i s o c c a s i o n I f o u n d n u m e r o u s i n c o r r e c t s t a t e m e n t s in s e v e r a l p u b l i c a t i o n s ) or by r e d o i n g w i t h

independ-

ent

numerical

codes

or

alternative

methods.

Sometimes

specialization

or

t r e a t m e n t of s i n g u l a r c a s e s t u r n e d out t o be u s e f u l . I hope and am q u i t e c o n f i d e n t t h a t t h e number of r e m a i n i n g m i s t a k e s is a c c e p t a b l e and t h a t t h i s book will be welcomed a s a handy and c o n v e n i e n t c o m panion.

A. Foundations 1. Historical Development of the Coordinate Concept 1.1. Geography During the Neolithic age the transition f r o m nomadic to settled f o r m of e x istence required to divide the available land into parcels which were ed by individual persons or families. estate boundaries precisely.

seiz-

This made it necessary to define the

Simple land registers or f i e l d maps which pic-

t o r i a l l y represent the distribution of property do not s u f f i c e ;

one rather

needs an exact and unique quantitative specification. It is no longer possible to say when and in which culture area this was done f o r the f i r s t time,

but it is j u s t i f i e d to assume that a gradual deve-

lopment has taken place,

leading f r o m a more descriptive f o r m to a tabular

recording of important sites,

as e. g.

the corners of the plots of land or

conspicuous natural formations etc. People

soon

started

having

the

self-suggesting

thought

to

distinguish

pair of mutually orthogonal axes to which the positions are r e f e r r e d .

a

The

Cartesian coordinates defined in this way are not only the simplest method which

serves

the

purpose;

they

are

also

extremely

handy

and

flexible.

T h e r e f o r e over the millennia alternative ideas merely played a minor r o l e . Cartesian frames,

however,

t r i c t e d to a limited region,

are applicable only if the mapped area is r e s e. g.

single villages,

towns or districts.

With

growing skill and experience in shipbuilding and navigation more and more extended voyages of discovery and trade asked f o r a parametrization of the whole surface of the Earth with a single coordinate system. Probably relying on Chaldean sources and confirmed by the circular shapes of the horizon and the t e r r e s t r i a l shadow during lunar eclipses,

Pythagoras

(ca. 580 - ca. 500 BC) taught that the Earth is spherical. This f o r b i d s the use of Cartesian coordinates as soon as the curvature becomes noticeable. Instead, one applies a description with the help of two angles. Their d e f i nition depends on the rotation of the Earth. known in classical Greece,

This phenomenon was already

but lacked a rigorous proof until,

in 1851, Jean

Bernard L6on Foucault (1819-1868) carried out his famous pendulum experi-

10

ment.

1. H i s t o r i c a l Development o f the Coordinate Concept

The r o t a t i o n axis is f i x e d in space and physically

antipodal

points

on the t e r r e s t r i a l

surface

(North pole

distinguishes and South

two

pole).

The equatorial plane l i e s symmetrically between them. It passes through the centre of the Earth. As the f i r s t coordinate of some point Ρ on the s u r f a c e of the Earth we c o n v e n t i o n a l l y take i t s geographic latitude φ. It is the angle enclosed by the direction

f r o m the E a r t h ' s centre to Ρ and the equatorial

plane,

counted

p o s i t i v e to the North. The lines of constant φ are the circles of latitude, among them the equator (φ = 0 ) .

The orthogonal t r a j e c t o r i e s are also c i r c l e s ,

namely a l l which go

through both poles. They are called circles of longitude or meridians. One of

them is selected

arbitrarily.

It

the geographic

gets

longitude

0.

Then the longitude λ of any point Ρ is given by the angle between the prime meridian (λ = 0 ) but not a l w a y s ,

and the

circle

of

longitude

passing

through

P.

Usually,

Eastern longitudes are considered as p o s i t i v e .

The positioning of the z e r o meridian is not r e s t r i c t e d by any natural ference;

one may d e f i n e it at w i l l .

pre-

For that reason in the past a g r e a t

va-

r i e t y of d i f f e r e n t conventions has been applied. Most countries (but by no means a l l ) r e f e r r e d the geographic longitude to a w e l l d e f i n e d point which a l l o w e d f o r precise measurements. N o r m a l l y the meridian of

an important o b s e r v a t o r y ,

was e l e c t e d .

Nevertheless,

for

instance that of

some s t a t e s p r e f e r r e d other,

the capital

city,

less convenient

ar-

rangements. Only on the International Meridian Conference the z e r o

(Washington,

October

It passes through a passage

line was f i x e d bindingly.

(meridian circle) at the Royal Observatory of

Greenwich.

We c a l l the coordinate f r a m e consisting of λ and φ geographic. r a t u r e the term "polar coordinates" is used also, exclusively

to

a slight

modification

of

1884)

instrument

In the

lite-

but we shall r e s t r i c t

the geographic

coordinates,

is found if we r e p l a c e the latitude φ by the polar angle 90°-

The

latter

is also known as the "colatitude". Latitude measurements can be c a r r i e d out by simple means; e. g. , it s u f f i c e s to d e r i v e the elevation of the North pole which equals φ f r o m the daily tremes (upper and lower culminations) of circumpolar Another

method

which

is

mainly

applied

to

("shoot") the Sun or a star with a sextant.

nautical

ex-

stars. tasks

is

to

observe

The aim is to f i n d the

angle

f o r m e d by the d i r e c t i o n to the c e l e s t i a l body and to the horizon. When w e neglect several small c o r r e c t i o n s like refraction, which is caused

1.1.

Geography

11

by t h e d i f f r a c t i o n of

light

in t h e

atmosphere,

or the d i f f e r e n c e

t h e visual a n d t h e mathematical horizon ( s e e b e l o w ) ,

between

w h i c h d e p e n d s on

the

h e i g h t of t h e o b s e r v e r a b o v e s e a l e v e l , w e i m m e d i a t e l y g e t t h e e l e v a t i o n of the sighted If

its

object.

position

in t h e

sky

is

tabulated

in

the

l a t i t u d e is derived easily f r o m the culmination On t h e o t h e r h a n d ,

(almanacs),

yearbooks

the

height.

b e c a u s e of t h e a r b i t r a r y f i x i n g of t h e z e r o m e r i d i a n t h e

l o n g i t u d e c a n b e d e t e r m i n e d o n l y by c o m p a r i s o n of t h e r e s u l t s of t w o tions.

This

centuries,

requires

a precise

knowledge

of

the

i t c o u l d n o t be a c h i e v e d t o e v e r y o n e ' s

In p a r t i c u l a r

the transport

of

actual

time.

sta-

earlier

satisfaction.

c l o c k s by s h i p w a s e x t r e m e l y

T h e r e f o r e in 1714, on N e w t o n ' s s u g g e s t i o n ,

In

problematic.

the English p a r l i a m e n t o f f e r e d a

p r i z e of 2 0 0 0 0 p o u n d s s t e r l i n g f o r a m e t h o d w h i c h a l l o w e d t o m e a s u r e i n t e r continental

longitude

differences

(Europe-America)

with

an

error

of

not

more than half a degree. A f t e r long discussions, son ( 1 6 9 3 - 1 7 7 6 ) ,

t h e a w a r d w a s g i v e n t o t h e c l o c k - m a k e r J o h n Harri-

who had c o n s t r u c t e d

a chronometer w h i c h w a s f i x e d in

a

C a r d a n i c f r a m e a n d w o r k e d q u i t e p r e c i s e l y even on h e a v y s e a s . Up t o t h e n o n e h a d t a k e n r e s o r t t o d i f f e r e n t p r o c e d u r e s of w h i c h w e s h a l l m e n t i o n o n l y a f e w of t h e m o s t i m p o r t a n t On s o l i d g r o u n d

there

is the possibility

ones. to carry

of m a n y p o i n t s by trlangulation. To t h i s e n d ,

out relative

positioning

one d e f i n e s a net of trian-

gles in w h i c h one s i d e a n d a l l a n g l e s c a n be e v a l u a t e d .

If

the radius

t h e E a r t h i s k n o w n ( i t c a n be c a l c u l a t e d f r o m t h e d a t a t h e m s e l v e s ) ,

of

longi-

t u d e d i f f e r e n c e s may be deduced as well. At s e a ,

however,

t h i s c a n n o t be d o n e s i n c e f i x e d p o i n t s a r e l a c k i n g .

o n e m a d e u s e of c e r t a i n

astronomical

places (nearly) at the same time.

phenomena which occur

Valuable are,

in

among o t h e r s ,

Here

different

stellar oc-

cultations by t h e Moon a n d eclipses of t h e Jovian satellites w h i c h h a v e b e c o m e o b s e r v a b l e a f t e r t h e i n v e n t i o n of t h e

telescope.

With t h e c o n s t r u c t i o n of electric telegraphs a b o u t 1830 by C a r l Gauß (1777-1855) a n d Wilhelm E d u a r d Weber (1804-1891) a n d n e a r l y

Friedrich simultane-

o u s l y by P a w e l L w o w i t s c h Schilling (1786-1837) t i m e s i g n a l s c o u l d b e t r a n s f e r r e d over large distances,

and the d i f f i c u l t y was overcome

Today longitude measurements, the poles,

e x c e p t in t h e i m m e d i a t e v i c i n i t y of o n e of

a r e a s p r e c i s e a s t h o s e of t h e l a t i t u d e .

t h i s p o i n t in s e c t .

1.4,

completely.

We s h a l l c o m e b a c k

w h e r e we discuss some r e c e n t l y developed

which a r e applied mainly to geodesy.

to

methods

12

1. H i s t o r i c a l Development o f the Coordinate Concept

In astronomy a number of reference frames are needed to describe positions.

From a f o r m a l mathematical point of view,

stellar

they are all equiva-

lent to the geographic coordinates. T h e r e f o r e we shall just summarize them without f u r t h e r comment. More detailed information is found in the specialist literature,

f o r instance Bucerius [1967] or Neutsch & Scherer [1992].

The astronomical coordinates are roughly subdivided into topo-, heliocentric ones,

geo-,

and

according as the reference point f r o m where the angles

are counted lies at the position of the observer,

the Earth's or the Sun's

centre. The most easily realized topocentric coordinates are given by the horizontal system.

It is based on the direction of

local g r a v i t y ,

cally downward to the nadir. Diametrically opposed to this, the observer,

pointing

verti-

straight above

is the zenith. The plane through the origin which is perpen-

dicular to the line connecting zenith and nadir defines the

(mathematical)

horizon. It d i f f e r s f r o m the apparent horizon at sea by the perspective e f f e c t due to the curvature of the Earth's surface. On the continents additional

devi-

ations are produced by the local bumpiness of the terrain. For astronomical purposes the mathematical horizon alone is decisive. The angle between the horizon and some heavenly body is its elevation h, which sometimes is replaced by the zenith distance ζ = 90°-h. coordinate is the orientation angle along the horizon.

The second

It is known as the

azimuth A. Unfortunately

in the various branches of

profession which are

with navigation several d i f f e r e n t conventions are in use. Thus,

concerned one has to

be careful when comparing results f r o m external sources. In astronomy i t s e l f the azimuth is reckoned f r o m the Northern point at the horizon via East, South, and West, while f o r example navigators occasionally p r e f e r to count f r o m either North or South to East or West. The position Ν 35° W

hence

corresponds

to

the

azimuth

A = -35°

or,

equivalently,

A = 315°. In aeronautics and aviation even other rules are employed,

etc.

A d e f i n i t e disadvantage of the horizontal system is that the coordinates of a star change permanently because of the rotation of the Earth, even if we neglect the minor positional shifts produced by proper motion, and parallax.

(They are in the order of arc seconds).

ready in antiquity other reference systems have been introduced, oriented according to the direction of tension

intersects

the

fictitious

the Earth's rotation

celestial

sphere

aberration,

For this reason

in

axis.

North

al-

which are and

Its

ex-

South

1.1.

Geography

13

Poles;

t h e g r e a t c i r c l e which is s i t u a t e d s y m m e t r i c a l l y t o t h e s e t w o p o i n t s

is t h e

(celestial) equator.

The d e v i a t i o n of a s t a r f r o m t h i s line,

reckoned positive to the North,

is

i t s declination δ. The a n g l e c e n t r e d a t t h e N o r t h Pole which e m b r a c e s t h e d i r e c t i o n t o t h e o b j e c t and t h e m e r i d i a n is t h e hour angle β. It

increases

in p r o p o r t i o n t o t h e t i m e while t h e d e c l i n a t i o n is i n d e p e n d e n t of t i m e and t h e l o c a t i o n of t h e o b s e r v e r .

The rotating equatorial system t h u s

defined

t h e r e f o r e is much s i m p l e r t h a n t h e h o r i z o n t a l f r a m e . It h a s been known a l r e a d y t o Hipparchus (ca. 180 - ca. 125 BC). He a l s o did t h e n e x t s t e p and r e p l a c e d t h e h o u r a n g l e by a q u a n t i t y

which

is i n v a r i a n t and d e p e n d s on t h e s t e l l a r p o s i t i o n a l o n e . To t h i s end he s u b t r a c t s f r o m t h e h o u r a n g l e f i r s t t h e s o - c a l l e d sidereal time. T h i s i s a u n i f o r m m e a s u r e of t i m e which i n c r e a s e s by 360° d u r i n g one r e v o l u t i o n p e r i o d of t h e E a r t h ( c l o s e t o 23 h 56 m 04 s ). s t a n t in t i m e ,

but varies f r o m place to place.

t h e g e o g r a p h i c a l l o n g i t u d e of t h e o b s e r v a t o r y , of t h e s t a r ,

we g e t a p r o p e r

subtract

coordinate

n a m e l y i t s right ascension a.

An a r b i t r a r y a d d i t i v e c o n s t a n t , r i d i a n on t h e E a r t h , The l a t t e r

The d i f f e r e n c e is c o n -

If we f u r t h e r m o r e

is t h e

s i m i l a r t o t h e e s t a b l i s h i n g of t h e z e r o m e -

is f i x e d such t h a t t h e vernal equinox l i e s a t α = 0.

point

on t h e

ecliptic,

i. e.

the

apparent

orbit

Sun, w h e r e i t moves f r o m t h e s o u t h e r n t o t h e n o r t h e r n h e m i s p h e r e .

of

the

In o u r

t i m e s t h i s h a p p e n s every y e a r a b o u t March 20. Right a s c e n s i o n and d e c l i n a t i o n t o g e t h e r f o r m t h e (inertial) equatorial system. E x c e p t f o r t h e a b o v e - m e n t i o n e d r e f e r e n c e f r a m e s s e v e r a l o t h e r s a r e in c o m mon u s e , f o r i n s t a n c e t h e ecliptic coordinates. They d i f f e r f r o m t h e e q u a t o r i a l o n e s only in t h e choice of t h e f u n d a m e n t a l p l a n e which in t h e s e n t c a s e is t h e e c l i p t i c . latitude.

pre-

The p a r a m e t e r s a r e c a l l e d e c l i p t i c l o n g i t u d e

The f o r m e r is a g a i n c o u n t e d w i t h r e s p e c t t o t h e v e r n a l

and

equinox,

in t h e d i r e c t i o n of t h e ( a p p a r e n t ) s o l a r m o t i o n . Both s y s t e m s a r e i n c l i n e d t o each o t h e r by an a n g l e of a b o u t 2 3 ° 2 6 ' 2 1 " (in the year 2000),

t h e obliquity of the ecliptic. We s h a l l n o t d i s c u s s

here

t h e f o r m u l a s r e q u i r e d f o r t h e t r a n s i t i o n f r o m one s e t of c o o r d i n a t e s t o t h e o t h e r ; t h e y can be looked up in N e u t s c h & S c h e r e r [1992] o r , w i t h n u m e r i c a l e x a m p l e s and d e t a i l e d i n s t r u c t i o n s , in C u r n e t t e & Woolley [19741. Among m a ny a l t e r n a t i v e s ,

t h e y may a l s o be o b t a i n e d f r o m t h e i n v e s t i g a t i o n s of

part

Β of t h i s book. By t h e way, in t h e l i t e r a t u r e j u s t c i t e d one may f i n d m o r e p r e c i s e i n f o r m a t i o n on t h e t e m p o r a l

v a r i a t i o n of

the position

of t h e e q u i n o x e s .

This

is

14

1. H i s t o r i c a l Development of t h e C o o r d i n a t e Concept

c a u s e d by t h e precessional motion of t h e E a r t h ' s a x i s , Hipparchus.

also discovered

T h e s a m e h o l d s t r u e f o r t h e v a r i a b i l i t y of t h e e c l i p t i c a l

quity.

It g o e s b a c k t o g r a v i t a t i o n a l perturbations by Moon a n d Sun.

Since

all

astronomical

coordinate

systems,

including

( t h e y r e f e r t o t h e s y m m e t r y p l a n e of o u r G a l a x y ) ,

the

galactic

a r e of t h e s a m e

t h e s e s h o r t a n d r u d i m e n t a r y r e m a r k s s h o u l d be s u f f i c i e n t .

by

obli-

ones type,

1.2.

A f f i n e and P r o j e c t i v e

Geometry

15

1.2. Affine and Projective Geometry

In h i s p r i n c i p a l

work

"Elements", Euclid (Eukleides, c a .

3 0 0 BC)

develops

geometry a s a s e l f - c o n t a i n e d l o g i c a l s y s t e m of t h e o r i e s w h i c h is b u i l t on a f e w b a s i c a s s u m p t i o n s (axioms). A m o n g t h e 13 b o o k s of t h e " E l e m e n t s " m a i n l y t h e f i r s t f o u r a r e of to us.

T h e y p o s s i b l y go b a c k t o e a r l i e r

of Chios ( 2 n d h a l f

authors,

of t h e 5 t h c e n t u r y BC),

in p a r t i c u l a r

interest

Hippocrates

and a r e concerned with t h e

geo-

m e t r y in t h e (Euclidean) plane. R u l e r and c o m p a s s e s a r e t h e only a l l o w e d m e a n s to c o n s t r u c t new g e o m e t r i c objects,

a n d t h e y h a v e t o b e a p p l i e d in a d e f i n i t e w a y ( n a m e l y in o r d e r

d r a w s t r a i g h t l i n e s o r c i r c l e s of w h i c h t h e c e n t r e a n d o n e p e r i p h e r a l a r e given).

Coordinates are not introduced.

T h e c o r r e c t n e s s of a

t i o n i s u s u a l l y p r o v e n w i t h t h e h e l p of c o n g r u e n c e t h e o r e m s f o r Only R e n i

Descartes ( 1 5 9 6 - 1 6 5 0 )

quantitatively.

tried

to

characterize

points

This enabled him to d e s c r i b e g e o m e t r i c a l

to

point

constructriangles.

in t h e

plane

o b j e c t s and

their

i n t e r r e l a t i o n s n u m e r i c a l l y a n d t o r e p l a c e t h e E u c l i d e a n a r g u m e n t s by s i m p l e calculations.

Later,

were deduced.

from

this

discussed certain precursory In m o d e r n

approach

the

Cartesian coordinate systems

Though they a r e called a f t e r him,

terms,

the

Descartes himself

merely

steps.

introduction

of

Cartesian

coordinates

identifies

p o i n t s in t h e E u c l i d e a n p l a n e w i t h t h e e l e m e n t s of t h e t w o - d i m e n s i o n a l

the (Eu-

2

clidean) vector space IR . of

points

conditions,

coinciding while

Straight

with them,

circles

are

equations (for the details, In m a n y r e s p e c t s , On t h e o n e h a n d ,

cf.

lines

which

are turn

associated chapter

represented out

with

to

by t h e

be s o l u t i o n s

special

kinds

of

collection of

7).

t h i s i d e a h a s b e e n of e m i n e n t

influence to

mathematics.

t h e t r e a t m e n t of g e o m e t r i c p r o b l e m s by c o o r d i n a t e s

facilitates the constructions,

on t h e o t h e r h a n d ,

c e s s i b l e o n l y in t h e e x t e n d e d

theory.

One of t h e m i s t h e q u e s t i o n ,

w h i c h f i g u r e s in t h e p l a n e c a n be d r a w n

ruler

and compasses.

linear

quadratic

often

numerous aspects are

It r e d u c e s t o f i n d i n g out w h e t h e r

ac-

with

the coordinates

of

a l l p o i n t s t o b e c o n s t r u c t e d c a n b e c a l c u l a t e d f r o m t h e g i v e n o n e s by

solv-

ing

basic

of

linear

or

quadratic

equations,

i.e.

by

combining

a r i t h m e t i c a l o p e r a t i o n s w i t h t h e e x t r a c t i o n of s q u a r e

roots.

the

four

1. H i s t o r i c a l Development of the Coordinate Concept

16

This automatically leads to number theoretical problems. We just mention a f e w famous classical examples which have been shown to be insoluble within the framework of

Euclidean constructions:

The Delian problem (to find a

cube whose volume is twice as much as that of a given one), the trisection of an arbitrary angle and squaring the circle. the highlights in this f i e l d is the construction of the regular 17-

One of

gon by Gauß. His investigation exerted a strong impetus on the further development of modern number theory. S t i l l another expansion,

lying closer to the heart of geometry i t s e l f ,

made possible by the introduction of coordinates, (plane algebraic)

nition and classification of

was

namely the precise d e f i -

curves.

We shall

discuss

it

thoroughly in chapter 7. Here we merely state that the curves of second degree are in essence identical with the conic sections which had already been studied systematically e a r l i e r by Apollonius of Perge (ca. 260 - ca. 190 BC). Even much more important than these achievements

is the

epistemological

gain due to Descartes' formulation. The above representation t r a n s f e r s the 2 a f f i n e plane IR

into a model of the Euclidean plane.

In this way the con-

sistency problem of Euclidean geometry reduces to that of the real numbers. We do not want, however, to comment on the dispute which was kindled between

d i f f e r e n t philosophical

intuitionism, Instead,

schools within mathematics

(constructivism,

etc. ) by this question.

we p r e f e r to consider another d i f f i c u l t y which is connected

with

one of Euclid's axioms, namely the (in)famous parallel postulate. It may be stated as f o l l o w s : "For each ( s t r a i g h t ) line g and every point Ρ not coinciding with g,

there

is one and only one line h which contains Ρ and is disjoint to h.

It

is

called the p a r a l l e l to g through P. " This assumption is incomparably more complicated and much less evident than all other axioms of Euclid. This suggested that the statement just mentioned could possibly

be a consequence of the remaining postulates.

then be unnecessary,

or better:

It

would

should be considered as a geometric

theo-

rem. Every attempt to prove it f a i l e d , though in some cases this was by no means clear.

A thorough analysis,

gaps or inconsistencies.

however,

revealed f o r all published

arguments

Most of them silently used an assumption which in

some sense was expressed in simpler terms than the original parallel postulate,

but logically said the same.

1.2. A f f i n e and P r o j e c t i v e Geometry

At any rate,

17

in this way several equivalent f o r m s of the proposition under

discussion were discovered,

as f o r example the well-known theorem on the

angle sum in a planar triangle (180°). Gauß was the f i r s t who concluded the parallel axiom might perhaps be independent of the others.

In order to more clearly present the considerations

to f o l l o w , we shall work with modern terminology. In particular, we denote as absolute geometry those parts of without making use of parallels,

Euclidean theory which can be proved

and thus do not require the incriminated

postulate. The question thus can be expressed b r i e f l y as f o l l o w s : a proper part of Euclidean geometry (and if

so,

Is absolute geometry

which statements does it

include?) or are both identical with each other? Actually, dent.

the f i r s t alternative is true:

Gauß proves

the axiom of parallels is indepen-

this by seeking realizations

of

absolute geometry

for

which the parallel postulate does not hold. Without knowledge of Gauß' unpublished investigations

three decades b e f o r e ,

Nicolai

ky (1792-1856) and Jänos Bblyai (1802-1860) f o l l o w e d path.

Two

versions of

Ivanovic essentially

Lobachevsthe

this non-Euclidean geometry are conceivable

same (there

may be either infinitely many parallels to g passing through Ρ or none; cordingly one speaks of hyperbolic or e l l i p t i c geometry).

ac-

Very simple mo-

dels have been found by Christian Felix Klein (1849-1925) and Jules-Henri Poincarfe (1854-1912).

They represent

the

(hyperbolic)

non-Euclidean

plane

by a part of the Euclidean one and hence v e r i f y the consistency of the l a t ter if that of the f o r m e r is taken f o r granted. Thus the existence of Euclidean geometry implies that of its non-Euclidean counterpart (the e l l i p t i c variant can be constructed similarly). These considerations again show the immense influence of the coordinate approach on our understanding of geometry. that

the s p e c i f i c properties

of

Nevertheless it soon turned out 2 the a f f i n e plane IR involve the necessity

to discuss certain exceptions separately.

For instance,

equation of a straight line g in a Cartesian ( x , y ) - f r a m e ! in various ways,

as e . g .

let us examine the We may express it

the well-known p o i n t - s l o p e - f o r m or the intercept

f o r m ; see chapter 7. Neither is able to describe all straight lines. to the y - a x i s and its parallels; origin

The f i r s t cannot be applied

the second f a i l s if g is incident with the

(0,0).

The complication caused by this is undesirable. the

inevitable

discrimination

between

parallel

The same also applies and

intersecting

pairs

to of

18

1. H i s t o r i c a l Development of the Coordinate Concept

lines in a f f i n e geometry. To avoid both drawbacks,

Jean Victor Poncelet (1788-1867) extends the a f -

f i n e plane by a number of "ideal" of "improper" points. As we shall learn in chapter 7, each of them is incident with all lines of of

parallels,

but not with the others.

a particular

All improper points together

set form

the "infinite" (straight) line. The addition plane,

of

the ideal

and the special

instance,

elements enlarges the a f f i n e to the

cases described

above "vanish

projective

into thin air".

For

two lines which are parallel in the a f f i n e interpretation now in-

tersect at infinity. But we have achieved much more. In the p r o j e c t i v e interpretation, two a r b i t r a r y straight lines always have precisely one point in common,

exactly in

correspondence with the still valid observation that two points determine a unique connecting line. Anyhow, in the geometry of the p r o j e c t i v e plane there is a p e r f e c t symmetry between points and lines,

expressed by the duality principle of

Poncelet

(theorem 7. 1. 1). The price to be paid f o r this improvement is extremely low. sists in the necessity

It mainly con-

to use three homogeneous or projective

coordinates

instead of the two a f f i n e (Cartesian) ones. The parameters themselves are irrelevant; to

the

what counts is their proportion.

so-called

barycentric calculus

of

This idea can be traced back

August

Möbius (1790-

Ferdinand

1868). The description of

points,

lines and incidence relations

in the

projective

plane does not leave anything to be desired. Things change, however,

if we

want to consider curves of higher degree. In the p r o j e c t i v e theory they are characterized as zero sets of homogeneous polynomials ( i . e .

forms). A prob-

lem is that not every algebraic equation with real c o e f f i c i e n t s has a solution in R. To

circumvent

the

closed f i e l d , e . g .

difficulty

one

has

to

replace

R by

an

algebraically

that of the complex numbers (C). In the p r o j e c t i v e plane

over C all considerations can be carried through systematically without e x ceptions. As long as continuity arguments are irrelevant, we could construct a geometry equally well over any other algebraically tic 0,

for

instance over A (set of

addition the existence of

models of

all

closed f i e l d of

algebraic

numbers).

characteris-

This shows in

Euclidean and non-Euclidean

which d i f f e r essentially f r o m the "naive" approach.

theories

1 . 2 . A f f i n e sind P r o j e c t i v e Geometry

Of

course,

dimensional

one

neither

structures

need

nor

(planes).

19

will

restrict

the

investigation

When w r i t t e n in c o o r d i n a t e s ,

to

t h e r e is

s o l u t e l y no d i f f i c u l t y t o t r a n s f e r a l l t e r m s a n d r e l a t i o n s t o h i g h e r sional spaces. In t h e

two-

We s h a l l d i s c u s s t h i s m a t t e r a t t h e end of c h a p t e r 7.

following

period

generalizations

in d i f f e r e n t

directions

have

developed. They gave r i s e to c o m p l e t e l y new and e x t e n s i v e m a t h e m a t i c a l ciplines

ab-

dimen-

(present-day

algebraic

geometry,

braic manifolds, schemes and so on).

theory

of

Unfortunately,

have enough room to account f o r them properly.

varieties,

i. e.

been disalge-

in t h i s book w e do n o t

20

1. H i s t o r i c a l Development of the Coordinate Concept

1.3. D i f f e r e n t i a l Geometry The f i r s t attempts to develop some kind of

infinitesimal calculus go

far

back to classic antiquity. Archimedes (ca. 287-212 BC) succeeds in evaluating volumes and surface a r e as

of

simply-shaped

solid

bodies

like

spheres,

circular

cylinders

and

cones. To that end, he employs the so-called exhaustion method. Clearly these early precursors of

integration theory cannot be viewed

as

systematic procedures. The same holds true f o r the work of Johannes Kepler (1571-1630) or Bonaventura Cavalieri (1598-1647). Both, Kepler's "Faßregel" (invented to determine the volume of wine casks, thus serving a very important practical purpose),

and C a v a l i e r i ' s principle,

proposed in 1635,

are

transformations of the Archimedian contemplation to more general bodies. At about the same time, d i f f e r e n t i a l calculus is developed by several maCrucial intermediate steps are the methods to find the tan-

thematicians.

gent to a curve at a given point by Kepler

(applications to celestial

me-

de Fermat (1601-1665).

But these considerations

also

are based mostly on ad-hoc-considerations though gradually a general

con-

chanics! ) and Pierre

cept begins to emerge. The synthesis of both trends is contained in the contemporary but independent investigations of

G o t t f r i e d Wilhelm Leibniz (1646-1716) and Isaac New-

ton (1643-1727). Newton

gets his f i r s t results concerning

d i f f e r e n t i a l calculus

ver publishes anything about it. His "Principia", mechanics, style.

(or,

as he

the theory of fluxions) around 1671, but during his l i f e t i m e ne-

called it:

in which he founds modern

do not require limiting processes and argue in classical

Later,

as a counterpart,

Greek

the theory of fluentes (= integration)

fol-

lows. Although Leibniz only in 1684 comes to d i f f e r e n t i a t i o n and 1686 to the integral concept,

he quite early brings his achievements to the attention

of

the s c i e n t i f i c community. T h e r e f o r e the extremely unpleasant and disgusting priority

quarrel,

which

the f o l l o w i n g years,

is picked by some people

is groundless.

of

limited

intellect

(who at least in the beginning do not take part in the b a t t l e ) pursue tally d i f f e r e n t paths.

in

This is the more so since both authors to-

1.3.

Differential

21

Geometry

From the t e c h n i c a l

point

of

view

Leibniz'

interpretation

turned out

to

be

f a r b e t t e r because it is much more g e n e r a l l y a p p l i c a b l e than the m o r e physical

ideas

of

Newton.

differentiation

(d)

A large

part

and i n t e g r a t i o n

of

his n o t a t i o n ,

(J"),

is s t i l l

like

the

in use t o d a y .

symbols

for

This

an-

is

o t h e r r e a s o n f o r the success of his i n t e r p r e t a t i o n in comparison t o t h a t

of

his opponent. Only in the l a s t f e w decades N e w t o n ' s a n a l y s i s has seen a p a r t i a l sance"; The

some r e c e n t developments can be v i e w e d as i t s

new

calculus

is a l m o s t

hydrodynamics by Leonhard

field

(1736-1813), bi

are

Euler

Pierre

Simon

applied

(1707-1783),

as to c e l e s t i a l

and o t h e r s as w e l l latter

immediately

descendants.

physics,

especially

Bernoulli

Daniel

mechanics.

Laplace

to

Eminent

(1749-1827),

"renais-

contributors

Joseph

to

(1700-1782) to

the

Lagrange

Louis

W i l l i a m Rowan Hamilton (1805-1865) and Carl Gustav Jacob Jaco-

(1804-1851).

During t h i s p e r i o d tions

of

one r e a l

the or

investigations

complex

are

variable.

in the

justification

rentiable

or

mappings.

explanation

only

restricted

The f u n c t i o n concept

d e v e l o p e d beyond an i n t u i t i v e understanding. out

large

to

func-

itself

is

Most s c i e n t i s t s c o n s i d e r

smooth,

i.e.

infinitely

not

with-

often

diffe-

The c o n f u s i o n caused by the u n r e f l e c t e d usage o f

d e f i n e d t e r m s is c l a r i f i e d only

ill-

gradually.

We o w e the n e c e s s a r y s y s t e m a t i z a t i o n and s c h e m a t i z a t i o n of the r e l e v a n t m a thematical

objects

mainly

to

Cauchy

Augustin-Louis

(1789-1857)

and

Karl

T h e o d o r Wilhelm WeierstraS (1815-1897). It is now most i m p o r t a n t to e s t a b l i s h a t h e o r y of higher dimensional and,

in connection w i t h this,

out in d i f f e r e n t

f u n c t i o n s of

many v a r i a b l e s .

dehnungslehre"

(Graßmann [1844])

spaces in any ( f i n i t e ) On the

other

hand

proposes

in his f a m o u s " L i n e a l e

an

algebraic

Initially assumed. sential

(smooth)

an e x p l i c i t

mathematicians

develop,

which

of

Ausvector

partially

stimulated

1.4),

by

differential

surfaces. embedding

in t h r e e - d i m e n s i o n a l

Gauß is the f i r s t who f r e e s himself

tool

theory

dimension.

many

p r o b l e m s and open questions of t h e o r e t i c a l geodesy (sec. geometry of

carried

directions.

Hermann Günther GraBmann (1809-1877)

Firstly,

spaces

This is

enables

him t o

do so,

are

of

Euclidean

space

this restriction.

intrinsic

IR

The

(Gaussian)

is es-

coordi-

nates. Even if dered

external

important,

q u a n t i t i e s which depend on the embedding a r e s t i l l Gauß points out that inner

properties

have a much

consimore

1. H i s t o r i c a l Development of the Coordinate Concept

22

fundamental meaning.

This is because they can be defined and in principle

also measured within the surface i t s e l f .

(The demand of practical

bility is always clearly visible in Gauß' work).

applica-

The abstraction f r o m sur-

rounding space is a major step f o r w a r d in comparison with older theories. Finally.

Bernhard Georg Friedrich Riemann (1826-1866) puts both threads t o -

gether and builds a unified theory of functions. His definition of the word "manifold", however,

is not exactly identical with today's.

ferences is the assumption of a metric.

By and large,

One of the d i f -

his conception

cor-

responds to what we nowadays call a Riemannian space. One cannot say any more whether Riemann admits manifolds whose atlases (= collections of local coordinate systems) contain more than one chart or if he r e s t r i c t s to those spaces f o r which one global reference f r a m e s u f f i ces. The reason is that he presents his theory to the public in his inauguration lecture,

Riemann [1854].

He has to show consideration f o r the non-

mathematicians in the audience and hence avoids too technical deductions. But this is not the decisive f a c t o r .

All essential assumptions are

already

included, even if several more decades have to go by until the modern manif o l d concept emerges, mainly through contributions f r o m El win Bruno Christ o f f e l (1829-1900),

Eugenio Beltrami (1835-1900),

Ricci-Curbastro

Gregorio

(1853-1925) and Tullio Levi-Civitä (1873-1941). During this phase a tendency to increasing abstraction (e.g.

weaker

attempt

to

differentiability

discuss

all

conditions

ramifications

history up to the beginning of the

20 t h

of

etc.) the

and

is observed. development;

century, c f .

generalization We shall

not

concerning

its

Scholz [1980].

Meanwhile there is a trend towards topology, though already very early a l gebraic aspects play a leading role,

at least occasionally.

We merely men-

tion the absolute calculus of £ l i e Joseph Cartan (1869-1951). with the technical tools f o r many geometric

It endows us

investigations.

In this connection the ideas exhibited by Marius Sophus Lie (1842-1899) in a series of very influential papers on d i f f e r e n t i a l equations are of

utmost

importance to us. The basic innovation is to link up algebraic

and topological

resp.

(diffe-

r e n t i a l ) geometric structure. To this end he considers a smooth manifold on which a product is defined which obeys the group laws.

Multiplication and

inversion are required to be smooth operations. These o b j e c t are now called Lie groups. They very o f t e n come into play naturally

when

dealing

with

mathematical

branch of physics can do without them.

or

physical

problems.

Today

no

1.3. Differential Geometry

23

A l r e a d y Lie [1970] h i m s e l f n o t e s t h a t an a n a l y t i c a t l a s c o m p a t i b l e w i t h g i v e n one c a n a l w a y s be f o u n d ( c f .

chapter

3).

On t h e o t h e r

hand,

the much

w e a k e r d i f f e r e n t i a b i l i t y c o n d i t i o n s s u f f i c e . T h e c u l m i n a t i o n of t h e

accord-

i s t h e p o s i t i v e s o l u t i o n of H u b e r t ' s 5 t h Problem, s e e

section

ing r e s e a r c h 6. 7.

T h e s t a r t i n g p o i n t f o r Lie a n d h i s s u c c e s s o r s is t h e n e a r l y b i j e c t i v e t i o n s h i p b e t w e e n Lie g r o u p s and Lie algebras. As v e c t o r s p a c e s ,

the

relalatter

a r e much e a s i e r a c c e s s i b l e t h a n t h e m a n i f o l d s t h e m s e l v e s . For

example,

groups,

the

classification

w h i c h may be c o n s i d e r e d

o t h e r s c a n be c o n s t r u c t e d , bras.

of

the

as the

finite

dimensional

simple

Lie

"building blocks" f r o m which

i s r e d u c e d t o t h e a n a l o g o u s t a s k f o r Lie

all

alge-

I t s s o l u t i o n w a s a c h i e v e d by Wilhelm K a r l J o s e p h Killing (1847-1923)

and C a r t a n ( K i l l i n g [1888-1890],

Cartan

[1952-1953]).

24

1. H i s t o r i c a l Development of t h e C o o r d i n a t e Concept

1.4. G e o d e s y a n d Cartography

The m a i n p r o b l e m s of geodesy a r e t h e d e t e r m i n a t i o n of f i g u r e and

gravity

f i e l d of t h e E a r t h . After earlier tentative considerations,

t h e E a r t h m i g h t be a disk s w i m m i n g

in t h e o c e a n , Pythagoras and Aristotle ( 3 8 4 - 3 2 2 BC) p r o p o s e d i t s

spherical

s h a p e . Though a c c e p t e d by m o s t s c h o l a r s , t h i s s u g g e s t i o n w a s d i s p u t e d c o n t r o v e r s i a l l y d u r i n g t h e whole of a n t i q u i t y and well i n t o t h e Middle Ages. The f i r s t r e l e v a n t q u a n t i t a t i v e r e s u l t c o n c e r n i n g t h e s i z e of t h e E a r t h obtained

by Eratosthenes of Cyrene (ca.

284 - ca.

2 0 0 BC).

His

was

starting

p o i n t w a s t h e o b s e r v a t i o n t h a t in t h e end of June t h e s u n l i g h t w a s r e f l e c t ed in a deep f o u n t a i n n e a r Syene (Assuan).

The s o l a r

culmination

summer solstice at this place thus occurred close to the zenith, t h e same t i m e in A l e x a n d r i a t h e Sun s t o o d a b o u t 7°12',

i.e.

around

while

at

one f i f t i e t h of

the complete circle to the South. From the

approximately

known d i s t a n c e

between the two

locations

which

a m o u n t s t o c a . 5 0 0 0 s t a d i a , he deduced t h e c i r c u m f e r e n c e of t h e E a r t h t o be nearly 250000 stadia.

Within t h e p r e c i s i o n of m e a s u r e m e n t ,

he l a t e r

cor-

r e c t e d t h i s value t o 2 5 2 0 0 0 s t a d i a , g e t t i n g a r o u n d n u m b e r , namely 7 0 0 s t a dia, f o r t h e l e n g t h of one d e g r e e (Hoppe [1966]). U n f o r t u n a t e l y t h e r e a r e two obstacles to t r a n s l a t i n g this into the system. First,

metric

it is n o t known which of t h e v a r i o u s s t a d i u m s in use a t t h e

t i m e E r a t o s t h e n e s employed. F u r t h e r m o r e , we c a n n o t say any m o r e how he c a l culated the distance.

P r o b a b l y he d e r i v e d it f r o m t r a v e l t i m e s a n d / o r

road

lengths. The a c t u a l d i s t a n c e a s t h e c r o w f l i e s b e c a u s e of t h e u n a v o i d a b l e b e n d s

is

clearly considerably shorter.

If and how E r a t o s t h e n e s t o o k t h i s e f f e c t i n t o

consideration

the

and

whether

distance

of

4325

stadia

proposed

Wolf [1973] ( t h e a u t h o r does n o t i n f o r m us a b o u t t h e b a s i s of h i s t i o n s ) is b e t t e r ,

may be l e f t u n d e c i d e d .

In any c a s e ,

by

calcula-

one g e t s a v a l u e f o r

t h e E a r t h ' s c i r c u m f e r e n c e which is in a c c o r d a n c e w i t h t h e t r u e n u m b e r w i t h in 5. . . 10 %. In view of t h e p r i m i t i v e m e t h o d s which w e r e a t command t h i s is a r e m a r k a b l e

Eratosthenes'

accuracy.

It t o o k a n u m b e r of c e n t u r i e s u n t i l f u r t h e r s t e p s f o r w a r d could be made. Only in 1669 C h r i s t i a a n Huygens (1629-1695) e n r i c h e d geodesy by a f u n d a m e n -

1.4. Geodesy and Cartography

t a l l y new aspect.

25

He noted that the daily rotation of the Earth generates

centrifugal forces.

This leads to an enlargement of

(a ) in comparison with the polar

radius (a p ).

the equatorial

radius

Hence the t e r r e s t r i a l

sur-

f a c e should resemble a flattened e l l i p s o i d much more than a sphere. Newton calculated f r o m the available data an expectation value of the f l a t tening f = (a E -a p )/a E of about 1:230. He assumed the solid Earth to be p e r f e c t l y elastic. This is not a good approximation, as we know today.

There-

f o r e the real f l a t t e n i n g is less than the theoretical. To decide the controversy if the f i g u r e of the Earth is spherical (a p = a E ) or can be described better

by an oblate (a

prolate spheroid

the Paris Academy of

(a p > a^.),

ρ

< a ) or, Ε

on the contrary,

Sciences dispatched

a

two

l a r g e - s c a l e expeditions. They got order to carry through measurements of as large parts of the meridian as possible,

one in the vicinity of the equator

(Peru), the other in a northern t e r r i t o r y

(Lapland).

In the course of this, numerous points along the meridian were marked and the r e l a t i v e positions defined by triangulation.

A direct measurement of an

arbitrary triangle side yields the scale f a c t o r and consequently the length of the arc (which a f t e r w a r d s is transformed to sea-level). A comparison with the astronomically determined latitude d i f f e r e n c e between the northernmost and southernmost grid points e. g. gives the distance c o r responding to one degree. America.

In Norrbottsn this is much more than in Latin

This immediately corroborates the flattening of the Earth.

P a r t i a l l y due to lack of experience and sloppiness, many problems arose during both expeditions. A detailed and readable account of

the d i f f i c u l t i e s they had to

can be found in Wolf [1973], In this book the reduction methods,

surmount intermedi-

ate and f i n a l results are listed also. In

the

17th,

18th

initiated similar

and

early

19th

triangulations.

centuries,

several

European

sovereigns

They were aimed at constructing

of f i r s t order net points with best possible accuracy.

systems

Other less important

points were selected and added later by relative positioning. Among the many land surveyings of this kind, some are of eminent historical significance.

The f i r s t genuine triangulation is the Dutch Survey (meridian

between Alkmaar and Breda) which started in 1614 under the supervision of Willebrord Snell (Snellius,

1591-1626).

He already employed essentially

methods which became standardized later. extremely tedious and laborious, yet.

The numerical calculations

all

were

since logarithm tables were not available

1. H i s t o r i c a l Development of the Coordinate Concept

26

Snell's procedure has been applied on many other occasions as well,

so we

need not comment on them in greater detail. New ideas were brought in only much later. Mainly the Hannover Survey under GauB has to be acknowledged in this respect. are introduced in one stroke: to be mentioned. observations

Here two important

Firstly the invention of

novelties

the heliotrope has

It consists of a pivoted mirror which allows f o r

over

very

long distances by r e f l e c t e d

sunlight.

precise

In this

way

much larger triangles can be measured directly. Furthermore, the instrument serves as an optical telegraph and thus accelerates the transmission of

in-

f o r m a t i o n and orders considerably. Even much more important, however, is the consequent employment of the e x tremely useful method of least squares.

It had been discovered by Rudjer

Josip Boskovic (1711-1787) and was developed further by Adrien-Marie Legendre (1752-1833). Gauß [1823] himself brings the idea to perfection.

In com-

bination with his law of error propagation, it is the rigorous mathematical foundation of the great survey of the Kingdom of Hannover. All f i r s t order triangles

are

subjected

much higher degree of

to

the

analysis

simultaneously,

thus

achieving

a

consistency.

Friedrich Wilhelm Bessel (1784-1846) combines all available results of

sur-

veys and deduces f r o m them the closest spheroidal approximation to the true f i g u r e of the Earth (by a least-square f i t ) . Later investigations, bers.

based on improved data, yield slightly d i f f e r e n t num-

Nevertheless Bessel's spheroid is used up to the present in Western

Europe as the r e f e r e n c e surface f o r the GauB-Kriiger coordinates (see below and chapter 29), while the Eastern European countries p r e f e r the ellipsoid of

Krassowski.

Of the remaining models of the Earth which we shall discuss elsewhere in more detail,

we have to mention the so-called International Ellipsoid,

Hayford [1909]. It is also the fundament of many o f f i c i a l map systems. veral

international

organizations

like the IAU or the IUGG propose

cf. Setheir

own r e f e r e n c e systems. The exact dimensions can be looked up in chapter 29 Since,

caused by topographic peculiarities,

strongly f r o m any spheroid,

the t e r r e s t r i a l

surface d i f f e r s

it is not at all important which of the above-

mentioned surfaces is taken as the basis f o r cartography and positioning. Much smoother are the surfaces of constant total potential U of the Earth. U is the sum of a purely gravitational part U

and a component U

ing

by

9

the

centrifugal

acceleration

Laplace [1799] finds that U

produced

the

diurnal

obeys the potential equation AU

ζ

describrotation.

= 0 and can

27

1 . 4 . Geodesy and C a r t o g r a p h y

be c o n v e n i e n t l y

developed w i t h r e s p e c t

to a suitably

chosen

base

of

the

s p a c e of spherical harmonics ( c h a p t e r 23). The f i r s t 10 or 15 o r d e r s (a f e w h u n d r e d t e r m s ) of t h i s s e r i e s can be d e duced r e l i a b l y quantify the

f r o m t h e o r b i t s of large-scale

structure

artificial of

satellites.

the gravitational

All t o g e t h e r

they

field.

per-

Local

t u r b a t i o n s which depend on i r r e g u l a r i t i e s of t h e m a t t e r d i s t r i b u t i o n in t h e Earth's crust

could be seen only in much h i g h e r o r d e r s .

In l i m i t e d

areas

(most f a m o u s e x a m p l e : t h e n o r t h e r n slope of t h e H a r z Mountains) plumb line deviations of 10" and m o r e f r o m t h e " e x p e c t e d " value a r e q u i t e common. One of

t h e equipotential surfaces U = c o n s t . , namely t h e p a r t i c u l a r

passing through a well-defined point,

is known as t h e geoid. In t h i s

n e c t i o n gravimetric measurements a r e a l s o of g r e a t

one con-

importance.

A l t i t u d e d e t e r m i n a t i o n s a r e c a r r i e d out r e l a t i v e t o t h e geoid a l o n g t h e

lo-

cal U - g r a d i e n t .

fix

U n f o r t u n a t e l y t h e r e is no u n i v e r s a l a g r e e m e n t how t o

the zero height either.

The s t a t e s of W e s t e r n E u r o p e r e f e r t o an

average

w a t e r - l e v e l in t h e p o r t of A m s t e r d a m ( c l e a r e d f r o m t i d a l and m e t e o r o l o g i c a l e f f e c t s ) . On t h e o t h e r hand, E a s t e r n E u r o p e a n c o u n t r i e s u s e t h e

Kronstadt

g a u g e . The l a t t e r is a b o u t 2 dm h i g h e r t h a n t h e f o r m e r . In Germany t h e z e r o level (Normalnull, NN) by c o n v e n t i o n lies e x a c t l y 37 m below a c e r t a i n m a r k n e a r B e r l i n ("Normalhöhenpunkt"), and so on. Until r e c e n t l y t h e t r i g o n o m e t r i c p r o c e d u r e w a s t h e only way t o a c h i e v e a p o s i t i o n i n g a c c u r a c y of c e n t i m e t r e s over d i s t a n c e s of s e v e r a l h u n d r e d metres.

(The t y p i c a l s i d e l e n g t h s of t r i a n g l e s of f i r s t o r d e r a r e c a .

kilo30 to

60 km). In t h e l a s t f e w y e a r s new and t o t a l l y d i f f e r e n t m e t h o d s w o r k i n g t o a s i m i l a r d e g r e e of p r e c i s i o n have been f o u n d . We j u s t m e n t i o n in p a s s i n g t h e d i rect

determination

of

distances

with

laser

light

which is r e f l e c t e d

at

a

p r i s m a t i c m i r r o r . From the precisely measurable travel time f r o m the source t o t h e r e f l e c t o r and back a g a i n ,

one i m m e d i a t e l y g e t s t h e d i s t a n c e of

both

points. Thus i n s t e a d of t r i a n g u l a t i o n s ,

trilaterations can be used a s w e l l . The a c -

c u r a c y is n o t l i m i t e d by t h e clock, but by t h e unknown v a r i a t i o n of t h e r e f r a c t i o n i n d e x of t h e a i r which can only be e s t i m a t e d .

Uncontrollable

ef-

f e c t s of t h e w e a t h e r make t h e c a l c u l a t i o n of refraction and t h e velocity of light w i t h i n t h e a t m o s p h e r e d i f f i c u l t . The s a m e h o l d s t r u e f o r t h e l a s e r r e f l e c t o r s which have been e s t a b l i s h e d on t h e Moon d u r i n g s e v e r a l s p a c e m i s s i o n s . a n c e up t o a f e w cm. Here, h o w e v e r ,

They a l s o a l l o w t o f i n d t h e

beyond t h e a b o v e - m e n t i o n e d

dist-

problems

1. H i s t o r i c a l Development of the Coordinate Concept

28

The classical Lunar theories are at best

t h e r e is one more obstacle.

to p r e d i c t the Moon's position with an e r r o r of about 1 m, w h i l e i n t e g r a t i o n which takes r e l a t i v i s t i c time

intervals)

a little

better.

forces

into account does

Nevertheless

there

is

(over

still

able

numerical limited

a gaping

of one or t w o o r d e r s of magnitude between observational and

hole

calculational

accuracy. Radar ranging b a s i c a l l y

applies the same idea.

system in t w o d i f f e r e n t ways:

It is used in the

planetary

e i t h e r the signal is passively r e f l e c t e d by a

s u f f i c i e n t l y l a r g e body (Moon, Mars, Venus), or one employs a s p a c e c r a f t as a m p l i f i e r and a c t i v e

transponder.

satellite geodesy has undergone a rapid development. A f e w dozen

Recently,

time signal t r a n s m i t t e r s in e x a c t l y known Earth o r b i t s serve as a standard. If f o r an o b s e r v e r on the ground three or more of them are above the h o r i zon at the same time,

it is possible to deduce the exact position f r o m the

a r r i v a l times of the r a d i o

waves.

The method c a p t i v a t e s because of

its s i m p l i c i t y which makes it possible

f i n d o n e ' s position even in rough t e r r a i n (mountains etc. ) or r e g i o n s out f i x e d landmarks (deserts, maintenance

of

a score

it is not inexpensive.

of

Greenland,

satellites

Antarctica).

with

to

with-

Since it r e q u i r e s the

high-precision

(atomic)

clocks,

The best-known network of this kind is probably

the

Global Positioning System (GPS). Another

high-precision

positions cm,

of

large-size

technique, radio

which

telescopes

allows

to

determine

the

on d i f f e r e n t continents

is the s o - c a l l e d Very Long Baseline Interferometry CVLBI).

It consists

in simultaneously

observing

the radiation

g a l a c t i c ) r a d i o source f r o m various sites. on tape,

all

stations numerically

of

The received

t o g e t h e r with accurate time marks.

r e s u l t s of

Later,

a distant signal

is

its p r o j e c t i o n onto the line of

The r e s o l u t i o n

of

all

and calculate the i n t e r f e r e n c e

recorded

effects. or,

ra-

sight.

these procedures

i n e v i t a b l e t o take the temporal

(extra-

one has to compare the

These y i e l d the baseline length between each pair of o b s e r v a t o r i e s ther,

relative

up to ± 1

is e x t r e m e l y

variability

of

high.

It

the t e r r e s t r i a l

is

therefore

surface

into

consideration. The most important periodic e f f e c t s are tidal forces e x e r t e d by Moon and Sun.

They are p e r c e p t i b l e

even in the " s o l i d " Earth,

to t i d e s with amplitudes of a f e w decimetres. tion of the involved heavenly bodies,

where they g i v e

rise

Since they depend on the mo-

they may be eliminated

numerically.

F u r t h e r m o r e , secular e f f e c t s are d i r e c t l y observable nowadays. The most i m -

29

1 . 4 . Geodesy and C a r t o g r a p h y

p r e s s i v e a m o n g t h e m is t h e continental drift p r e d i c t e d by A l f r e d Wegener (1880-1930). theory

Though

his

(Wegener [1915]),

professional in i t s

colleagues

modern

disguise

at

first

ridiculed

this

a s plate tectonics it

is

one of t h e b e s t - e s t a b l i s h e d and m o s t a c t i v e b r a n c h e s of geophysics. The i n d i v i d u a l l i t h o s p h e r i c p l a t e s move w i t h r e s p e c t t o each o t h e r w i t h t y p i c a l v e l o c i t i e s of a f e w c m / y e a r .

Over t i m e i n t e r v a l s of a d e c a d e o r

t h i s s u m s up t o q u i t e l a r g e a m o u n t s .

so,

In c e r t a i n p l a c e s even 10 c m / y e a r

oc-

cur. T h i s is m o r e t h a n one m i g h t t h i n k . At t h e b o r d e r of t w o n e i g h b o u r i n g p l a t e s the

effect

can

occasionally

be

made

visible.

For

instance,

the

western

p a r t s of I c e l a n d b e l o n g t o t h e N o r t h American, t h e e a s t e r n p a r t s t o t h e E u rasian plate.

Both d r i f t a w a y f r o m each o t h e r by ca.

2 cm a y e a r .

Simple

m a r k i n g s on p a r a p e t s of b r i d g e s and s u c h l i k e (shown on s i g h t s e e i n g a r e s u i t a b l e t o make t h e a c c u m u l a t e d m o t i o n e a s i l y

tours)

discernible.

The r e q u i r e m e n t s of geodesy lead Gauß to w r i t e a s e r i e s of p a p e r s on t h e theory of surfaces.

He i n t e n t i o n a l l y

starts

out

from a totally

different

p o i n t t h a n a l l h i s p r e d e c e s s o r s did. 3 it w a s g e n e r a l l y a s s u m e d t h a t a s u r f a c e 3 is embedded in IR .

B e f o r e Gauß,

It is d e f i n e d a s t h e s e t of a l l s o l u t i o n s of an e q u a t i o n like F ( x , y , z) = 0. Here F d e n o t e s an a r b i t r a r y s m o o t h f u n c t i o n of t h e t h r e e C a r t e s i a n

coordi-

nates. In c o n t r a s t t o t h i s , Gauß n o t e s t h a t g e o d e t i c m e a s u r e m e n t s as e. g.

triangu-

l a t i o n s a r e r e s t r i c t e d t o t h e s u r f a c e of t h e E a r t h or i t s i m m e d i a t e ty.

For t h a t

vicini-

r e a s o n he p u t s t h e e m p h a s i s of his s u r f a c e t h e o r y on

p r o p e r t i e s which do n o t depend on t h e s u r r o u n d i n g s p a c e .

He,

unlike

those Rie-

mann, n e v e r t h e l e s s h o l d s t o t h e e x i s t e n c e of an embedding. Gauß' a p p r o a c h s t a r t s out f r o m a p a r a m e t r i z a t i o n of t h e s u r f a c e g w h i c h , r e q u i r e s two inner o r Gaussian co-

corresponding to its two-dimensionality, ordinates u , v .

The i m p l i c i t

relation

F(x,y,z) = 0 thus

f o r the surface points: assumed t h a t rentiable).

χ = x(u,v);

all f u n c t i o n s x , y , z

is r e p l a c e d

y = y(u,v);

are smooth

S o m e t i m e s only p a r t s of 3

can

by e x p l i c i t

ζ = z(u, ν).

(i.e.

formulas

It is

silently

infinitely often

be d e s c r i b e d in t h i s w a y .

diffeThen

m i n o r m o d i f i c a t i o n s a r e n e c e s s a r y , but they do not c a u s e any s e r i o u s d i f f i culty. Just

this

mainly

formulation

initiated

dimensional case.

by

opens

the

Riemann.

road Gauß

to

differential

himself

only

geometry which

considers

the

is

two-

30

1. H i s t o r i c a l Development of the Coordinate Concept

R e s t r i c t i n g the Euclidean m e t r i c of the embedding s p a c e R 3 to g, we g e t an e x p r e s s i o n f o r the s q u a r e of the a r c element, namely d s 2 = d x 2 + dy 2 + dz Z . If we s u b s t i t u t e the defining equations of x , y , z ,

this b o i l s down to a ho-

mogeneous second d e g r e e polynomial in the c o o r d i n a t e d i f f e r e n t i a l s du and St dv, the 1 fundamental form of 3· According to the d e f i n i t i o n , it is p o s i tive d e f i n i t e and inter a l i a a l l o w s to deduce a v a r i a t i o n a l

condition

for

t h e geodesies o n g .

The sectional curvatures at a c e r t a i n point of g lead by a theorem of J e a n Frederic

Frenet

(1816-1900)

to

another

(possible

indefinite)

quadratic

f o r m , which is known a s the 2 n d fundamental form of g. As a m e a s u r e of the deviation of the s u r f a c e f r o m i t s t a n g e n t i a l plane we may use the Gaussian curvature, i. e. the r a t i o of the determinant of

the

second f u n d a m e n t a l f o r m to t h a t of the f i r s t . The importance of this discovery

cannot be o v e r e s t i m a t e d .

For the

first

time we have found an a b s o l u t e invariant of the s u r f a c e (a quantity which is independent of the choice of the Gaussian c o o r d i n a t e s ) . Probably the most i n f l u e n t i a l r e s u l t of s u r f a c e theory is the f a m o u s theorems egregium, see Gauß [1827], surface

can be c a l c u l a t e d

It b a s i c a l l y s t a t e s t h a t the c u r v a t u r e of

f r o m the m e t r i c and i t s f i r s t

two

a

derivatives

with r e s p e c t to the i n t r i n s i c c o o r d i n a t e s u, v. An i n t e r e s t i n g c o r o l l a r y is t h a t two s u r f a c e s g and S can be developed on (smoothly and i s o m e t r i c a l l y t r a n s f o r m e d into) each other only if the G a u s s ian c u r v a t u r e s in c o r r e s p o n d i n g points of g and δ have the same v a l u e s . This l e a d s to an isometric = length-preserving b i j e c t i o n between g and It

is

simultaneously

angle-preserving

(conformal) a n d

area-preserving.

The

r e v e r s e of the l a s t p r o p o s i t i o n is a l s o t r u e , c f . c h a p t e r s 3 and 4. Already one year e a r l i e r , (Gaussian) curvature,

Gauß found a f o r m u l a f o r the i n t e g r a l

of

the

extended over a t r i a n g l e in g which is bounded by g e -

odesies. Following c o n t e m p o r a r y r o m a n t i c Zeitgeist, he c a l l s it, somewhat b o m b a s t i cally,

"theorema elegantissimum". Its a p p l i c a t i o n s go f a r beyond the r e a l m s

of geodesy and d i f f e r e n t i a l geometry. The integral curvature of a c l o s e d s u r f a c e g, namely the i n t e g r a l over g of the G a u s s i a n c u r v a t u r e ,

can be reduced to this by t r i a n g u l a t i o n .

The o u t -

come is an e x p r e s s i o n which contains only topological invariants like the Euler number o r t h e genus.

This f i r s t connection between local and global a s p e c t s has not been published

by Gauß (except

for

some r e m a r k s

in his l e t t e r s

to

several

col-

1.4. Geodesy and C a r t o g r a p h y

leagues),

31

such t h a t it h a s t o be i n d e p e n d e n t l y f o u n d and p r o v e d a g a i n much

l a t e r by P i e r r e Ossian Bonnet (1819-1892). This r e s u l t , differential

since t h e n known a s t h e theorem of GauB-Bonnet, h a s geometry

like n o t h i n g

else.

Many e x t e n s i o n s

and

shaped

generaliza-

t i o n s up t o t h e p r e s e n t day e n r i c h e d algebraic topology e n o r m o u s l y . But l e t us r e t u r n t o t h e main t o p i c of o u r d i s c u s s i o n !

Often a

pictorial

r e p r e s e n t a t i o n w i t h t h e help of maps is m o s t a p p r o p r i a t e t o i l l u s t r a t e

po-

sition-dependent

de-

tails.

data.

In p a r t i c u l a r ,

this holds f o r local

geographic

A chart (in g e o g r a p h y ) is a o n e - t o - o n e mapping f r o m (some p a r t o f ) 2

t h e E a r t h ' s s u r f a c e o n t o a r e g i o n w i t h i n t h e E u c l i d e a n p l a n e R . The l a t t e r may be t h o u g h t of as a s h e e t of p a p e r o r s o m e t h i n g s i m i l a r . We d e m a n d t h e a s s i g n m e n t t o be i n f i n i t e l y o f t e n d i f f e r e n t i a b l e (in b o t h d i r e c t i o n s ) . Obviously,

there

are

countless

possibilities

to

obey

these

T h e r e a r e no g e n e r a l r u l e s which p r o c e d u r e is t h e b e s t .

conditions.

Depending on t h e

p u r p o s e one m u s t a p p l y d i f f e r e n t c r i t e r i a . We t h e r e f o r e have t o be s o m e w h a t vague and r e s t r i c t o u r s e l v e s t o a couple of p r i n c i p a l m a t h e m a t i c a l

remarks.

F o r t o p o l o g i c a l r e a s o n s a s i n g u l a r i t y - f r e e r e p r e s e n t a t i o n of t h e w h o l e t e r r e s t r i a l s u r f a c e by a s i n g l e c h a r t does n o t e x i s t ,

even if we m e r e l y

requi-

r e c o n t i n u i t y of t h e map and i t s i n v e r s e . F o r t h e moment we a s s u m e t h e E a r t h t o be s p h e r i c a l . The ( G a u s s i a n ) c u r v a t u r e of i t s s u r f a c e is e v e r y w h e r e p o s i t i v e ,

but t h a t of t h e p l a n e i s z e r o .

On a c c o u n t of t h e t h e o r e m a e g r e g i u m i t is i m p o s s i b l e t o c o n s t r u c t a c h a r t which m a p s a f i n i t e s e c t i o n of t h e s p h e r e into t h e p l a n e .

Every c h a r t

of

t h e E a r t h t h u s i n e v i t a b l y is s p o i l t w i t h distortion. N e i t h e r can we a c h i e v e c o n f o r m i t y and a r e a - p r e s e r v a t i o n a t t h e same t i m e , s i n c e t h i s would be e q u i v a l e n t t o i s o m e t r y .

On t h e o t h e r hand,

i t is well

p o s s i b l e t o f u l f i l l one of t h e s e t w o c o n d i t i o n s . Of t h e n u m e r o u s c o n f o r m a l m a p s we h e r e j u s t s k e t c h t h e t w o most important;

historically

f o r more i n f o r m a t i o n see Hoschek [19691.

The s i m p l e s t m a p p i n g p r o b a b l y is t h e stereographic projection. It h a s been d e s c r i b e d a l r e a d y by Hipparchus but may be of even o l d e r o r i g i n . by c h o o s i n g t h e

equatorial

p l a n e a s t h e image r e g i o n

and

We g e t i t

projecting

all

p o i n t s on t h e E a r t h in p e r s p e c t i v e ( f r o m t h e S o u t h p o l e ) . We s h a l l d i s c u s s t h i s t r a n s f o r m a t i o n in g r e a t e r d e t a i l in c h a p t e r 7 and a p ply i t a f t e r w a r d s t o a l a r g e number of g e o m e t r i c p r o b l e m s . Much of i t s i m p o r t a n c e s t e m s f r o m t h e f a c t t h a t it maps c i r c l e s t o c i r c l e s . B e t t e r s u i t e d f o r n a v i g a t i o n is a n o t h e r c h a r t which g o e s back t o

Gerardus

Mercator ( G e r h a r d

which

Kremer,

1512-1594).

He f i r s t

uses

a cylinder

is

32

1. H i s t o r i c a l Development of t h e C o o r d i n a t e Concept

t a n g e n t i a l t o t h e E a r t h a t t h e e q u a t o r a s an i n t e r m e d i a t e i m a g e s u r f a c e on w h i c h he p r o j e c t s f r o m t h e c e n t r e of t h e E a r t h . r o l l e d o u t in t h e

T h e r e a f t e r the cylinder

is

plane.

T h i s Mercator projection i s c o n f o r m a l

as well,

but

is r e g a r d e d

higher

s a i l o r s b e c a u s e i t h a s t h e a d v a n t a g e t h a t t h e c i r c l e s of l o n g i t u d e a n d i t u d e a r e r e p r e s e n t e d by o r t h o g o n a l s y s t e m s of s t r a i g h t l i n e s . f i e s to mark out the course.

The two t e r r e s t r i a l poles,

This

however,

by lat-

simpli-

are shift-

ed t o i n f i n i t y ; h e n c e t h e u s e f u l n e s s of M e r c a t o r ' s m a p p i n g in t h e

(ant)arc-

tic sea is limited. As an e x a m p l e of an a r e a - p r e s e r v i n g m a p we only m e n t i o n t h e one p r o p o s e d by J o h a n n H e i n r i c h Lambert (1728-1777) w h i c h p r o j e c t s r a d i a l l y gent plane at the North pole.

L i k e a l l c h a r t s of t h i s k i n d ,

onto

the

tan-

it s u f f e r s f r o m

t h e d r a w b a c k of e x t r e m e l y h e a v y d i s t o r t i o n s in t h e m a r g i n a l r e g i o n s .

It w a s

a l s o k n o w n a l r e a d y in a n t i q u i t y . To r e d u c e t h i s e f f e c t ,

over t h e c e n t u r i e s many s u g g e s t i o n s have been

which we a r e not going to discuss

made

here.

In a d d i t i o n t o t h e a r e a - o r a n g l e - p r e s e r v i n g m a p s t h e r e a r e o t h e r s w h i c h do not

possess

any

of

the

properties

but

minimize

the

total

distortion

(in

some sense). The a s s u m p t i o n t h a t t h e E a r t h is s p h e r i c a l s u f f i c e s f o r most g e o g r a p h i c a p plications.

To t a b u l a t e

the

coordinates

of

the

fundamental

trigonometric

p o i n t s of h i g h - p r e c i s i o n s u r v e y s on t h e o t h e r h a n d r e q u i r e s a c l o s e r x i m a t i o n t o t h e t r u e s h a p e of o u r p l a n e t . t o be a c c o u n t e d f o r . cuss

briefly

some

of

In p a r t i c u l a r ,

appro-

the oblateness

has

At t h e end of t h i s o v e r v i e w w e t h e r e f o r e w a n t t o d i s the

projections

which

are

apt

to

chart

(flattened)

spheroids. T h e m o s t o b v i o u s i d e a of c o u r s e i s t o u s e t w o a n g l e s , ographic coordinates. is r o t a t i o n a l l y

symmetric

around

the geographic

longitude,

b u t on a c c o u n t of

its figure axis,

ans t h e r e are various possibilities f o r the The angle a t plane

and

the centre

the

in a n a l o g y t o t h e g e -

S i n c e t h e E a r t h in t h e p r e s e n t g e n e r a l i z e d c a s e

direction

of t h e E a r t h to

the

point

since it c a n n o t be m e a s u r e d d i r e c t l y .

the

to

keep

meridi-

latitude.

which to

we a r e a l l o w e d

t h e e l l i p t i c i t y of

still

be

is s p a n n e d represented

by t h e is

equatorial

not

suitable

It i s b e t t e r t o t a k e t h e elevation of

the (North) pole. This choice is n o r m a l l y p r e f e r r e d to o t h e r c a n d i d a t e s , t r a t e on i t . nent

s o we s h a l l

A common f e a t u r e w i t h t h e s p h e r i c a l c o o r d i n a t e s is t h e

necessity

to

calculate

trigonometric

functions.

This renders

concenpermathe

ap-

1.4. Geodesy and Cartography

plication laborious.

33

T h e r e f o r e Söldner c o n s t r u c t s a c h a r t which

t h e c i r c l e s of l o n g i t u d e and l a t i t u d e by v e r t i c a l

and h o r i z o n t a l

spectively.

isometrically.

E q u a t o r and all m e r i d i a n s a r e mapped

represents lines,

The Söldner coordinates d e f i n e d t h a t w a y u n f o r t u n a t e l y l a c k c o n f o r m i t y . order to overcome this defect,

Gauß m o d i f i e s t h e idea.

central meridian as a reference basis. rudimentary; scripts.

He d i s t i n g u i s h e s

His i n v e s t i g a t i o n s ,

however,

re-

In a

remain

o n l y K r ü g e r [1912] c l o s e s s o m e g a p s in G a u ß ' u n p u b l i s h e d m a n u -

He e s t a b l i s h e s a c o m p r e h e n s i v e t h e o r y of t h i s transversal Mercator

projection.

Mainly

in t h e

German

literature

one u s u a l l y

speaks

of

Gaue-

Krüger coordinates. T h e y a r e t h e f u n d a m e n t of many o f f i c i a l s u r v e y s . We d e v o t e c h a p t e r 29 of t h i s book t o a c o m p r e h e n s i v e a n a l y s i s of t h i s t o p i c , c l u d i n g a d e t a i l e d a c c o u n t of a l l r e l e v a n t

formulas.

in-

34

1. H i s t o r i c a l Development of t h e C o o r d i n a t e Concept

1 . 5 . Special Coordinates

E v e r y c h a r t of a m a n i f o l d d e f i n e s a c o o r d i n a t e s y s t e m .

The s a m e h o l d s t r u e

f o r a n y d i f f e o m o r p h i s m b e t w e e n t w o r e g i o n s of a E u c l i d e a n s p a c e . T h e v a r i e ty

of

possible

reference

haustive case-by-case

frames thus

investigation

tion seems d i f f i c u l t .

is p o t e n t i a l l y

infinite,

cannot be achieved.

Nevertheless

a few general

and

Even a

rules exist

b e o b e y e d w h e n e v e r one w a n t s t o c h o o s e a s u i t a b l e

an

which

should

parametrization:

(1) T h e invariance group of t h e c o o r d i n a t e s m u s t c o r r e s p o n d t o t h e t r i e s of t h e p r o b l e m . labour,

T h i s u s u a l l y r e s u l t s in a c o n s i d e r a b l e

since the calculation

of u n i n t e r e s t i n g

ex-

classifica-

or needless

symme-

saving

of

information

is avoided. If, f o r example, well

as the

a d i f f e r e n t i a l e q u a t i o n has to be i n t e g r a t e d which,

initial

about some axis,

or boundary

conditions,

is r o t a t i o n a l l y

i t is a l m o s t a l w a y s a d v i s a b l e t o c h o o s e t h e

angle with respect

to the axis

permits a dimensional

a s one of

the coordinates.

as

symmetric azimuthal This

often

reduction.

On a c c o u n t of Noether's theorem a s i m i l a r a r g u m e n t a p p l i e s if t h e r e a Lie g r o u p of s y m m e t r i e s ;

(2) A n o t h e r r u l e of t h u m b i s t o p r e f e r t h e i s an a l t e r n a t i v e .

is

c f . c h a p t e r 6.

The r e l e v a n t

criterion

"simpler" coordinates of

complexity,

p e n d s on t h e k i n d of q u e s t i o n one w a n t s t o a n s w e r .

if

there

however,

de-

T h e r e f o r e we s h a l l

be c o n t e n t w i t h t h i s somewhat vague f o r m u l a t i o n . U s u a l l y orthogonal c o o r d i n a t e s a r e f o u n d m o r e c o n v e n i e n t t h a n

non-or-

b e c a u s e metric and Christoffel symbols ( a n d a

fortiori

thogonal

ones,

the standard differential operators grad, f e w e r nonvanishing components.

rot,

div a n d Δ a l s o )

contain

But in low d i m e n s i o n a l s p a c e s t h e

f e r e n c e is r e l a t i v e l y u n i m p o r t a n t .

It g r o w s o n l y s l o w l y w i t h t h e

dif-

dimen-

sion. T h e a d d i t i o n a l a m o u n t of l a b o u r c a u s e d by t h e l a r g e r n u m b e r of

terms

sometimes

given

problem.

is c o m p e n s a t e d

by a m o r e s u i t a b l e d e s c r i p t i o n

of t h e

In t h i s c a s e n o n - o r t h o g o n a l f r a m e s c a n a l s o b e c o m e u s e f u l in

practise. An i n s t r u c t i v e e x a m p l e a r e t h e Weyl coordinates w h i c h we s h a l l t h o r o u g h l y in p a r t s D a n d G of t h i s book.

They a r e a p p l i c a b l e

discuss success-

1.5. Special

fully

Coordinates

whenever

group.

we

35

are

provided

with

a

(nontrivial)

finite

symmetry

T h e y a r e i n t i m a t e l y l i n k e d w i t h m o d e r n numerical integration me-

thods, e s p e c i a l l y f o r s p h e r i c a l

domains,

w h i c h w i l l o c c u p y u s in

part

F. In s p i t e

of

these

f i c a t i o n of

d i f f i c u l t i e s which

the conceivable

v a r i o u s c a t e g o r i e s can be c a r r i e d A natural above,

are

coordinates,

obstacles at

least

to

a systematic

classi-

a rough subdivision

into

out.

s p l i t - u p h a s been mentioned a l r e a d y .

For the r e a s o n s

o r t h o g o n a l c o o r d i n a t e s y s t e m s (i. e. t h o s e w i t h m u t u a l l y

l a r axes) a r e used a l m o s t exclusively.

discussed

perpendicu-

As w e s h a l l d e m o n s t r a t e ,

this is o f -

t e n a c o n s e q u e n c e of h i s t o r i c a l d e v e l o p m e n t s o r p e r s o n a l t a s t e a n d n o t

al-

ways objectively justified. Besides t h a t ,

o t h e r o r d e r i n g s c h e m e s a r e of i n t e r e s t .

It i s n a t u r a l t o

dis-

c r i m i n a t e b e t w e e n algebraic a n d transcendental s y s t e m s in IRn, a c c o r d i n g w h e t h e r t h e c o o r d i n a t e s a r e a l g e b r a i c f u n c t i o n s of t h e s t a n d a r d components or not.

The f o r m e r a r e n o r m a l l y e a s i e r t o h a n d l e b e c a u s e

need not evaluate complicated expressions. l e r a t i o n of t h e r e q u i r e d tems,

calculations.

This is connected with an

one

acce-

A p a r t f r o m Cartesian a n d Weyl

t h e confocal coordinates and many o t h e r s b e l o n g t o t h i s

to

Cartesian

sys-

class.

F o r t h e a b o v e - m e n t i o n e d r e a s o n s g e n e r a l l y v a l i d r e c o m m e n d a t i o n s c a n n o t be given; way. her

t h u s in t h e f o r m u l a r y ( p a r t H) a l l s y s t e m s a r e p r e s e n t e d in t h e s a m e T h i s i s i n t e n d e d t o m a k e it e a s i e r f o r t h e r e a d e r t o d e c i d e on h i s o r

own.

T h e c r i t e r i o n f o r t h e i n c l u s i o n in t h e l i s t w a s t h e a p p l i c a b i l i t y

to a

lar-

g e r n u m b e r of p r o b l e m s . T h i s a d m i t t e d l y i s a n u n s y s t e m a t i c s e l e c t i o n m e t h o d i n f l u e n c e d by p e r s o n a l

bias.

In t h e l i t e r a t u r e m o s t l y t h e o r t h o g o n a l c o o r d i n a t e s a r e i m p r o p e r l y r e d . In p r a c t i s e ,

prefer-

t h e y a r e much m o r e i m p o r t a n t t h a n t h e o t h e r s . T h i s i s why

t h e y d o m i n a t e t h e t a b l e s in p a r t H. But in a d d i t i o n a f e w m o r e

parametriza-

t i o n s h a v e b e e n i n c l u d e d w h i c h a r e e x p e c t e d t o be u s e f u l in f u t u r e

applica-

tions. As i n d i c a t e d

above,

i t is i m p o s s i b l e t o d i s c u s s

v e c o o r d i n a t e s t o a n y d e g r e e of c o m p l e t i o n .

all potentially

Nevertheless the

constructi-

comparatively

f e w s y s t e m s w h i c h a r e e m p l o y e d f r e q u e n t l y h a v e a n u m b e r of f e a t u r e s in c o m mon. They a r e d i s t i n g u i s h e d m a t h e m a t i c a l l y by c e r t a i n s y m m e t r i e s .

The

majority

of t h e m h a s b e e n d i s c o v e r e d o r i n v e n t e d a l r e a d y a l o n g t i m e a g o , s o m e e v e n in a n t i q u i t y .

Of c o u r s e ,

the argument

is a l s o t r u e

in t h e o p p o s i t e

direc-

36

1. H i s t o r i c a l Development of t h e C o o r d i n a t e Concept

t i o n : One u s u a l l y w a n t s t o m a n a g e w i t h t h e b e t t e r - k n o w n m e t h o d s b e f o r e one t r i e s s o m e t h i n g new. Among t h e o l d e s t n u m e r i c a l d e s c r i p t i o n s of p o i n t p o s i t i o n s a r e t h e Cartesian a s w e l l a s t h e polar coordinates. We p r e s e n t e d t h e i r h i s t o r y a l r e a d y in the

foregoing 3

space R

sections.

into a direct

If

we

(in m o d e r n 1 2

sum R ®R

terms)

split

and use t h e c a n o n i c a l

the

configuration

parameter

on

the

f i r s t summand, b u t p o l a r c o o r d i n a t e s on t h e second, we g e t an i n t e r m e d i a t e f o r m , n a m e l y cylinder coordinates. Other decompositions systems of s i m i l a r

can be used e q u a l l y w e l l ,

shape.

thus getting

m o r e hybrid

T h e i r e x p l i c i t c o n s t r u c t i o n c a u s e s no

problems,

n e i t h e r does t h e c a l c u l a t i o n of t h e m e t r i c c o m p o n e n t s , t h e C h r i s t o f f e l s y m bols,

e t c . We n e v e r t h e l e s s included s e v e r a l of t h e m in t h e t a b l e s .

All m e n t i o n e d c o o r d i n a t e f r a m e s have,

aside f r o m orthogonality,

one m o r e

common f e a t u r e which we should e m p h a s i z e . A l o t of i m p o r t a n t p h y s i c a l p r o b lems lead t o o r d i n a r y

or p a r t i a l

differential equations

of

second

order,

e . g . t h e m o t i o n of p o i n t m a s s e s in N e w t o n i a n m e c h a n i c s o r t h e t e m p o r a l d e v e l o p m e n t of f l u i d s in h y d r o d y n a m i c s (s. c h a p t e r 5). If

the

coefficient

differential

matrix

equation

of

the

is d e f i n i t e ,

highest

order

semidefinite

derivatives

or

of

indefinite,

a

partial

one

respec-

t i v e l y s p e a k s of e l l i p t i c , p a r a b o l i c or h y p e r b o l i c p r o b l e m s . The

simplest

elliptic

differential

equation

of

second

degree

is

the

one

which is obeyed by Newton's gravitational potential U, namely AU = 0. Here Δ d e n o t e s t h e Laplacian. It is d i s c u s s e d a t l e n g t h in c h a p t e r s 3, 22, and 23.

A c a r e f u l a n a l y s i s of t h o s e

(orthogonal)

coordinates

in R

which a d m i t

a

separation of t h e j u s t d e f i n e d potential equation or p o s s i b l y even t h e much m o r e g e n e r a l Hamilton-Jacobi equation ( c f . c h a p t e r 5) by p r o d u c t s o r s u m s g o e s back t o S t ä c k e l .

A d e t a i l e d a c c o u n t of h i s r e s u l t s f o l l o w s in c h a p t e r

27.

E x c e p t f o r t h e a b o v e - m e n t i o n e d C a r t e s i a n , s p h e r i c a l and c y l i n d r i c a l f r a m e s , t h e l a r g e and p r a c t i c a l l y e x t r e m e l y i m p o r t a n t f a m i l y of Stäckel coordinates contains several other well-known types,

among t h e m ellipsoidal, conical

a n d torus coordinates.

A d i f f e r e n t , o c c a s i o n a l l y a p p l i e d method t o c o n s t r u c t new o r t h o g o n a l f r a m e s is t o c h o o s e an a r b i t r a r y s m o o t h r e a l f u n c t i o n on t h e domain one w a n t s t o represent as the f i r s t coordinate. priately. rarely,

A f t e r w a r d s it h a s t o be amended

One of many e x a m p l e s a r e t h e Roche coordinates w h i c h , a r e used in b i n a r y s t a r a s t r o p h y s i c s .

approif

In t h i s c a s e t h e N e w t o n

only po-

1.5. Special

Coordinates

37

tential serves as the initial function. A m o d i f i c a t i o n is t o s t a r t w i t h s e v e r a l f u n c t i o n s w h o s e g r a d i e n t s a r e

eve-

rywhere mutually perpendicular.

rea-

lize this

We s h a l l h a v e m a n y a n o p p o r t u n i t y t o

idea.

All t h e s e s y s t e m s h a v e in common t h a t t h e y a r e direct. T h i s m e a n s : w e may e x p r e s s t h e s t a n d a r d p a r a m e t e r s of t h e E u c l i d e a n s p a c e e x p l i c i t l y in of t h e

terms

coordinates.

T h e s i n g l e e x c e p t i o n a r e t h e Weyl coordinates p r o p o s e d by N e u t s c h & F i c h t n e r [1988] Coxeter

in

order

to

solve

problems

with

finite

invariance

is e x a c t l y the opposite;

though it

t h e Weyl c o o r d i n a t e s a s p o l y n o m i a l s in t h e C a r t e s i a n

is possible to components,

g e n e r a l o u t of t h e q u e s t i o n t o s o l v e f o r t h e m . T h i s , h o w e v e r , t e r much,

write

it is

in

does not m a t -

b e c a u s e of t h e s p e c i a l p r o p e r t i e s of t h e Coxeter groups. We s h a l l

classify them

in p a r t

D of

this

book.

like m e t r i c ,

C h r i s t o f f e l symbols or

div,

are

Δ,

representable

In f a c t ,

the

easily

all

differential

and

explicitly.

essential

information

operators

grad,

For

the

rot,

practically

i m p o r t a n t c a s e s ( d i m e n s i o n s 2 t o 4) w e s h a l l c a r r y t h i s t h r o u g h in 30.

(of

type).

Here the situation

and

groups

chapter

38

2. N o t a t i o n and Conventions

2. Notation and Conventions 2.1. Sets and Topological Spaces

To begin w i t h ,

we have to a g r e e on c e r t a i n pieces of n o t a t i o n ,

since in the

l i t e r a t u r e on our s u b j e c t a l a r g e number of d i f f e r e n t conventions and s y m bols a r e in use.

A l l d e f i n i t i o n s and theorems given in t h i s chapter

will

be

r e t a i n e d throughout the t e x t and used without f u r t h e r comments or e x p l a n a tions. More about them can be found in e v e r y i n t r o d u c t o r y book; frain

from

proofs.

reproducing

the m o s t l y

trivial

and g e n e r a l l y

t h e r e f o r e we known or

Only in a f e w s e l e c t e d cases w e sketch the u n d e r l y i n g

We apply the usual n o t a t i o n scheme f o r sets. by a l i s t of

i t s elements,

A set can e i t h e r be

I x,y,z

described

I

like

χ has the p r o p e r t y

This means:

idea.

e.g.

=

or a c h a r a c t e r i z a t i o n

re-

obvious

An o b j e c t y is contained in Μ if

6(x)

and only

if

it possesses

the

p r o p e r t y S. We then w r i t e y e M; o t h e r w i s e y t M. The Cartesian product o f the s e t s A, B,C, . . .

is

Α χ Β χ C χ

(a,b,c,... ) •

The l o g i c a l plies"),

a € A, b 6 B, c e C,

{

symbols have the usual meanings:

'

N

an a r b i t r a r y

1

set

Ω is

a bijection

of

Ω onto

itself.

p e r m u t a t i o n s o f Ω f o r m the synmetric group S(£i) of Ω. Its isomorphism does not depend on Ω, but only on i t s c a r d i n a l i t y .

All class

F o r that r e a s o n one a l s o

writes S

i f # Ω = η < o°. η The n o t a t i o n (a , a , . . . , a , a ) means the p ermutation K 1 2 k-i k at

to

a

( l s i s

ments o f Ω ( i f length k ( o r cycles are If

(the

π e S(n) tors

shorter:

while

is mapped

a k-cycle).

to

aj

each

remaining

ele-

Such a p e r m u t a t i o n is c a l l e d a cycle Cycles of

length 1 a r e f i x e d points;

of)

Ω

is

finite,

we

can

as a f i n i t e product of t r a n s p o s i t i o n s . uniquely

According

to

determined by π, which

of

the

write

in c o n t r a s t

two

A c y c l e of

every

of 2-

permutation

The r e q u i r e d number of to

possibilities

p e r m u t a t i o n π even or odd. The signum of it, sgn(n), spectively.

and a l l

carries

transpositions.

cardinality

is not

mod 2).

k-1),

any) a r e i n v a r i a n t .

which

fac-

i t s p a r i t y ( = number call

the

is then + 1 or - 1,

occurs,

we

re-

length k thus is an even (odd) p e r m u t a t i o n

whenever

46

2. N o t a t i o n and Conventions

k is odd (even). other! ε

abc...

Be a w a r e that length and signum vary oppositely

to

each

Instead of sgn(a, b, c, . . . ), we sometimes also use one of the symbols abc... or ε

The assignment π ι—> sgn(n) is a homomorphism f r o m c a t i v e group { ± 1).

Its kernel ( = the set of

into the m u l t i p l i -

even permutations)

is the al-

ternating group on Ω. It is denoted Α(Ω) or A π. For η £ 2 is A a normal subgroup of index 2 in S . η η T w o cycles commute with each other if they are d i s j o i n t . Every permutation π of Ω = {1

η ) can be w r i t t e n as a product of mutually d i s j o i n t

cycles.

This r e p r e s e n t a t i o n is unique up to the order of the f a c t o r s and the t r a r y choice of in question,

we

the initial only

element in each cycle.

need to

pick

an element

To f i n d the

a j e Ω,

call

arbi-

partitioning

its

image

a,

whose image is a^, and so on. A f t e r a f i n i t e number of steps we reach the starting

point

= a^ again.

The f i r s t

cycle then is ( a ^ . - . a ^ ) ,

g e t the others we repeat the procedure with any element b ence set Ω Ν ^ } ,

of the

and

to

differ-

as long as not all l e t t e r s of Ω have been considered y e t .

If

in this canonical cycle decomposition the number of i - c y c l e s equals k , k k k we say that π has the (cycle) structure ||l , 2 , 3 . . . . ||. We also o c c a s i o nally use f o r reasons of write

||l2,

2, 3|| instead of

simplicity ||l2,

21,

31,

obvious abbreviations;

for

example

we

4°, 5°, 6°, 7°||, etc.

The a l t e r n a t i n g group A η is simple f o r η a 5. In this case Sπ has no other n o n t r i v i a l normal subgroup than A . As the only a l t e r n a t i n g group of f i n i t e π rank, A possesses a nontrivial normal subgroup, namely the (Klein) four 4

It consists of the unit element and the t h r e e double 2 transpositions and is isomorphic to 2 .

group (Vierergruppe).

The subgroups of S(n) are the permutation groups on Ω. The orbit or block of transitivity of χ 6 Ω under the action of G s ε ( Ω ) is the set χ

G

=

{ x*

I

? * G }

G is transitive on Ω whenever x G = Ω holds. y

The o r b i t s under the conjuga-

tion χ ι—» χ , x , y e G, are the conjugacy classes of G. The group ring K[G] of the group G over a f i e l d Κ is the set of sums of the shape KtG]

=

{

Σ

reG

k^ r

k

j 6

K,

r e G J

all

formal

2.2.

with

Groups

the

47

multiplication

rule

^j^-l^y

= ( k ^ H y ^ )

e K[G].

This

indeed

is a r i n g w i t h 1.

Amongst the w e l l - k n o w n general theorems of

group theory we merely

T h e o r e m 2.2.1 (Homomorphism theorem): L e t φ e H o m ( G , H ) be a g r o u p h o m o m o r p h i s m .

Then the k e r n e l o f

Kern φ

is n o r m a l

in G;

t h e image o f

χ e G

cite:

φ,

?>(x) = 1

φ,

Img φ

is a s u b g r o u p o f

H,

and w e

have

Img φ

Vice versa,

every normal

=

G / Kern φ

s u b g r o u p is the k e r n e l

of

a suitable

homomorphism.

48

2. N o t a t i o n and Conventions

2.3. Matrices

In t h i s s e c t i o n we p r e s e n t a f e w e l e m e n t a r y f a c t s f r o m l i n e a r The

set

of

all

(m, n)-matrices

w i l l be d e n o t e d by K

(m,n)

.

(m

rows,

η columns)

with

Here Κ a l w a y s is a r i n g ,

algebra.

coefficients

i n DC

which u s u a l l y w i l l

be

t h e f i e l d of r e a l (K = IR) or c o m p l e x n u m b e r s (IK = C). The components o r entries of t h e m a t r i x Μ e K ( m , n ) a r e M^ e Κ (1 s i s m, 1 < j < n). The transpose of Μ t h u s is t h e m a t r i x l M e K ( n ' m ) w i t h t

M

= Μ

ij

Jl

We s y m b o l i z e t h e complex-conjugate of a number o r m a t r i x by a b a r : M, t h e a n d Β 6 Κ1-"' by j u x t a p o s i t i o n :

matrix product o f A e DC If m = η, w e s a y Μ

AB e Κ

i s quadratic. T h e zero matrix o f d i m e n s i o n

'

(n, n) i s

de-

or simply 0, t h e identity o r unit matrix by 1 o r 1. The n o t a n η t i o n 0 w i l l a l s o be a p p l i e d t o t h e zero vector. n o t e d by 0

Diag(\i

λ ) is t h e diagonal matrix of dimension ( n , n ) w i t h c o e f f i c i e n t s

λ , n a m e l y t h e m a t r i x Μ e DC(n,n) f o r which λ

ι

if

i

Μ ij otherwise h o l d s . The m a t r i x Μ is monomial if it c o n t a i n s in each r o w and e a c h column e x a c t l y one c o e f f i c i e n t o t h e r t h a n z e r o . equal

If t h e s e n o n z e r o e l e m e n t s a r e

all

t o 1, w e g e t a permutation matrix.

A m o n o m i a l m a t r i x is r e p r e s e n t a b l e in t h e f o r m Μ = D -Ρ = Ρ"Dg w i t h a p e r m u t a t i o n m a t r i x Ρ and d i a ge o n a l m a t r i c e s D and D . The f a c t o r s P, D , and 1 2 ι D a a r e u n i q u e l y d e t e r m i n e d by M. The f u l l monomial group of dimension η over K, which c o n s i s t s of a l l m o n o mial m a t r i c e s , is i s o m o r p h i c t o (K x ) n : S . The n o r m a l s u b g r o u p (DCx)n c o n ti tains the diagonal matrices, permutation

t h e f a c t o r is r e a l i z e d by t h e s u b g r o u p of

all

matrices.

A scalar matrix i s a d i a g o n a l m a t r i x of t h e f o r m λ · 1 = Diag(A

λ),

i.e.

2.3.

Matrices

49

a m u l t i p l e ( w i t h f a c t o r λ e K) of t h e i d e n t i t y m a t r i x . The v e c t o r s p a c e Κ

w i l l n o r m a l l y be a b b r e v i a t e d

as Κ ,

its

elements,

t h e row vectors a r e s i m p l y c a l l e d vectors over Κ (of dimension n). The column vectors o b t a i n e d

l i e i n t h e dual space K < n , l ) .

by t r a n s p o s i t i o n

We w r i t e f ( v ) o r s o m e t i m e s a l s o v f f o r t h e image of t h e v e c t o r ν u n d e r a linear mapping or f u n c t i o n f . The c o m p o s i t i o n of t h e m a p s f , g , h , . . .

(first

f , t h e n g, t h e n h, and so on) gives hogof(v)

=

h(g(f(v)))

vf9h

=

The a d v a n t a g e of t h e l a t t e r n o t a t i o n should be obvious: (1) f , g, h o c c u r in t h e n a t u r a l o r d e r of a p p l i c a t i o n ;

and

(2) we need no b r a c k e t s . F u r t h e r m o r e r o w v e c t o r s a r e e a s i e r t o p r i n t t h a n column v e c t o r s ,

because

t h e y r e q u i r e l e s s r o o m . F o r t h e s e r e a s o n s we have d e s c r i b e d t h e e l e m e n t s of Kn a s r o w v e c t o r s .

E v i d e n t l y t h e use of column v e c t o r s (with t h e

a c t i n g f r o m t h e l e f t ) would lead t o an e q u i v a l e n t q u i t e common,

e s p e c i a l l y in t h e o l d e r

T h e trace (Spur) o f

the

(n.n)-matrix

description.

matrices

It is

also

literature. Μ

is

π 1=1 w h i l e det(M) m e a n s i t s d e t e r m i n a n t .

Μ is singular if det(M) = 0, and regu-

lar (invertible) o t h e r w i s e .

latter

In t h e

case

the 1

c h a r a c t e r i z e d by e a c h of t h e t w o e q u a t i o n s Μ Μ In g e n e r a l ,

inverse is Μ

= 1 and Μ

1

1

.

is

Μ = 1.

Mk is t h e k t h power of Μ (k e Z; w h e n e v e r k is l e s s t h a n

Μ m u s t be r e g u l a r ) .

It

zero,

We have M° = 1 and Μ1 = M.

The dimension of a v e c t o r s p a c e V over Κ (= c a r d i n a l i t y of any K - b a s i s of e.g.,

dim^tK11) = n.

V) w i l l be d e n o t e d by dim^tV),

thus,

m i s u n d e r s t a n d i n g t o be f e a r e d ,

we omit t h e s u b s c r i p t "K" and simply

When t h e r e is no write

dim V. The g r o u p of

all

invertible K-linear

transformations

of

an

n-dimensional

v e c t o r s p a c e V = Kn over t h e f i e l d Κ is GL(V) = GL(K n ). It is i s o m o r p h i c t o t h e g r o u p of r e g u l a r (n, n ) - m a t r i c e s w i t h c o e f f i c i e n t s in K, which we s h a l l call

the

(general) linear group GL (Κ). π

h o w e v e r o f t e n be i r r e l e v a n t .

This

meticulous

distinction

will

50

2. Notation and Conventions

Moreover, group)

are

the normal subgroups SL(V) = SL(K ) and SL (K) (special η determined

by the

additional

restriction

linear

"determinant = 1" in

the corresponding GL. The f a c t o r groups are isomorphic with the multiplicative group of the ground f i e l d :

The λ-eigenspace (λ e C) of a complex ( n . n ) - m a t r i x Μ is the set

ER(X,M)

veC •

ν Μ= λ ν

{

and the multiplicity of λ is the dimension of ER(X, M): mult M U)

dim ER(A,M)

Numbers λ e C with mult^tX) > 0, hence ER(A,M) * 0 (null space) are called eigenvalues of M. When counted with multiplicities,

they f o r m the spectrum

of the matrix M:

EW(M)

λ s C

mult^U) >

The sum of all eigenvalues of Μ (with multiplicities) equals its trace,

the

product of the eigenvalues is identical with the determinant. The i t h row of the matrix Μ e K ( m ' n ' is equal to the image of the i t h basis vector under the linear map φ: Km —> Κ

represented by M. The kernel of Μ

or φ is

Kern Μ

Kern φ

while the image of Κ

=

|veK™

v^ = ν Μ = 0 J

is the subspace Img Μ = Img φ of Κ

ER(0, M)

spanned by the

rows of M. The dimension of the image is the rank of Μ or φ, Rank Μ =

Rank φ

=

dim Img φ

Rank Μ thus is the maximum number of linearly independent rows (or columns) in M. We moreover have

2.3.

Matrices

51

Theorem 2.3.1:

( a ) Assume t h a t U and V a r e v e c t o r s p a c e s over Κ and φ: U —> V is l i n e a r . The f a c t o r s p a c e of U w i t h r e s p e c t t o t h e k e r n e l of φ is

canonically

i s o m o r p h i c w i t h t h e image of φ: U / Kern((p)

a

Img(*>)

(Homomorphism theorem f o r v e c t o r s p a c e s ) .

£

V

For t h e d i m e n s i o n s t h e

equa-

tion dim Kern(p) + dim Img(qp)

=

dim U

holds. (b) The s p a c e (A,B> g e n e r a t e d by t w o s u b s p a c e s A and Β in V h a s t h e d i m e n sion dim κ" 1 )ο(κϊΓ 1 )

holds and the coordinate changes λ « λ

1

as well as κ°κ

1

were assumed to be

smooth. For that reason the above criterion is independent of the charts ic and λ.

Hence it describes an geometric characteristic of the map f

In d i f f e r e n t i a l perties;

geometry,

one is interested

only

statements which require a special choice of

of much less, if any relevance.

itself.

in such "intrinsic" the coordinates

proare

76

3. Manifolds

3.3. Curves and Tangent Spaces The characterization of

all 1-dimensional manifolds does not present any

difficulty:

Theorem 3.3.1: ( a ) Every connected one-dimensional manifold is diffeomorphic to either the real axis IR or the circle S1. (b) Both cases exclude each other: IR and S1 are not even equivalent topologically.

Thus there are only two essentially d i f f e r e n t types of connected manifolds of dimension 1: the set IR of real numbers and the periphery S1 of the unit circle. For reasons of space limitation, we refrain from including the simple proof here. This result suggests

Definition 3.3.1: A curve in the manifold Μ is a smooth mapping r : I —» Μ from a connected 1-dimensional manifold I into M. Depending on whether I is homeomorphic to IR or to S1, we shall call the curve f open or closed. In the former case we shall suppose - except if we explicitly stated otherwise - that I is an open interval in IR.

For the time being Μ always denotes a (smooth real) n-dimensional manifold. The set of all curves passing through a f i x e d point χ € Μ splits up naturally into equivalence classes:

3.3.

C u r v e s and Tangent S p a c e s

77

Definition 3.3.2: Two open c u r v e s y : I

—» Μ a n d

—> Μ a r e tangent t o e a c h o t h e r

I

χ e Μ if f o r s u i t a b l e p a r a m e t e r v a l u e s t

v

v

=

1

w

€ I and t 1 2

e I

in

2

x

•

h o l d s a n d a m a p ic e tt c o n t a i n i n g χ c a n be f o u n d w i t h d

r

dt

The

classes

of

the

( K

ι

° V t=t

equivalence

=

< * rΙΚΟ») J -jT2 lt=t

dt

1

relation

given

by

2

this

condition

are

the

tangent vectors t o Μ w i t h base point x. T h e y f o r m t h e tangent space TM . χ The t a n g e n t

v e c t o r of t h e c u r v e y: I —> Μ a t t h e parameter t 6 I, t h a t

a t t h e place χ = y ( t ) ,

is

i s d e n o t e d by y ' ( t ) 6 TM .

By a p p l i c a t i o n of t h e chain rule i t i s e a s y t o d e m o n s t r a t e t h a t t h i s

parti-

t i o n i n g i n t o t a n g e n t v e c t o r s d o e s n o t d e p e n d on t h e c h a r t κ. S o m e t i m e s a n a l t e r n a t i v e d e s c r i p t i o n of t h e t a n g e n t v e c t o r s i s of

practical

value:

Theorem 3.3.2: Two curves γ

ι

and y

2

in Μ w i t h

V V

=

y

2(V

x

=

a r e t a n g e n t a t t h e p o i n t χ if a n d o n l y if f o r a l l s m o o t h ( o r even a l l functions f:

Μ —> R

d t

is f u l f i l l e d .

1

lt-t

d t

1

2

U-t

2

C1-)

78

3.

Manifolds

T a n g e n t v e c t o r s a r e d i f f e r e n t i a l o p e r a t o r s w h i c h c a n be a p p l i e d t o

smooth

f unctions:

Definition 3.3.3: S u p p o s e χ 6 Μ a n d ν e ΤΜ χ .

The directional derivative of a C ° ° - f u n c t i o n f

w i t h r e s p e c t t o t h e v e c t o r ν is f(r(t+h))-f(r(t)) v(f)

=

11m h-»o

h

w h e r e y i s an o p e n c u r v e in Μ w i t h y ( t ) = χ and y ' ( t ) = v.

I t i s c l e a r t h a t t h e s p e c i a l c h o i c e of y is

irrelevant.

T a n g e n t s p a c e s h a v e a n a t u r a l v e c t o r s p a c e s t r u c t u r e i m p o s e d by t h e

charts:

Theorem 3.3.3: ( a ) C o n s i d e r a ( r e a l ) m a n i f o l d Μ of d i m e n s i o n n. T h e n f o r a l l χ 6 Μ w e h a v e the

isomorphism

TM

X

s

Rn

w h e r e a d d i t i o n of t a n g e n t v e c t o r s a t χ and m u l t i p l i c a t i o n by r e a l

num-

b e r s a r e d e f i n e d in t h e o b v i o u s w a y w i t h t h e h e l p of an a r b i t r a r y

chart

κ containing x. (b) The v e c t o r of

s p a c e s t r u c t u r e t h u s f o u n d (in c o n t r a s t

t h e v e c t o r s ! ) is i n d e p e n d e n t

of κ.

This means:

i n d u c e s a l i n e a r m a p p i n g of ΤΜ χ i n t o i t s e l f . of t h e Jacobi matrix of t h e

to the

It i s g i v e n by t h e

transition.

Proof: Trivial and s t r a i g h t f o r w a r d calculation;

coordinates

Every c h a r t

l e f t to the

reader.

change action

3.3. Curves and Tangent Spaces

79

A chart κ also induces a natural basis on the tangent space at any point in the domain U , f o r we have

Theorem 3.3.4: Let Μ be a manifold of dimension η, κ e k[X)

=

(x 1

a chart. We set f o r all χ € U^ x")

6

Rn

To each i e {1 n) there is an open interval coordinate line through x, that is the curve κ : ,, ν K^lt)

=

κ

-1/ 1 (x

a x 1 , such that the —» Μ with

ith

i-l . i+1 n, X ,t.x ,...,x )

is defined f o r all t e I . Its tangent vector κ ^ ( χ ' ) at the point χ w i l l

be

denoted by d . ( a ) In the κ-coordinates we get the relation a d

=

θχ1 The l a t t e r

has to

be interpreted

as the

directional

derivative

along

the coordinate lines, at a

1

f

=

( b ) The coordinate derivatives (3 lar,

a (f) 1

=

— „1 ax

θ ) f o r m a basis of TM . In particuη χ

l

every tangent vector ν e TM

can be represented uniquely as a l i -

near combination of the a ^ ν

=

v1 a

ι

( c ) The v-action on the set of all smooth functions is a derivation:

v(fg)

=

ι v1 a Cfg)

=

ι v1

9 t f 8 )

—

ax

=

v(f)-g + f-v(g)

80

3.

Manifolds

3.4. Vector Bundles

B e f o r e being able to d e f i n e v e c t o r shall

have to

carry

and,

out an a u x i l i a r y

more g e n e r a l l y ,

construction

which

tensor f i e l d s , is

we

interesting

in

its own r i g h t . It does not make sense to speak of manifold

M,

since

vectors

are

the "set of

meaningful

only

all

tangent

vectors"

in combination

with

of

a

their

base points. Though tangent spaces at two d i f f e r e n t points x, y e Μ are i s o morphic to each other,

a "canonical" b i j e c t i o n between the elements in TM

X

and those in TM y does not e x i s t , Mainly f o r this reason we have to introduce the s o - c a l l e d tangent bundle as a substitute f o r the union of all tangent spaces of M:

D e f i n i t i o n 3.4.1: L e t Μ be a m a n i f o l d , ( a ) The tangent bundle of Μ is

TM

=

I

χ e Μ, ν

e TM

Cb) The projection of the bundle is the map which attaches to each pair in TM i t s f i r s t component,

i.e.

the base point:

π :

TM

(χ,ν)

»Μ I—> χ

The outstanding p r o p e r t y of TM is

Theorem 3.4.1: ( a ) The tangent

bundle TM of

C°°-manifold i t s e l f

an n-dimensional

(smooth)

m a n i f o l d Μ is a

(with the dimension doubled).

Its a t l a s Μ consists of the induced charts κ with κ e 1ί , d e f i n e d via TM Μ

3.4. Vector Bundles

κ(χ,ν)

81

=

(χ

χ ;v

ν )

e

R

χ IR

=

IR

Here

k(x)

=

(x 1 , . . . , χ " )

are the coordinates of the base point and (v 1 the v e c t o r ν with respect to the basis of TM

v n ) the components of which is canonically

giv-

en by the chart κ: ν

=

ν1 a

ι

( b ) The bundle p r o j e c t i o n π: TM —» Μ is i n f i n i t e l y o f t e n d i f f e r e n t i a b l e .

Proof: We consider t w o charts α and β of Μ whose domains are U and U , r e s p e c ~ ~ « Ρ t i v e l y . Then the induced charts α and β are w e l l - d e f i n e d on the p r e - i m a g e s it β°α

(U ) and it α

(U ) of U and U under π. The smoothness of the t r a n s i t i o n β α β f r o m α t o β is t r i v i a l f o r the x - c o o r d i n a t e s . For the v-components

the analogous statement f o l l o w s d i r e c t l y f r o m the d e f i n i t i o n of the

partial

d e r i v a t i v e s 9 . This shows that ι

U TM is

an a t l a s

of

the

manifold

=

TM.

·( ic κ € U 1 (_ I Μ J The

unlimited

differentiability

of

it is

evident.

Structures

similar

with m a n i f o l d s .

to

the

tangent

bundles

frequently

occur

in

connection

It is t h e r e f o r e reasonable to f o r m a l i z e the above c o n s t r u c -

tion in o r d e r to include more general situations as w e l l . This can be done on d i f f e r e n t levels. lowing t e r m i n o l o g y s u f f i c e s .

For our present

intentions the

fol-

3. Manifolds

82

Definition 3.4.2: Let Μ be an n-dimensional manifold. An N-dimensional vector (space) bundle (N e IN) on Μ is a t r i p l e (Μ,Β,π),

composed of

( 1 ) the base space Μ i t s e l f ; (2) a second manifold Β (= "bundle") of dimension n+N; and f i n a l l y ( 3 ) a smooth s u r j e c t i v e mapping π: Β —» Μ, called the bundle projection. We assume in addition that a nonempty family of trivializations Φ^ (i 6 I ) exists with the properties ( a ) Every

is a diffeomorphism f r o m i r ' N l ^ ) to U ( χ RN. Here the U

£ U

are open sets. ( b ) The U n c o v e r M:

U U leu

= Μ

1

( c ) The t r i v i a l i z a t i o n s are compatible with the fibres ττ" 1 (χ):

where ( d ) For

denotes the natural projection f r o m U^xR" onto U^

each two

of Φ »Φ J i

subscripts

i, j 6 0 f u l f i l l i n g

UnU

* 0,

the

restriction

to some f i b r e if'tx)

=

{χ> χ RN

is a (vector space) isomorphism of RN. It thus induces a transformation in GL(R H ).

The tangent bundle is t h e r e f o r e just the simplest special case of this definition. The terminology connected with vector bundles may seem somewhat technical, but its are

importance

applicable

to

lies vector

in its

economy.

spaces,

as

Practically

for

instance

all

operations

constructing

the

which dual

space, tensor products, Graßmann algebras or direct sums etc. , can be c a r ried over without change to vector bundles. der the f i b r e s i T ' t x ) ,

It is only necessary to consi-

χ € Μ, separately.

Since the bundle space Β is a manifold i t s e l f ,

we are able to

introduce

3.4. Vector Bundles

,00

83

C - f i e l d s of vectors,

tensors, and so on. There is nothing more to be done

than to reformulate our f o r m e r approach a l i t t l e . The appropriate tool is

Definition 3.4.3: A (smooth) section of a vector bundle (Μ,Β,π) is a C^-map Β

which, when combined with the projection, gives rise to the identity on the base space M: Id

Μ

It is customary and convenient to w r i t e the image of χ under ν

,s

comes out. This is equivalent with the required e x p r e s s i o n . Similarly,

i (ω)

(b) is found with the intermediary s t e p

=

f

Σ )

dx '

λ

.. .

dx *

λ

1

a

(dx * )

i

λ

dx

1 , 1

λ

. . .

a

dx

k

which is an immediate consequence of the a n t i d e r i v a t i o n p r o p e r t y of i , r e V

o r d e r i n g and a p p l i c a t i o n of i (dx a ) V Statement

=

dxa(v)

=

v(xa)

=

va

(c) at l a s t f o l l o w s f r o m the elementary o b s e r v a t i o n

(proven by

i n s e r t i o n of the d e f i n i t i o n s ) t h a t f o r every d i f f e r e n t i a l f o r m ω, L (du) V holds.

Consequently,

d e r i v a t i v e commute.

=

d(L ω) V

when r e s t r i c t e d to Λ(Μ),

Cartan d i f f e r e n t i a l and Lie

The same is t r u e with d«i

since the C a r t a n r u l e dd = 0 implies d ° ( d o i +i od) V V

=

d ° i d = V

V

+ i «d in the p l a c e of L , V V

( d « i +i od)»d V V

This r e d u c e s the p r o p o s i t i o n in question to the s p e c i a l c a s e of a O - f o r m ω. Under t h e s e c i r c u m s t a n c e s , ( d « i +i od)(u)

=

however, we have

d ( 0 ) + i (du)

=

i (du)

=

du(v)

=

ν(ω)

=

L (ω)

94

3. Manifolds

3.6. Partition of Unity

The i n t e g r a t i o n of a l t e r n a t i n g d i f f e r e n t i a l f o r m s of maximum rank is a d i rect

generalization

of

t i a b l e ) f u n c t i o n s in

the Riemann integral of

(infinitely

often

differen-

IRn.

The underlying idea is quite simple.

We f i r s t consider the special

which the m a n i f o l d is covered by a single chart κ: Μ —* ical point ξ 6 Μ to χ = (χ 1

IRn

case

in

mapping the t y p -

x n ) e IRn.

The f o r m ω 6 Λ η (Μ) we want to i n t e g r a t e may be expressed in c o o r d i n a t e s g i ven by κ as

ω(χ)

where f :

=

fix1

x n ) dx 1 λ . . .

λ dx n

κ(Μ) —» R denotes some suitable smooth f u n c t i o n . We set

J

ω

J...J

=

Μ

fix1

x n ) dx 1 . . .

dx n

Χ6Κ(Μ)

(to be i n t e r p r e t e d as a Riemann i n t e g r a l ) . The general situation,

when one chart alone does not s u f f i c e to map M,

much more complicated. above

case

striction

by

is

Under these circumstances we have to reduce to the

splitting

up the integral

discussed b e f o r e .

If

the

re-

there are i n f i n i t e l y many contributions,

we

are f a c e d with a convergence problem.

into

parts

which

satisfy

We solve it by constructing a " p a r t i -

tion of unity" ( d e f i n i t i o n 3 . 6 . 2 ) . This can be achieved on every manifold, even with some additional properties

(theorem 3 . 6 . 3 )

which

guarantee

that

only

a finite

technical

number

of

terms have to be summed up. These remarks should g i v e the reader an intuitive f e e l i n g f o r the ideas und e r l y i n g the i n t e g r a t i o n theory of The

rest

of

this

these approaches.

section

aims

manifolds.

at

a correct

mathematical

elaboration

of

It has been included f o r the sake of completeness and may

be skipped without hesitation if

one is w i l l i n g

to accept

the existence

of

a p a r t i t i o n of unity as a f a c t . We next have to introduce a f e w pieces of notation f r o m s e t - t h e o r e t i c a l

to-

3.6. Partition of Unity

95

pology.

Definition 3.6.1: Let Μ be a topological space. An (open) covering of Μ is a f a m i l y

U

=

I

i e D I

of open subsets L^ S Μ whose union is the whole of M. U is locally finite if every χ e Μ has an open neighbourhood which intersects only f i n i t e l y many U

nontrivially.

A covering

8

is a refinement of tt, if f o r all j e J there is an i e 0 with V^ £ I K

The

term "partial covering" has the obvious meaning. The topological space Μ is called paracompact if each open covering has an open locally f i n i t e partial covering, a f i n i t e partial covering. The support supp(f) of

and quasicompact if each covering has

A quasicompact Hausdorff space is compact.

a continuous function f : Μ —> IR is the

topological

closure of the set of all χ e Μ with f ( x ) * 0.

We now come to the central notion which w i l l enable us to construct certain global

f i e l d s on manifolds as well

as to integrate

differential

forms

maximal rank:

Definition 3.6.2: ( a ) By a partition of unity on the topological space Μ we mean a set

Ζ

.00 of C -functions φ : Μ —» IR which s a t i s f y the f o l l o w i n g requirements: J

of

96

3. Manifolds

( 1 ) The set of all s u p p t ^ ) is locally f i n i t e ; ( 2 ) we always have

£ 0; and

( 3 ) f o r all x,

Σ vx) =

1

jeJ

( b ) Ζ is subordinate to the open covering

U

=

j Uf

i e 0 j-

if f o r each j e Jl it is possible to find an appropriate i 6 0 such that supp(^) £ U(.

The summation in ( a ) on account of

(1) contains only a f i n i t e number

of

nonvanishing contributions and thus is w e l l - d e f i n e d . Remarkably,

partitions of unity exist f o r all manifolds (and even conside-

rably more general spaces). With their help we shall overcome the d i f f i c u l ties mentioned at the beginning of this section. Except f o r the terminology,

we f o l l o w the argumentation of Warner [1983],

where even a l i t t l e more is proved. Our purposes, however, the f u l l strength of Warner's deductions,

do not require

and we can simplify parts of the

calculations in comparison to the just cited book. We start our considerations with a lemma which is interesting

in its own

right.

Theorem 3.6.1: Every open covering of a manifold Μ has a countable refinement which consists only of open sets with compact closures.

Proof: We set dim(M) = n. To any given χ € Μ we select a chart κ which has χ in its domain of definition, U^. The image of K(X) in the Euclidean space IRn. T h e r e f o r e ,

then is an open neighbourhood we can find a positive

real

3.6.

P a r t i t i o n of Unity

number c,

97

such that the closed ball Κ with radius ε and centre κ(χ) is a

subset of k(U ). κ

Since κ is homeomorphic, the pre-image κ x. The a r b i t r a r i n e s s of the choice of χ € Furthermore, the manifold Μ by definition able basis β = {Β , Β , Β , . . . } . We use the 1 2 3 Β

Β 6 8

(Κ) is a compact neighbourhood of Μ implies that Μ is paracompact. is a Hausdorff space with countnotation

compact

and consider a nonempty open subset U and some point u in U. On account of the paracompactness of Μ there is an open neighbourhood W of u with compact closure W . We may assume without r e s t r i c t i o n t h a t W" is contained in U. As an open s e t ,

W is the union of suitable basic s e t s B ^

which occurs in this representation,

For every

Bt

B t is a (closed) subset of the compact

set W" and thus compact as well. The relevant

are consequently members

of the c o l l e c t i o n S. Repeating this argument f o r all u e U, we are provided with an exhaustion of our neighbourhood U by ε - s e t s . Since U was an a r b i t r a r y (nonempty) open set, 6 i t s e l f is a basis of the topology of Μ (which,

as a subset of 8,

is

countable). We summarize the previous discussion: Μ possesses a countable basis e = {C^, C , C 3 > . . . ) all of whose members have compact closures. We t h e r e a f t e r

construct open subsets D^D^, D^, . . .

of Μ according to the

following instruction: F i r s t , we set m

= 1 and D = C . 1 1 1 Suppose we have already introduced m^ e Ν and D^ S M, where D k

=

C υ ... ν C 1 m k

In this c a s e ,

the compact topological closure D of D is covered by the k k s e t s D^ η C . Because of compactness, there is some μ 6 IN with D~ k

S

l

C υ ... υ C μ

We s e l e c t one such μ > m , call it m , and define k k+1

98

3. M a n i f o l d s

D k+l The s e q u e n c e D , D^, D 3 , . . .

=

C υ .. . υ C 1 μ

f o u n d t h r o u g h t h i s r e c u r s i o n is an open c o v e r i n g

of Μ ( s i n c e e h a s t h e same p r o p e r t y ) . All D^ D have c o m p a c t cclosui l o s u r e s D , and we have t h e i n c l u s i o n D~ £ D~ . Now k k k k+l c o n s i d e r an open c o v e r i n g

U =

I U

1 e D I

of M. F o r a l l k e IN, t h e d i f f e r e n c e D \ D is c o m p a c t and c o n t a i n e d k+2 k+l t h e open s e t D \ D . This i m pr l i e s t h a t K k+3 k

" [Dk+3NDk] I

{

1 6

in

° }

covers D \D k+2 k+l By c o m p a c t n e s s , we may c h o o s e a f i n i t e p a r t i a l c o v e r i n g .

Let us d e n o t e i t

by Bk- In t h e s a m e way we f i n d a f i n i t e p a r t i a l c o v e r i n g in t h e open c o v e r ing

U η Dg

of D z S D 3 ·

1 € II J

Clearly, δ

=

δ

ο

υδ

ι

υ S υ... 2

is a l o c a l l y f i n i t e r e f i n e m e n t of U. As a union of c o u n t a b l y many

finite

f a m i l i e s , Β is c o u n t a b l e i t s e l f , and every member of δ is open w i t h c o m p a c t closure.

Hence, δ i s t h e r e f i n e m e n t we w e r e looking f o r .

Though t h e p r o o f of t h e l a s t t h e o r e m is s o m e w h a t involved, t u r a l and s t r a i g h t f o r w a r d .

it is quite n a -

The n e x t i n t e r m e d i a t e r e s u l t i s much s i m p l e r .

is u s e f u l t o s t a t e i t a l s o a s a s e p a r a t e lemma.

It

3.6. Partition of Unity

99

Theorem 3 . 6 . 2 : For each point χ of a manifold Μ and every open neighbourhood U of x, a compact f:

set

Κ with

χ e Κ c U and

a nowhere

negative

smooth

function

Μ —» R with compact support exist which vanishes outside U and has the

constant value 1 on K.

Proof: It s u f f i c e s to prove the assertion f o r tained in U,

an arbitrary

x-neighbourhood

con-

since the conclusion then immediately carries over to U i t -

self. We choose a chart κ with U

κ

3 χ and replace U by U π U . The action of κ κ

t r a n s f e r s the problem into Rn, and we may r e s t r i c t our investigation to the special case Μ = IRn. Furthermore,

we extend all functions to Rn by setting

them equal to 0 outside the original domains. It is moreover allowed to assume χ = 0 (apply an appropriate translation! ). If necessary, we reduce U further to the open ball

y

y e R"

with radius 2 ε and centre 0. Here,

< 2 ε

ε is a suitably selected positive

real

number. A f t e r these preparations,

we set

y e R"

and seek an arbitrary

y

s ε

nonnegative C -function h: IR —> IR, which is

larger

than zero f o r positive arguments and vanishes everywhere else, f o r instance

exp(-t h(t)

)

(t>0)

= 0

Then the function f : R —> R, defined by

(tsO)

3. Manifolds

100

=

h r c " l y | ] h[2C-|y|] + h[|y|-e]

does the job.

We now come to the key result of this section.

T h e o r e m 3.6.3:

Suppose u

=

{ u

I i

e

0 }

is an open covering of the (smooth) manifold M. Then there e x i s t s a countable partition

Η =

{ « V W · · · )

of unity, which is subordinate to U, such that all functions φ have compact supports.

Proof:

Like in the proof of theorem 3 . 6 . 1 we construct a sequence D^D , D 3 , . . . of open subsets with the properties discussed there:

each D~ is compact and

contained in its successor D . Ali D together cover M. We add to this 6 k+l k family an initial member, namely Dq = e>. Let χ e M. We introduce N(x) as the largest number υ € IN , f o r which χ t D~ ο ν and choose an i(x) e 0 with χ e U . B y theorem 3. 6. 2 we can find a nonl (χ)

negative real-valued C -function f

on Μ whose support is compact and lies

in U η |D~ \D I ι i (x)+2 l (x)J Beyond this,

f

attains in a certain neighbourhood W of χ the constant

3.6.

Partition of Unity

101

value 1. We now f i x k and let χ run through the compact set ing W

D k \D k

· The correspond-

f o r m a covering of D^XD^ of which we pick out a f i n i t e partial c o -

vering. This leads to a f i n i t e collection X of functions f f o r every k. Assembled k χ in their natural order, they provide us with a countable f a m i l y of f u n c tions

X

Χ υ Χ ν Κ υ 1 2 3

=

if we simply w r i t e φ

{ * ΐ ' * 2 · ν · ·

}

instead of f

f o r brevity. The associated neighbour*J hoods f o r m a locally f i n i t e open covering J

·[ W ,W ,W , . . . ]· \ ι a a J of M. The sum

* -

Σ

JeW

is thus w e l l - d e f i n e d and positive on M. The normalized functions

ψJ

=

*y φ

then are a partition of unity with the desired additional

properties.

102

3. Manifolds

3.7. Oriented Manifolds

The t o p o l o g i c a l tions

f r o m IR

group GL(IRn) s GL (IR) of all i n v e r t i b l e linear t r a n s f o r m a n onto i t s e l f s p l i t s into t w o connected components which w e

call GL + (IR n ) ss GL + (R) and GL~(IRn) a GL'(IR). η η the sign of the determinant.

The marks "+" and " - "

indicate

The set GL + (R n ) of orientation preserving maps is a normal subgroup of

in-

while the orientation reversing t r a n s f o r m a t i o n s f o r m

the

dex 2 in GL(R n ),

single coset GL (IRn). This decomposition corresponds with a b i p a r t i t i o n of the ( o r d e r e d ) bases of IRn into

two

orientation classes.

The basis b 1

b η is said

to

posi-

be

tively or negatively oriented, depending on which component of GL(IRn) c o n tains the linear map t r a n s f e r r i n g

( f o r all i ) the i t h unit v e c t o r

e(

to

the

basic element b 1. (The accompanying m a t r i x has the r o w s b I b n ). By this d e f i n i t i o n , the p o s i t i v e class is the one in which the standard

ba-

sis e 1, . . . , e η lies.

The g e n e r a l i z a t i o n of this idea to manifolds is p e r f e c t l y

obvious.

Definition 3.7.1: ( a ) The charts κ and λ of the m a n i f o l d Μ have the same orientation are oriented equally) if the Jacobi determinant of the chart λοκ

1

(i.e.

transition

is e v e r y w h e r e positive.

(b) In case this condition

is f u l f i l l e d f o r

pairs κ,λ of

all

charts

in

the

M - a t l a s 51, we call It (and Μ i t s e l f ) oriented. ( c ) Finally,

the m a n i f o l d (M,11) is orientable if

an oriented

atlas U' of

Μ

e x i s t s which is compatible with U. Remark: The

Euclidean space IRn,

oriented,

the

sphere S n ,

and

the

torus 1Γη = s'x. . . xS 1

while the Möbius strip and the Klein bottle belong

known n o n - o r i e n t a b l e manifolds.

to

the

are

best

We shall not need the l a t t e r t w o m a n i f o l d s

and thus do not discuss them in more d e t a i l . In order to g i v e at least one instructive example,

we mention without p r o o f

3.7. Oriented Manifolds

the real

that

projective

103

space P n (R)

is

orientable

for

odd,

but

not

for

even n.

Every chart κ of

an n-dimensional manifold Μ which represents some point

χ € Μ induces an o» r i e n t a t i o n on TM * s R . The analogous statement f o r the cotangent

space Τ Μ

is true as w e l l .

Charts which are

contained

o r i e n t a t i o n class of κ c l e a r l y lead to the same o r i e n t a t i o n as κ.

in

the

We imme-

d i a t e l y conclude f o r tangent and cotangent bundles:

Theorem 3.7.1: *

The o r i e n t a b i l i t y of one of the three manifolds Μ, TM, and Τ Μ implies the o r i e n t a b i l i t y of the others.

We can also c l a s s i f y the d i f f e r e n t i a l f o r m s in a similar way:

Definition 3.7.2: L e t Μ be an n-dimensional m a n i f o l d and ω € Λ η (Μ) an a l t e r n a t i n g d i f f e r e n t i al f o r m of maximum rank on M. We call ω a volume form if ω ( χ ) * 0 f o r e v e r y χ e M. A l l volume f o r m s can be derived f r o m a p a r t i c u l a r one, e . g . ω, by m u l t i p l y ing with a nowhere vanishing smooth function f :

Μ —> R. All volume f o r m s on

Μ are thus r e p r e s e n t a b l e as f u with suitable f .

We say that ω and f u are in

the same orientation class whenever the quotient function f is p o s i t i v e . Locally,

we choose the o r i e n t a t i o n of d i f f e r e n t i a l f o r m s a l w a y s in c o r r e s -

pondence with that of the 1 - f o r m s . Thus the volume f o r m d x 1 A . . . A d x n is considered as p o s i t i v e ( n e g a t i v e ) if the coordinate d »i f f e r e n t i a l s in this o r d e r d e f i n e a p o s i t i v e ( n e g a t i v e ) basis of Τ Μ.

dx1,...,dxn

A d i r e c t examination whether a given atlas U is oriented can be quite ous. ble

It is even more d i f f i c u l t to f i n d an oriented a t l a s which is with

great

U.

For

interest.

that

reason,

the

following

orientability

tedi-

compati-

criterion

is

of

104

3. Manifolds

Theorem 3.7.2: A m a n i f o l d Μ is o r i e n t a b l e if and only if there is a volume f o r m on M.

Proof: We assume that each chart κ in the atlas I of Μ has a connected domain U . κ This is not a severe l i m i t a t i o n , because every a t l a s can be brought to this special

f o r m if

we replace the charts with their r e s t r i c t i o n s

on the

con-

nected components. A volume f o r m ω g i v e s r i s e to a smooth r e a l - v a l u e d f u n c t i o n which is d e f i n ed on K ( U ) via κ |dx*A. . . Adx Adx j n

Here,

(χ1

it is in the connected region K(U ) either

s t r i c t l y negative. is

f ω

χ η ) is the coordinate v e c t o r of the point χ 6 M. Since f

not vanish,

κ

=

positively

strictly

Depending on which of these a l t e r n a t i v e s occurs, or

negatively

oriented.

It

is

clear

that

this

does

positive

or

we say,

defines

an

o r i e n t a t i o n on M. Vice v e r s a , tion f

f o r each t w o charts κ and λ a p o s i t i v e

if 11 is oriented,

func-

with

(AoKf'i^dxV .

can be found.

This implies,

.Adx"j

=

f dx 1

Λ

...

Λ

dx

if we put

ω

=

*I κ ^dx'ft. . .

ω λ

=

λ ^dx1A. . .

Adxnj

and *

Adx"j

the r e l a t i o n

ω

=

λ and the

quotient

function f ° λ

(λ f ) ω

=

(foA) ω

κ is positive

κ in U

η U^.

We merely

have

to

3.7. Oriented Manifolds

105

construct a partition

{116 °} of unity which is subordinate to the covering

u

=

{ υκ

I KS « }

given by the atlas and calculate the sum ω (We must choose the κ

=

L

)

such that

U

φ κ

11

2 Trg((p ) holds). The v e r i f i c a t i o n i that ω is a everywhere positive volume f o r m may be l e f t to the reader as a trivial

exercise.

By similar procedures one can often build global forms or f i e l d s f r o m cally given

"pieces" with the help of

a suitable partition

of

unity.

method is one of the most important tools of d i f f e r e n t i a l geometry.

loThis

3. Manifolds

106

3.8. Integration of Differential Forms

By f a r the most r e l e v a n t gration

on m a n i f o l d s .

application of

As

indicated

the f o r e g o i n g section is the

above,

the

integral

of

a

inte-

differential

f o r m is a vast extension of the classical Riemann i n t e g r a l on (compact

do-

mains in) IRn. We assume the r e a d e r

to be f a m i l i a r with the d e f i n i t i o n of

the l a t t e r

as

the common l i m i t of upper and l o w e r approximating sums and the most element a r y p r o p e r t i e s required f o r its e x p l i c i t The e x e c u t a b i l i t y iour

of

of

evaluation.

the ideas in question depends s t r o n g l y

the Riemann

integral

under

coordinate

changes.

foundation is the substitution rule of theorem 3.8.1.

Its

on the

behav-

mathematical

As a p r e p a r a t i o n

for

the intended g e n e r a l i z a t i o n we need

Definition 3.8.1: We i n t e r p r e t

the Riemann integral of

Β c r " denotes a compact domain,

a smooth f u n c t i o n f :

Β —» IR,

where

as the integral of the η - f o r m ω e Λ η (Β),

given by

ω(χ)

=

f(x'

x " ) dx 1

A

...

Λ

dx n

and w r i t e

J ω

=

Β

J. . . J f (x 1

xn)

dx^.-dx"

Β

Note that the o r i e n t a t i o n plays an important part here since the

Riemann

volume element dx 1 . . . dx n is independent of the order of the f a c t o r s ,

while

the volume f o r m d x 1 A . . . A d x n must be multiplied by - 1 if we impose an odd permutation on the coordinates. rect

only

for

orientation

T h e r e f o r e the f o l l o w i n g statement is

preserving

sign would have to be inserted.

transformations;

otherwise

a

corminus

3.8.

I n t e g r a t i o n of D i f f e r e n t i a l Forms

107

Theorem 3.8.1: Under

an

orientation

preserving

invertible

continuously

differentiable

transformation τ : Β —> IRn, where Β c IRn is a compact domain, the substitution rule

J

Φ

=J

τ(Β)

τ* φ

Β

holds f o r every n - f o r m 0 e Λ (B ) defined on the image BT = τ(Β).

Proof: We denote the orientation ("signum") of an IRn-basis by sgn(. . . ). the typical

point

tution rule of

χ = (χ 1

real

x n ) to y = (y 1

y n ),

If τ maps

the f a m i l i a r

analysis yields the proposition a f t e r

a short

substicalcula-

tion:

J

φ

=

sgnCdy1

dy n )

τ(Β)

J

fty1

y n ) dy 1 Λ . . .

Λ dy n

yer(B) .

1

=

sgn(dx , . . . , d x n ) χίΒ

=

J

aty1

yn)

^

*">

f o x ( x ) det

f ° x ( x ) df1 Λ ... Λ dfn

=

dx

Λ ...

Λ dx n

J τ*ω

xeB

It

is sensible

to

introduce

the

integration

of

differential

forms

in

two

steps:

Definition 3.8.2: Let Μ be an n-dimensional manifold, II an oriented atlas on M, and ω e Λ η (Μ) an η - f o r m with compact support. ( a ) Consider

f i r s t the special

case that there

is a chart κ covering

the

whole of M, which assigns the coordinates ( x 1 , . . . , x n ) e IRn to the point χ e M. I f ,

furthermore, f : ic(M) —> IR denotes the smooth function cha-

3. Manifolds

108

r a c t e r i z e d by ω(χ)

=

fix1

x n ) dx1

Λ

...

Λ

dx"

we set J ω =

J.. . J f (x 1

Μ ( b ) In general,

xn)dx1...dxn

κ(Μ)

we seek a locally f i n i t e countable partition of unity with

compact supports, f o r instance

•

z

{ v w · · · }

which is subordinate to the covering

Ε

=

{ Uk

I

κ . «

}

of Μ by the domains of the charts. T h e r e a f t e r we choose f o r each k e IN some chart κ κ^ e 1U, whose domain U^ contains the support of φ T h e n k integral of ω is

ίω = Μ

the

Σ Κω

keIN υ k

This looks much uglier than it is. Of course, we have to show that the d e f inition is meaningful. Existence

and finiteness of

the Riemann integral

in (a) f o l l o w

from

the

compactness of K(supp(u)) and the continuity of the integrand. We may replace κ by any other chart λ, say, which meets the requirements, without a f f e c t i n g the integral,

cf.

theorem 3.8.1. The expression in ( a ) is

thus w e l l - d e f i n e d . That a partition exists,

of

unity with the properties mentioned under

f o l l o w s f r o m theorem 3 . 6 . 3 .

(b)

really

The same argument as b e f o r e demons-

trates that the partial integrals have f i n i t e values which do not depend on the employed charts κ .

3.8.

I n t e g r a t i o n of D i f f e r e n t i a l Forms

109

If we had taken a d i f f e r e n t partition of unity, f o r instance

' = {ω = Σ ielN jelN

Σ /

ω

Σ ίψ3ω

=

jelN ielN V }

Hence (b) is also a correct definition.

jslN

Part (a) is a special

case of

(b);

thus we must also show that both statements are in accord with each other. To achieve this,

put II = {κ} in the general formula and use the

partition

of unity which consists only of the single function φ: Μ —» {1>.

Applying theorem 3. 8. 1 to each partial integral separately, get the extremely

we immediately

important

Theorem 3.8.2: Assume τ: Μ —> Ν is a diffeomorphism between two oriented manifolds of dimension n. Then f o r each η - f o r m ω with compact support on N, the substitution rule

ίω = Ιτ*ω Ν holds.

Μ

110

3. Manifolds

3.9. Stokes' Theorem

the

in-

t e g r a l on an oriented m a n i f o l d has s t i l l another p e c u l i a r i t y of eminent

Apart

f r o m the substitution

im-

portance.

rule investigated

in the last

section,

It is known as the "theorem of Stokes" though Stokes himself

proved a very r e s t r i c t e d version of

only

it.

B e f o r e w e can s t a t e the proposition in a f o r m which serves our purposes, need some t e r m i n o l o g y .

Our next d e f i n i t i o n is a s l i g h t and natural

we

exten-

sion of what we had studied e a r l i e r .

Definition 3.9.1: We denote by IR" the positive half-space in Rn, that is the set of

all

vec-

t o r s whose f i r s t coordinate is nonnegative:

R;

=I

(χ 1

€ IR

χη)

A regular domain in an oriented manifold Μ of dimension η is a subset Β ε Μ in which to each boundary point χ e bd(B) a chart κ: U

—» Rn can be found

with κ ( χ ) = 0 and

k(U nB) κ

=

K(U ) η IRn κ

+

The meaning of this condition is of course that the boundary of Β is s u f f i ciently

"tame".

By d i r e c t

insertion

of

the d e f i n i t i o n s

(left

to

the

reader

as an easy e x e r c i s e ) one v e r i f i e s

Theorem 3.9.1: The boundary of a r e g u l a r domain Β £ Μ is an oriented submanifold of

codi-

mension 1 in M. Its o r i e n t a t i o n is obtained f r o m that of Μ by t r a n s f e r r i n g f r o m IRn a c c o r d ing to the f o l l o w i n g rule:

3.9.

Stokes'

Theorem

We c l a s s i f y ,

as

usual,

111

the

standard

basis

o r i e n t e d and do the same f o r the b a s i s e

e1 e

2

en

of

IRn as

positively

of bd(IR ) s IR

n

n

n

+

The r e s t r i c t i o n κ| , of an oriented c h a r t κ of Μ to bd(B) then has 'bd(B)

the

same o r i e n t a t i o n as κ i t s e l f . While bd(B) means the topological

space of the boundary points of B,

we

w r i t e SB f o r the j u s t defined (oriented) manifold with point set bd(B).

S t o k e s ' theorem reads in the required form:

Theorem 3.9.2: Let Μ be an n-dimensional oriented manifold and α e Λη 1 (M) with support.

compact

Then we have f o r each r e g u l a r domain Β

{α

=

{da

3B

Β

E s p e c i a l l y f o r Β = Μ, t h i s yields J da

=

0

Μ

Proof: The argumentation is l a r g e l y p a r a l l e l to t h a t of theorem 3 . 8 . 2 ;

we t h e r e -

f o r e discuss only the main d i f f e r e n c e s . The

proposition

clearly

implies

the

existence

of

all

integrals.

We

shall

obtain t h i s as a b y - r e s u l t of the proof below. First,

we consider the covering of Β induced by the c h a r t s of an o r i e n t e d

atlas

I t s r e s t r i c t i o n to the boundary is a covering of 9B with

gous p r o p e r t i e s .

analo-

T h e r e a f t e r we s e l e c t a subordinate p a r t i t i o n of unity on Β

(getting

a t the same time one of 3B) e x a c t l y

section,

cf.

like t h a t

of

the

foregoing

theorem 3. 8. 2.

Both sides o f the equation then decompose into several

partial

and it will s u f f i c e to e s t a b l i s h the f o r m u l a in question f o r each tion s e p a r a t e l y .

integrals, contribu-

This leads to a reduction to the special case t h a t the

in-

t e g r a t i o n domain is covered by a single c h a r t κ: U —» IRn, say. Without r e -

112

3.

striction,

Manifolds

w e may thus impose the a d d i t i o n a l c o n d i t i o n Β £ U . κ

An a p p l i c a t i o n o f κ shows t h a t it is a l l o w e d to consider Β as a r e g u l a r

do-

main in IP". We e x t e n d the f o r m s to be i n t e g r a t e d t o the w h o l e of R n by s e t ting

them

equal

to

0

outside

the

original

domains.

Then

the

following

s t a t e m e n t r e m a i n s t o be p r o v e d : E v e r y α 6 Λ η 1 ( R n ) w i t h compact support obeys the f o r m u l a

J

=J

α

9R n

da

Rn

+

+

We s p l i t a a f t e r having introduced the standard c o o r d i n a t e s of Rn into tial contributions

Α

_ , 1 f i x k

=

k

Χ

η·.

J

, 1 dx

w h e r e k runs f r o m 1 through n.

Λ

. . . Λ

The f

, k-l dx

. da

simple

Fubini

k

, , \k-l

=

theorem w e

, k+1 dx

Λ

. . . Λ

k . 1 dx „ k

Λ

, η dx

It is

clearly

From

3f

(-1)

Λ

a r e smooth f u n c t i o n s .

s u f f i c i e n t to v e r i f y the t h e o r e m f o r a s i n g l e α^.

and the

par-

like

. . . Λ

j η dx

dx

get

the

first

part

of

the

proposition

after

a

transformation.

The second equation is an immediate consequence of 3M = es.

Remark: Our

proof

of

the

theorem

of

Stokes

can be c a r r i e d

over

literally

to

the

s t a t e m e n t which is obtained by r e p l a c i n g the smoothness of α by the w e a k e r c o n d i t i o n t h a t α is a C 1 - f o r m of

rank (n-1).

is s t i l l continuous ( C ° ) and hence

integrable.

The Cartan d e r i v a t i v e da then

We make use of t h i s c o n s i d e r a b l e e x t e n s i o n in the c h a p t e r s 6 ( c o m p l e x lysis;

Cauchy's theorem) and 22 (potential

much m o r e

powerful

them in the sequel.

generalizations

are

ana-

theory).

In the

literature

even

discussed,

but w e

shall

need

not

4.1. The M e t r i c Tensor

4.

Riemannian

113

Spaces

4.1. The Metric Tensor

As announced at the beginning of chapter 3, we want to r e s t r i c t our investigations

to

manifolds

which

are

endowed

with

an

invariant

It can be r e a l i z e d by an element of arc length ds,

length.

measure

of

like in Euclide-

an space Rn where it may be calculated via

ds 2

=

(dx1)2 + . . . +

(in Cartesian coordinates x 1 ) . 2 The special f o r m of ds is not preserved if

(dxV

1 we r e p l a c e the χ by

another

f r a m e , but a change of charts induces a linear t r a n s f o r m a t i o n of the c o t a n gent space of merely

have

IRn. Thus the d i f f e r e n t i a l s dx 1 to

be

multiplied

with

the

(at some f i x e d point χ e IRn)

invertible

Jacobi

matrix

of

the

transition. This operation c a r r i e s ds 2 over to a quadratic f o r m whose signature (+ n , is also invariant. vanish,

i. e.

In g e n e r a l ,

the o f f - d i a g o n a l

coefficients will

no

longer

in the new coordinates y 1 there w i l l be contributions by

ed t e r m s " dy'-dy^ (i *

"mix-

j). 2

In any case, w e represent the squared length element ds the " m e t r i c "

g.

In contrast

that g is i n d e f i n i t e .

to

Riemann h i m s e l f ,

by a (0, 2 ) - t e n s o r ,

we a l l o w

the

possibility

Only the requirement that the determinant of g is no-

where z e r o is indispensable.

The precise f o r m u l a t i o n thus is

Definition 4.1.1: ( a ) A metric on the n-dimensional (smooth) m a n i f o l d Μ is a nowhere d e g e n e r a t e (0, 2 ) - t e n s o r f i e l d g whose signature is constant on M. Under these conditions,

the pair

(M, g ) is c a l l e d a Riemannian manifold

or a Riemann space. ( b ) If

g

ab

chart,

are

the

coordinates

of

the

metric

we denote the c o e f f i c i e n t s of

also f o r m the components of a tensor,

with

respect

the inverse m a t r i x

to by

a

suitable

gab.

which is of rank ( 2 , 0 ) .

They

We call

114

4. Riernanntan Spaces

the metric associated with the g a b contravariant,

the other type (g

ab

)

covariant. As the example _ab d

Τ indicates,

c

=

g

ap

g

cq

Τ

bqd ρ

we can " s h i f t " tensor components and thus switch f r o m con-

t r a · to covariant expressions and vice versa.

Note the index positions

and that we used the same letter f o r both modifications. ( c ) The (arc) length of an open curve y: I —» Μ between the parameter v a l ues t , t

e I £ R is given by a curve integral

over the length element

ds. It can be evaluated in arbitrary coordinates χ 1 j

ds

2

=

g

. a

ab

dx

,

dx

xn f r o m

b

via the formula t l ( r ; t ,t ) 1

r

=

2

. x°xbdt

2

/g

I

ab

t1 where,

as usual, the dots denote derivatives with respect to the curve

parameter t. The curve y is timelike tive,

2

is positive, spacelike if it is nega2 and null or lightlike whenever ds = 0. In the definition of the if

ds

arc length we only admit timelike curves such that the radicand is a l ways £ 0. ( d ) A geodesic connecting the points x, y 6 Μ is a curve running f r o m χ to y whose length, in comparison with "neighbouring" curves having the same property,

is

stationary.

Equivalently,

the

first

variation

51

(with

f i x e d initial and f i n a l points) of 1 vanishes.

We know f r o m the calculus of variations that extremal problems like the one above defining the geodesies on Μ can be reduced to systems of d i f f e r e n t i a l equations. Without p r o o f , we state

ordinary

4.1. The Metric Tensor

115

Theorem 4.1.1: The variation problem

L ( x ' , x ' , t ) dt

=

0

with the smooth Lagrange function L ( x , x ; t ) and f i x e d boundary conditions x ! ( t ) and x l ( t ) is tantamount to the Euler-Lagrange equations

d dL

dL

dt dx

3x'

f o r all i.

From this we deduce a vector d i f f e r e n t i a l equation f o r the geodesies:

Theorem 4.1.2: The geodesic curves on M, dim Μ = n, are characterized in arbitrary coordinates (x 1

x n ) by the conditions x a + Ta

be

ib

=

0

(with the arc length as parameter). Here, we used the Christoffel symbols _ be

f o r the sake of

1 2

ad 8

Γ [8dc,b

_

+ 8bd,c

1 8bc,dJ

brevity.

Proof: The variational problem which defines the curve h?s the f o r m given in theorem 4.1.1. The Lagrange function is

4. Riemannlan Spaces

116

. L

ds dt

=

/

·a -b

=

X

if we f o r the moment leave the independent variable t undetermined. The d t h E u l e r - L a g r a n g e equation is

Thus, by insertion,

g

•b-c XX be, d

d

2 L

dt

g

_ Lg

3L

d ah

3x d

dt ax d

we get [öbxc + i V l be L d dJ 2 L

dc

"C T x + Lg

d

g

de

dt

Xc + g

X bd

2 L

-b . *b·c _ · b· c x + Lg χ χ + Lg xx bd de, b bd, c

L g dc X

r"

"b

L g bd X

2 L Having c a r r i e d through all d i f f e r e n t i a t i o n s ,

we are allowed to put the pa-

r a m e t e r equal to the arc length s. Then we have L = 1 and L = 0, and the f o r m u l a s i m p l i f i e s considerably:

g

db

" b ^ i r ^ 1-b-c x + g +g - g x x 2 [dc, b bd,c °bc,dj

=

_ 0

Here of course the dot has to be interpreted as d e r i v a t i v e with respect s. Multiplication by g a d yields the proposition.

As long hoods,

as we r e s t r i c t

our

investigation

to

sufficiently

small

neighbour-

the geodesies are the shortest lines connecting the end points.

For

l a r g e regions this statement in general does not remain true (example:

two

antipodal points on S ).

117

4.2. C h r i s t o f f e l Symbols and Covariant Derivat ion

4.2. Christoffel Symbols and Covariant Derivation

Under

coordinate

the Christoffel

transformations,

f o r e g o i n g section 4.1 do not behave like tensors.

symbols d e f i n e d

in

the

They obey the more com-

p l i c a t e d rule

ax 1 ax b ax°

a Γ

Jk

~

Γ bc

a V

a ? ai1 a ?

ax a ax b a i 1 a i *

as can be v e r i f i e d by a simple calculation. (l,2)-tensor Ta

be

ax a ax b

The corresponding

law f o r

any

would be

ax1 ax b ax c T1

=

Ta « » ax a 8xJ a i "

Jk

i.e.

only

the

first

term

of

the

formula

above.

The

comes f r o m the nonlinearity of the chart t r a n s i t i o n ;

second

contribution

it mainly consists

of

the d e r i v a t i v e of the Jacobi m a t r i x . N e v e r t h e l e s s the l e f t hand side of the geodesic equation we found in t h e o rem 4 . 1 . 2 is a v e c t o r = ( 1 , 0 ) - t e n s o r . variational

problem

which

led us to

This was to be expected, the f o r m u l a

in question

since

is

the

invariant

under chart changes. It f o l l o w s that the x a cannot f o r m the components of fact,

a tensor

either.

In

one easily c o r r o b o r a t e s this statement by d i r e c t insertion of the d e -

finitions. We are not going to d w e l l on these t r i v i a l

observations and p r e f e r t o

v e s t i g a t e the more general case of p a r t i a l d e r i v a t i v e s of

in-

tensors.

00

We know a l r e a d y that f o r each C - f u n c t i o n f on Μ the f of a (0, l ) - t e n s o r ,

namely the gradient df

If

we,

however,

are the components

partially

=

f

ιa

differentiate

dx a

a vector

field

ν € 5(M),

we

are

118

4. Riemannian Spaces

l e f t with components ν

which u n d e r t h e t r a n s i t i o n t o new c o o r d i n a t e s χ >b behave a c c o r d i n g t o

in p l a c e of t h e χ

v1

.J

=

θ

J

ax 1

v1

a2x'

ax ,J

ax-

axb.

a x ' a x " ax J

ax®

This r e l a t i o n r e m i n d s us of t h e a n a l o g o u s r u l e f o r t h e C h r i s t o f f e l s y m b o l s . A c a r e f u l i n s p e c t i o n shows t h a t t h e a d d i t i o n a l t e r m s in t h e e x p r e s s i o n ν

;b

=

ν

,b

+ Γ

be

ν

e x a c t l y c o m p e n s a t e each o t h e r . In o t h e r w o r d s : t h e ν

!b

form the components

of a t e n s o r of r a n k (1,1). The p r o c e d u r e j u s t c a r r i e d o u t can obviously be e x t e n d e d u n i q u e l y t o a d e r i v a t i o n on a l l t e n s o r f i e l d s :

Definition 4.2.1: The c o v a r i a n t d e r i v a t i v e of a t e n s o r f i e l d of r a n k ( r , s ) on t h e m a n i f o l d Μ r e l a t i v e t o t h e k^1* c o o r d i n a t e is t h e d e r i v a t i o n w i t h r e s p e c t t o "β" which f o r s c a l a r f u n c t i o n s f reduces to the p a r t i a l coordinate derivative f

and

to

,k

+ Γ

kc

ν

f o r a v e c t o r f i e l d v.

In g e n e r a l , we f i n d

Theorem 4 . 2 . 1 : The c o v a r i a n t

d e r i v a t i v e of a ( r . s ) - t e n s o r

f i e l d Τ by t h e i n d e x k is

t e n s o r of r a n k ( r , s + l ) which r e a d s in c o o r d i n a t e s

the

4.2. Christoffel Symbols and Covariant Derivation

a ... a Γ Τ1

b ...b ;k I s

a ... a Γ Τ1

=

a + Γ 1

b ...b ,k I s

p...a τ

r

Y" p r

119

b

.

a ...a τ

1

Γ

kbi

Here,

in the indicated manner,

f o r every upper (contravariant)

additive term has to be inserted,

and f o r each lower

index,

an

(covariant) index a

negative one.

Proof: It is already known that these formulas yield tensors of the correct ranks which,

when specialized to ( 0 , 0 ) - or (1,0)-tensors,

reduce to the

expres-

sions given in the theorem. Insertion into the Leibniz rule leads us to the desired result.

Covariant derivatives can also be introduced if on some manifold Μ there is 3 no metric, but only an a f f i n e connection. By this we mean a system of η smooth real functions (dim Μ = n) which may be chosen a r b i t r a r i l y

in one

coordinate frame. T h e r e a f t e r they have to be extended to other charts of Μ in correspondence with the transformation formula found earlier. We do not attempt to discuss further this possibility as

well

to

the

geodesic

equation

(because

the

latter

which is only

applicable

contains

the

Γβ

be ). The general theory will not be needed in the sequel. Concerning more detailed information, we r e f e r to the modern standard literature on d i f f e rential geometry.

The so-called Levi-Civitä connection originates f r o m the C h r i s t o f f e l bols, calculated a f t e r section 4.1, J.a be It is torsion free:

_ ~

1 ad Γ 28 [8dc,b

+ + gbd,c

Ί gbc,dJ

sym-

120

4. Riemannian Spaces

Another

peculiarity

is

of

great

importance,

since

it

allows

us to

change index s h i f t s with the help of the metric and covariant tion.

inter-

differentia-

The m e t r i c is namely "covariantly constant":

Theorem 4.2.2: All covariant d e r i v a t i v e s of the metric tensor vanish,

g

g

= 0

h

= η 0

ab

5" b;c

= 0

Proof: By theorem 4 . 2 . 1 and the d e f i n i t i o n of the C h r i s t o f f e l symbols we have

g

=

ab;c

g

ab,c

- r

d

ca

g - r db

d

g cb ad

=

0

The c o n t r a v a r i a n t components of g are (as c o e f f i c i e n t s of trix)

rational

f u n c t i o n s of

c o v a r i a n t l y constant. _a s b, c

=

the g

Finally,

and thus,

the inverse

by the chain rule,

ab the d e r i v a t i o n p r o p e r t y of " ; "

,· ad * (g g,h) db ;c

=

g

ad

.„ ;c

db

.ad g grth.„ db;c

+

=

implies 0

A r e l a t e d problem which also leads to covariant d e r i v a t i v e s is the l i z a t i o n of the f a m i l i a r concept t r y in

"parallel

t r a n s p o r t " of

ma-

likewise

elementary

generageome-

IRn.

On account

of

its

linear

lism of

two

vector

even i f

they

are

u(x)

attached

condition u ' ( x ) = v ' ( y ) .

structure, and v ( y ) to

it

is possible

to

in (Rn i r r e s p e c t i v e

different

positions.

In a m a n i f o l d M,

however,

of

This

the

paralle-

reference

frames,

define

is achieved

by

t h e r e is in general

the no

canonical r e l a t i o n s h i p between the tangent spaces at t w o d i f f e r e n t points χ and y.

121

4.2. Christoffel Symbols and Covarlant Derivation

The natural idea to explain parallelism of vectors through the above equation depends on the chart needed to introduce the coordinates and is thus not an intrinsic concept. Nevertheless,

in Riemann spaces (and even in manifolds with a f f i n e connec-

tion) we can at least define the parallelism of

a vector f i e l d with

itself

along a curve y:

Definition 4.2.2: A

vector

field

ν € 8(M)

on

the

Riemannian

manifold

(M,g)

is

parallel

transported ( i n the sense of Levi-Civitä) along the open curve y: I —> Μ if for

all t e I the directional

derivative of

vector y ' ( t ) is equal to zero. y(t)

v a with respect to the tangent

In suitable coordinates, f o r which =

(x 1 ( t )

xn(t))

holds, this condition hence reads

7

b

a

ν

ib

=

r>0

According to theorem 4.1.2, the geodesies subsume under this definition. We merely have to bring the relevant d i f f e r e n t i a l equation by introducing coordinates

ua

the

of the tangent vector of y to the f o r m 'a „a U + Γ

be

b e U U

= 0

or b fa u u

,b

. _a cΙ +Γ u be J

b a = u u ;b

= 0Λ

We thus obtain

Theorem 4.2.3: A curve y in a Riemann space is a geodesic if and only if its tangent tor y ' i s ( L e v i - C i v i t ä ) parallel transported along y.

vec-

122

4. Riemannian Spaces

It is also possible to express the component version of the formula f o r the Lie derivative with covariant rather than partial derivatives.

We only need

to replace the commas by semi-colons:

Theorem 4.2.4: The L i e derivative of an (r, s)-tensor f i e l d Τ with respect to ν € B(M) is, in coordinates,

Proof: A f t e r substitution

of the covariant d i f f e r e n t i a t i o n s by their defining e x -

pressions one notes that all C h r i s t o f f e l symbols cancel.

We are l e f t

with

the same equation in partial derivatives whose validity has been proved a l ready in theorem 3. 5. 5.

4.3. Normal Coordinates

123

4.3. Normal Coordinates We derive f r o m definition 3 . 3 . 3 that tangent vectors are d i f f e r e n t i a l

ope-

rators. A vector f i e l d on an n-dimensional manifold can t h e r e f o r e be i n t e r preted as a system of η ordinary d i f f e r e n t i a l equations in equally many dependent variables (one only has to w r i t e the condition in coordinates). solution consists of

a curve whose tangent vectors are determined

where by the f i e l d .

In geometric language,

The

every-

the appropriate description

is

given by

Definition 4.3.1: Let ν e S(M) be a vector f i e l d on the manifold M. By an integral curve of ν with i n i t i a l point χ € Μ we mean a curve y which is defined on some open interval I S O ? containing 0 which f u l f i l l s y(0)

=

χ

and

r ' Ct) f o r all parameter values

=

v(r(t))

tel.

If no proper extension of y exists which has the same property,

i. e. no in-

tegral curve y: Γ —> Μ of ν with I c I £ R, we call y maximal.

One of the most fundamental results in real analysis is ( c f .

Kamke [1964])

the observation that each system of ordinary d i f f e r e n t i a l equations whose c o e f f i c i e n t functions are infinitely o f t e n d i f f e r e n t i a b l e possesses f o r set

of

initial

conditions

a unique

solution.

The

integral

curve

is

any also

smooth and can be extended to a suitable neighbourhood. For reasons of space limitation, we do not want to investigate how to prove this proposition (a very simple argument employs the integration method of Picard and Lindelöf).

Instead,

we translate the theorem

in question

into

124

4. Riemannian Spaces

the language of d i f f e r e n t i a l geometry.

Theorem 4.3.1: Every vector f i e l d ν on a manifold Μ has f o r any χ € Μ a unique maximal int e g r a l curve with initial point x.

We are mainly interested in the application to geodesies.

Although the r e -

levant d i f f e r e n t i a l equation is of second order, we may reformulate it with a simple trick such that it attains the shape required f o r the above theorem. We just point.

have to choose a chart κ,

say,

of

Μ which contains the

initial

It attaches to a typical element (x, v) of the tangent bundle TM the

coordinate vector (x a ; v a ). With the help of the map , a a-. (χ ; ν )

ι—»

we define a vector f i e l d on TM which by theorem 4.3.1 has a unique (maximal) integral curve. On account of theorem 4.1.2 the l a t t e r is nothing else than the geodesic starting at χ with the tangent vector v ( x ) .

We hence have

Theorem 4.3.2: To each point χ in the manifold Μ and every ν

e TM

there is a uniquely

determined maximal geodesic line y with initial point y ( 0 ) = χ and tangent vector y ' ( 0 ) = ν . X

Let us f o r the moment denote the geodesic with initial conditions ( x , v ) by y

x,v

The result just found implies that y

x,v

is uniquely

defined in some

open interval around 0. Therefore, f o r all (x, v) e TM a positive real number p(x, v) exists such that the domain of y x,v includes the open

interval

between - p ( x , v ) and + p(x, v). From now on we f i x the point χ e Μ and select an arbitrary chart it £ containing x.

Assume that the induced chart κ ( c f .

ates with a typical ν e TM scalar product on

IRn

the coordinates (v 1

^

theorem 3. 4.1) associv n ) e IRn. The Euclidean

is, as usual, written in the f o r m

4 . 3 . Normal C o o r d i n a t e s

125

=

l

u

v

Due t o t h e c o m p a c t n e s s of t h e s t a n d a r d Sphere Π in TM

p

=

mir in

p(x,v)

= R , the number

= 1 j-

X e x i s t s and is > 0. A f t e r a r e p a r a m e t r i z a t i o n we m o r e o v e r f i n d y

x,sv

(t)

=

y

x,v

The i n t e r p r e t a t i o n of t h i s e q u a t i o n is t h a t ,

(st) w h e n e v e r one side is m e a n i n g -

f u l , t h e o t h e r is a l s o and both e x p r e s s i o n s coincide. In t h e p a r t i c u l a r c a s e t = 1 t h i s shows t h a t f o r any t a n g e n t v e c t o r ν € IRn w i t h s ρ

the point y

(1) e Μ is w e l l - d e f i n e d . We t h e r e f o r e o b t a i n

T h e o r e m 4.3.3:

To each p o i n t χ of t h e n - d i m e n s i o n a l Riemannian s p a c e Μ we can c o n s t r u c t a n e i g h b o u r h o o d V of t h e o r i g i n in TM such t h a t f o r a l l ν e V t h e g e o d e s i c X X X w i t h i n i t i a l c o n d i t i o n s (x, v) can be e x t e n d e d a t l e a s t up t o t h e p a r a m e t e r value 1. The a s s i g n m e n t ν

ι—> exp (v) X

d e f i n e s t h e exponential mapping exp : V X

=

X

y

x,v

(1)

—» Μ a t t h e p l a c e x.

It is

obvi-

are

those

ously C .

Of

primary

importance,

especially

R i e m a n n i a n m a n i f o l d s which f u l f i l l ρ

in a p p l i c a t i o n s X

to

physics,

= °o f o r any x. This m o t i v a t e s t h e i n -

t r o d u c t i o n of t h e f o l l o w i n g n o t a t i o n .

D e f i n i t i o n 4.3.2:

A R i e m a n n i a n s p a c e Μ and i t s m e t r i c g a r e c a l l e d geodesically complete if t o each (x, v) e TM t h e g e o d e s i c w i t h s t a r t i n g point χ and i n i t i a l

direction

126

4. Riemannian Spaces

ν can be extended up to a r b i t r a r i l y large real values of its curve parameter.

Almost all Riemann manifolds considered in this monograph s a t i s f y this assumption. We also note the t r i v i a l l y equivalent formulation

Theorem 4.3.4: A Riemannian manifold (M,g) is geodesically complete if and only if the e x ponential mapping exp

X

f o r each χ € Μ has the whole of TM as its domain of X

definition.

The aforementioned construction enables us to introduce distinguished c o o r dinate frames.

To achieve this,

we start

thanks to the implicit function theorem,

out f r o m the observation the map exp

is locally

that,

inverti-

ble. We need a f e w more conventions:

Definition 4.3.3: Let (M, g ) be a Riemann space of dimension η and χ some point in M. We choose a neighbourhood W £ V X X

of Ο e TM , f o r which exp is l nJj e c X X w

tive. The inverse mapping is f

:

U

» TM

The normal coordinates of the point y e U relative to the basis e v J χ I of TM are the numbers y given by the decomposition

f(y)

=

y1

e

e n

j

We w i l l r e f e r to the point χ itself as the origin of the frame thus characterized.

4 . 3 . Normal

Coordinates

127

T h e r e a s o n f o r t h e i n t r o d u c t i o n of t h i s c o n c e p t i s

Theorem 4.3.5: At t h e o r i g i n of any n o r m a l c o o r d i n a t e s y s t e m a l l C h r i s t o f f e l s y m b o l s

van-

ish.

Proof: F o l l o w s i m m e d i a t e l y f r o m a s i m p l e c a l c u l a t i o n w h i c h we s h a l l n o t b o t h e r t o w r i t e down in f u l l .

4. Riemannian Spaces

128

4.4. Curvature The selection of the charts used to describe a manifold Μ is largely trary.

arbi-

Many d i f f e r e n t approaches lead to equivalent representations.

the local ( d i f f e r e n t i a l ) properties defined by the atlas are of

Only

fundamental

importance.

Definition 4.4.1: Two atlases U and U' on the basic space Μ are compatible with each other if they induce the same d i f f e r e n t i a b l e structure. In this case we usually identify the manifolds (M,U) and ( Μ , Ι Γ ) .

Clearly this is an equivalence relation on the class of all conceivable

at-

lases of M. A more practical rephrasing is

Theorem 4 . 4 . 1 : The atlases Ϊ and II' of Μ are compatible if and only if tions k°k'

1

and κ ' « κ

1

all chart

transi-

with κ Ε II and κ' € Μ' are diffeomorphisms of

the

respective domains. A necessary and s u f f i c i e n t condition is that the union U υ IT

itself

is an

atlas.

It is o f t e n d i f f i c u l t to decide whether two given atlases are compatible or whether

two

different

descriptions

lead

to

the

"same"

manifold

(in

the

sense of definition 4.4.1). In the chapters to f o l l o w we shall consider numerous examples of coordinate frames, mainly of the Euclidean spaces, which give an ample confirmation of this statement. Although all compatible atlases are tantamount in principle,

one frequently

will find certain coordinates more convenient than others.

(This has been

the cause f o r writing the book on hand! ).

4.4. Curvature

129

Normally the best choice depends on the intended purpose; some particular manifolds,

however,

possess characterizations of exceptional

mathematical

simplicity. To give an example, this holds f o r charts of Riemannian spaces relative to which the metric has constant components.

We introduce a special notation

to cover this case which is suggested by the experiences from the theory of surf aces.

Definition 4.4.2: A chart κ of the Riemannian manifold (M,g) is f l a t if the components g the metric are constant in space,

i.e.

Ab

of

all Christoffel symbol ( r e f e r r e d to

κ) vanish identically. Analogously, a f l a t atlas is one which consists only of f l a t charts. Finally

the

space Μ itself

is called

flat

if

the

atlas

U^ is

compatible

with some f l a t atlas IT. Η The opposite to " f l a t " is "curved".

In a f l a t coordinate system the covariant derivatives reduce to the partial. Then f o r any vector f i e l d ν e S(M) ν

=

ν

=

;c;d

ν

=

,c,d

ν

,d,c

,d;c

holds. The implied relation a

V

= V

;c;d

a

;d;c

remains valid even in a curved frame, because as a tensor law it is independent of the coordinates. In curved spaces,

on the other hand, it cannot be true f o r all v.

Instead

we have ν

a

;c;d

=

ν •

a

,

>c,d

„a

+

cb

Γ

b

ν

,d

,

_a

+

Γ

db

b

ν

,c

-

_b

Γ

dc

ν

a

,b

(r a c b , d • r a de r e be - r a eb r e d e1l v b

I

an expression which normally is not invariant under the permutation c * d.

130

4. Riemannian Spaces

This strongly

suggests to consider the d i f f e r e n c e v a

- va

;c;d

its v-independent part as a measure of spatial curvature.

;d;c

or

rather

An easy calcula-

tion yields

Theorem 4.4.2: The second order covariant derivatives of any vector f i e l d ν € 8(M) on the Riemann space Μ s a t i s f y the condition

ν where R a

bed

a

;c;d

- ν

a

=

,d;e

_,a

R

bed

ν

b

are the components of a ( l , 3 ) - t e n s o r f i e l d , the Riemann (curva-

ture) tensor. Written down explicitly, R

bed

=

Γ

_

cb,d

these are .

db,c

ds

.

be

cs

db

The analogous rule f o r a 1 - f o r m ω e A 1 (M) is ω

b;c;d

-ω

=

b;d;e

- R

bed

ω

a

while f o r smooth functions F: Μ —> IR F

;c;d

- F

,d;e

holds.

Proof: The quantities Ra M d i f f e r e n c e va

bed

which have been introduced as an abbreviation in the

- va ;e;d

can be found without problems by a simple evalu;d;e

J

r

ation of the covariant derivatives. The remaining relations are gotten similarly by inserting the Here we have to apply the Leibniz rule of d i f f e r e n t i a l calculus.

definitions. That R a bed

are tensor components is a direct consequence of theorem 4. 2.1, the transformation law, and the mathematical properties of the v a .

The Riemann tensor has a lot of symmetries which can be understood best in

4.4.

Curvature

131

the completely c o v a r i a n t form.

Theorem 4.4.3: ( a ) The Riemann curvature of (M, g) can be c a l c u l a t e d f r o m R abed

=

g - Ζ ~ Κ + Ε 2 [ ac,b,d ad,b,c °bc,a,d bd,a,c 1 ef Γ 47 8 I 8ea, c

g

fb,d ~ 8 ea,d 8 f b , c

+

+

ec,a

fd,b

g

g

~ R

+ β

β

be,Γ

- ΰ Β + σ β -σ Β ea ,c °bd,f eb,c ad,f eb,d ac,f

+ Ä

Ä

ed, a

g

ac,e

g

bc,f

~ Κ

bd,f

8

ec,a

R

bd,f

~~ Ρ

fc,b

fd,b

ea,d

R

ed,a

R

ea ,d e f c , b

R

g

+ Κ

ea, c

R

8

eb,d e f c , a

+ R

ec,b

R

eb,c

~ R

ad,f

s

ed,b

σ

fd,a

R

ac,f

ad,,e g b c , f ]

(b) If we i n t e r c h a n g e a with b or c with d, t h e s e c u r v a t u r e components a r e multiplied by -1,

while R

(a, b) « ( c , d ) : R abed

=

abed

remains invariant

- R bacd

=

- R abdc

=

under the

permutation

+ R cdab

and so on.

Proof: Part

(a)

symbols.

is obtained f r o m theorem

4.4.2

by insertion

Proposition (b) is an immediate consequence,

of

the C h r i s t o f f e i

as an inspection

of

the individual t e r m s r e v e a l s .

We saw a l r e a d y t h a t the Riemann t e n s o r R®b space.

Turning

things

around,

the equation

condition which implies the f l a t n e s s have the very handy c r i t e r i o n

vanishes i d e n t i c a l l y Ra

= 0

is

an

in

flat

integrability

bed of the underlying manifold.

We thus

132

4. Riemannian Spaces

Theorem 4 . 4 . 4 : The Riemannian space (M,g) is f l a t if and only if R"

bed

=

r

- Γ

cb,d

_

db,c

ds

_

be

cs

db

is identically true.

Remarks: (a) All formulas discussed in this section which do not contain the metric tensor explicitly can be generalized to (non-Riemannian) manifolds with affine

connection,

but we shall not use this;

compare the

situation

with Neutsch & Scherer [1992], (b) In Euclidean space the Riemann curvature is identically zero, as the quantities derived from it (see below).

Nevertheless

as well the

just

found r e l a t i o n s are valuable, e . g . in order to check the c o r r e c t n e s s of the r a Ki- .>

Two identities s a t i s f i e d by the R ^

are of special interest:

Theorem 4 . 4 . 5 : The curvature tensor of a Riemann manifold is constrained by R"

bed

+ R"

cdb

+ R"

dbc

=

0

and R"

bcd;e

+ R"

bde;c

+ R"

bec;d

= 0

These equations are known as the 1 s t and 2 n d Bianchi i d e n t i t y ,

respective-

iy-

Proof: It is not reasonable here to carry out the necessary calculations in a r b i t r a r y coordinates since the large number of terms would be tiresome.

4.4. Curvature

133

A much better way to proceed is to f i x some point χ ε Μ in advance and employ Γ*

be

normal

(x)

amount

coordinates

cancel; of

with

only their

labour

origin

derivatives

considerably.

at

x.

Then

survive.

The trivial

all

Christoffel

This reduces the

details

of

the

symbols necessary

deduction

are

l e f t to the diligent reader.

We moreover enumerate several tensors related to R a

bed

Definition 4.4.3: The Ricci tensor of a Riemannian space is the contraction of the Riemann curvature tensor given by R

=

ab

R

acb

while the scalar curvature equals

η

R

=

ab

g

y.c

R

.

=

acb

g

ac

g

bd

_ R . „ abed

The Einstein tensor is the expression G

ab

=

R

ab

- - g 2

ab

R

From the Bianchi identities we derive a corollary which is fundamental f o r general r e l a t i v i t y ,

Theorem 4.4.6: The Einstein tensor is divergence f r e e :

134

4. Ri ernannt an Spaces

and symmetric, G

=

ab

G

ba

The l a t t e r property is (in Riemann spaces) shared by the Ricci tensor, R

=

ab

R

ba

Proof:

The symmetries listed in theorem 4 . 4 . 3 lead to R

=

g

R

bd

=

g

R

abed

=

R

cdab

db

and this in turn implies the analogous formula f o r G . Now the requested ab

d i f f e r e n t i a l equation is a consequence of theorem 4 . 4 . 5 . 0

= =

R ab + R ab + R ab be; a ea;b R a -b ; e2 R"

=

R

;e

We have

- R ba

- Rab

be; a

ae;b

and this yields the proposition without e f f o r t .

"Pure curvature" is provided by the Weyl tensor which is also obtained as a modification of R a

bed

Definition 4.4.4:

The Weyl tensor of an n-dimensional Riemannian space (n 2 3) is U*

bed

=

Ra •

bed

+ —

n-2

— — ( n-l) · (n-2)

[ g [ L

δ

bd

d

Ra g

c

be

- g

be

- δ"

c

Ra g

d

bd

+ 3a

c

R

bd

- 3a

d

R

be

R

It has the same symmetries as the Riemann curvature but, in c o n t r a s t to the latter,

all

its

contractions

vanish.

The

importance

of

the

Weyl

tensor

4.4. Curvature

135

stems f r o m the f a c t that two manifolds of dimension η s 4 with metrics can be mapped by a length-preserving map (i. e. are developable) onto each other if and only if the Weyl tensors in corresponding points are equal. In three dimensions,

Wa

vanishes identically; but we are not going to bed discuss the deeper properties of the Weyl curvature since we have no applications f o r them. We merely want to emphasize that the number of independent among the η components of the Riemann tensor is strongly reduced on account of the symmetry and the f i r s t Bianchi identity. It is exactly 1 - n

2 , 2 , , (n-1)

or, in turn, 1,6,20 f o r η = 2,3,4. In the special case of

the theory of surfaces (n = 2),

ents which do not vanish identically are R

1212

, R

1221

the only c o e f f i c i -

, R

2112

, R

2121

. A

tho-

rough consideration shows that the curvature tensor is exactly the (-21f o l d of the Gaussian curvature K: R Since

Κ

is

Gauß [1827], f actor).

essentially

the

only

=

- 2 Κ

curvature

invariant

of

a

surface,

cf.

such a result could have been expected (save f o r the precise

136

4. Riemannian Spaces

4.5. Volume

For

certain

investigations

concerning

tangent,

cotangent

or

other

vector

b u n d l e s o v e r some m a n i f o l d , t h e c o o r d i n a t e f r a m e s d e f i n e d by t h e c h a r t s do not

suffice.

Frequently,

however,

a generalization

which

suggests

itself

t u r n s o u t t o be h e l p f u l :

D e f i n i t i o n 4.5.1:

(a) A covariant n-Bein on an n - d i m e n s i o n a l m a n i f o l d Μ is a s y s t e m of η v e c tor fields e

e such t h a t t h e l o c a l v e c t o r s e ( x ) , . . . , e (x) a t a l l i n i n p l a c e s χ e Μ a r e l i n e a r l y i n d e p e n d e n t and t h u s f o r m a b a s i s of t h e t a n g e n t s p a c e TM . (b) C o r r e s p o n d i n g l y ,

a contravariant n-Bein is an n - t u p e l 1

f o r m s which yield in e v e r y point χ a b a s i s e ( x ) tangent

e1

e" of

n

e ( x ) of t h e

1co-

space.

(c) Two n - B e i n e e l

e° and e 1

e η a r e dual t o each o t h e r if t h e d u a -

lity condition < e ' ( x ) , e (x)>

(d)

is identically f u l f i l l e d .

Μ i s c a l l e d parallelizable i f t h e r e

=

i'j

i s a global n - B e i n o n M.

It is e v i d e n t t h a t t o every ( c o n t r a - o r c o v a r i a n t ) n - B e i n on Μ t h e r e is a uniquely d e t e r m i n e d dual one. of t h e dual b a s i s .

It can be f o u n d by a p o i n t w i s e

construction

(The s m o o t h dependence of t h e p o s i t i o n f o l l o w s f r o m t h e

i m p l i c i t f u n c t i o n t h e o r e m ) . T h e r e f o r e t h e t y p e of t h e n-Bein needed in p a r t (d) is i m m a t e r i a l . The p a r a l l e l i z a b i l i t y

of t w o m a n i f o l d s Μ and Ν i m p l i e s t h a t of t h e

direct

p r o d u c t Μ χ N. In p a r t i c u l a r ,

it is e a s i l y seen t h a t R and S 1 a r e p a r a l l e l i z a b l e ;

E u c l i d e a n space

thus

the

4 . 5 . Volume

137 ,n R'

and the

R χ ...

χ R

torus

.π Τ' have t h e s a m e p r o p e r t y . A much m o r e i n t e r e s t i n g q u e s t i o n is which spheres can be p a r a l l e l i z e d . leads to extremely

deep c o n n e c t i o n s

with

algebra

and a l g e b r a i c

It

topology

(K-theory). We s h a l l come back t o t h i s p r o b l e m in s e c t i o n 11.5 w h e r e

more

e x p l i c i t i n f o r m a t i o n is given. The f o l l o w i n g t e r m i n o l o g y is a d i r e c t a n a l o g y t o t h e c o r r e s p o n d i n g n o t a t i o n concerning coordinate frames.

D e f i n i t i o n 4.5.2:

A c o v a r i a n t n-Bein e called

e on t h e n - d i m e n s i o n a l R i e m a n n i a n s pr a c e Μ w i t h i n orthogonal if f o r all i * j t h e e q u a t i o n g i e ^ e ^ ) = 0

metric

g is

holds.

If we have in a d d i t i o n g(e , e ) = ± 1,

and c o n s e q u e n t l y t h e

condi-

tion gCe^e ) holds f o r

all

i , j 6 {1

n},

=

±δ

i

t h e n-Bein e , . . . , e is ortho1 η

we say t h a t

normal.

R e l a t i v e t o an o r t h o n o r m a l c o o r d i n a t e s y s t e m , a l l c o m p o n e n t s of t h e m e t r i c t e n s o r a r e e q u a l t o one of 1, -1 or 0 and f o r r e a s o n s of c o n t i n u i t y ant.

Hence a l l

C h r i s t o f f e l s y m b o l s and a f o r t i o r i

tensor vanish identically.

t h e Riemann

const-

curvature

O r t h o n o r m a l c o o r d i n a t e s t h u s e x i s t only in f l a t

manif o l d s . In m a r k e d c o n t r a s t t o t h i s , every given n-Bein can be o r t h o n o r m a l i z e d .

One

p a r t i c u l a r m e t h o d is t o apply t h e w e l l - k n o w n p r o c e d u r e of Gram and S c h m i d t s i m u l t a n e o u s l y in a l l p o i n t s of t h e s p a c e .

Obviously,

normal

of

n-Bein.

An i m p o r t a n t

application

t h i s y i e l d s an

this observation

is t h e

s t r u c t i o n of a d i s t i n g u i s h e d volume f o r m on an o r i e n t e d R i e m a n n i a n f o l d , t o which we come n e x t .

orthoconmani-

138

4. Riemannian Spaces

Theorem 4.5.1:

Let Μ be an oriented n-dimensional Riemann space. For each χ e M, we choose a (positively oriented) orthonormal basis a 1 , . . . , « " of the cotangent space Τ Μ and introduce the abbreviation X ωRt( Χ )\ = α 1 The assignment χ ι—»

Λ

...

Λ

n

α

determines a nowhere vanishing η-form

e Λη(Μ)

w h i c h i s c a l l e d t h e Riemann volume form of M.

The i n t e g r a l vol Μ

J".

of ωR over Μ is the (Riemann) volume of M. We likewise introduce the volume of a domain Β £ Μ via vol Β =

ω J R Β

in case the integral has a meaning.

Remark:

The l a s t definition is a proper extension of the former, since the i n t e g r a l on Β can e x i s t s even if Β is not a manifold. The closed n-dimensional unit ball Kn = I χ e IRn J |x| 5 1 I may serve as an example. It has a f i n i t e volume (theorem 4 . 5 . 3 ) though the boundary points which form the sphere Ωπ do not possess neighbourhoods homeomorphic to Rn, a s is required f o r points of a manifold. In f a c t ,

Κ

i s a manifold with boundary. In the f o l l o w i n g i n v e s t i g a t i o n s w e

4.5.

Volume

139

s h a l l n o t make use of t h i s c o n c e p t . For t h a t r e a s o n we r e f r a i n f r o m d i s c u s s i n g it in g r e a t e r

detail.

Proof:

» Had we s e l e c t e d a n o t h e r b a s i s of Τ Μ , like e . g . β o b t a i n e d t h e same

ω

Η(

χ

I

n β , we would have

)> b e c a u s e , f r o m t h e r u l e s of l i n e a r

β1 Λ ...

algebra,

= d e t ( x ) α 1 Λ ... Λ α η

Λ β"

w h e r e τ d e n o t e s t h e i n v e r t i b l e mapping of t h e c o t a n g e n t s p a c e o n t o

itself

which is u n i q u e l y d e t e r m i n e d by τ ί α 1 ) = β 1 . On a c c o u n t of t h e a s s u m e d o r t h o n o r m a l i t y of t h e a 1 and t h e β 1 , t h e m o d u l u s of d e t ( x ) e q u a l s 1; m o r e o v e r τ is o r i e n t a t i o n - p r e s e r v i n g ,

hence d e t ( x ) > 0.

T h i s c o m b i n e s t o d e t ( x ) = 1, and 1

α

n

λ1

Λ . . . Λ α

β

=

Λη Λ ... Λ

β

t h e b a s i s a 1 and ι z e r o s i n c e t h e r e a r e no n o n t r i v i a l l i n e a r r e l a t i o n s b e t w e e n t h e α . Consequently,

If

ωR (χ) is i n d e p e n d e n t

we i n t e r p r e t

α1,...,a"

as the

of t h e choice of

local

values

of

an

orthonormal

not

n-Bein

which is d e f i n e d in some n e i g h b o u r h o o d of χ ( o b t a i n e d f o r i n s t a n c e w i t h t h e help

of

the

Gram-Schmidt

method

d x 1 , . . . , d x n of a s u i t a b l e c h a r t ) ,

from

the

we see t h a t

U

coordinate

X R ( ) is

a

differentials

C°°-f u n c t i o n of

χ

and h e n c e an n - f o r m . For l a t e r applications,

we a l s o n o t e

Theorem 4 . 5 . 2 : As u s u a l , we d e n o t e t h e s t a n d a r d c o o r d i n a t e s of some p o i n t χ in E u c l i d e a n , 1 η η - s p a c e by χ χ . ( a ) The Riemann volume f o r m of IRn is n o t h i n g e l s e b u t t h e E u c l i d e a n volume element ω (χ)

=

dx 1

Λ

. . . Λ

dx"

140

4. Riemannian Spaces

( b ) On the ( n - l ) - s p h e r e

Ωη

=

{ x

6 R" I |x| = 1 }

x

S-1

u r may be brought to the f o r m

ω3(χ)

=

^

, i-l ,1+1 , π ( - 1 ) 1 x1 dx1 Λ . . . A αχ Λ dx Λ . . . Λ dx

Proof: Proposition ( a ) is t r i v i a l .

We introduce the abbreviation r f o r the Euclid-

ean distance of the point (x 1 , . . . , x n ) e IRn f r o m the origin.

It is calculat-

ed via Γ

2

=

, 1,2 (χ ) + . . .

, , n,2 + (χ )

which by d i f f e r e n t i a t i o n implies the relation . r dr

=

χ

I i i dx

+ ... + χ

The sphere Ω is by definition the set π equation r = 1 in Rn. Therefore

the

restriction

of

an ( n - l ) - f o r m

requested spherical volume f o r m provided dr holds.

Α

ω

of

=

ω = R

all

η , η dx solutions

of

the

algebraic

ω e Λ° 1(IRn) on Ω yields π

dx1 A . . .

the

A dx"

Through direct evaluation of the product we check that this condi-

tion is obeyed by ω = ω .

The total volume of Rn is of course oo. In contrast to this, compact domains always have f i n i t e measures. For the investigations to be resumed in part F (chapters 24 and 25) we need

4.5.

Volume

141

Theorem 4 . 5 . 3 : ( a ) The n - d i m e n s i o n a l b a l l Κ

of radius 1 has the volume

vol Κ 2k

-

k π j-jk!

f o r even dimension η = 2k and _ vol Κ 2k+i if

η = 2k+l is odd.

a l l dimensions,

k! _2k+l k -rrr,—r-rr 2 π (2k+l)!

=

Both f o r m u l a s combine into one which is v a l i d

for

namely

Μ

vol Κ

( b ) The s u r f a c e of Κ

is found f r o m

vol Ω '2k

, „ vol Ω 2k+l

(k-1)!

k! _2k+l k ,, 2 π (2k)!

=

depending on the p a r i t y of n. If w e p r e f e r the more g e n e r a l w i t h the Gamma f u n c t i o n ,

expression

we get

2 πη/2 vol Ω

= Γ,

2

( c ) Volume and s u r f a c e a r e connected by the Archimedian r e l a t i o n

νοΚΩ )

=

n·vol(K )

Proof: F o r the moment we denote the t o p o l o g i c a l l y c l o s e d m - d i m e n s i o n a l b a l l o f

ra-

142

4. Riemannian S p a c e s

dius r a s Κ (r). η The c a l c u l a t i o n of the volume of Κ (r) can, m t h a t of Κ (1) = Κ : m m vol Κ ( r )

=

by r e s c a l i n g ,

be reduced

to

r m vol Κ

In the p r o o f of theorem 4 . 5 . 2 we applied the r e l a t i o n ω^ = drAu g between the volume f o r m s of the Euclidean s p a c e and the sphere. This l e a d s to

vol Ω η i.e.

=

^ vol Κ ( r ) σΓ η

=

vol Κ (γ) ^ η βΓ

η·vol Κ

Archimedes' theorem. This in turn a l l o w s us to deduce the volume f o r -

mula f o r Ω f r o m t h a t of Κ . η π The i n t e r s e c t i o n of the hyperplane χ

= c e [-1,1] and Κ = Κ (1) is a ball η η

i s o m e t r i c to Κ (r) with the r a d i u s n-l

We obtain

vol Κ

=

vol Κ

dc

1(n-l)/2

-

C=-l

[-1

vol Κ dc n-l

C=-l

and thus f o r the r a t i o of the volumes f o r two s u c c e s s i v e n:

vol Κ vol Κ

1

-T

1-c"I i c=-l

(n-l)/2

π

dc

=

s i n α da a=0

Here we used the t r i g o n o m e t r i c s u b s t i t u t i o n c = cos a which s u g g e s t s s e l f . The only remaining problem is the evaluation of the i n t e g r a l

s i n α da a=0

it-

4 . 5 . Volume

143

The transformation η πα

Γ, 2 1 n-2 I I - cos α s i n α

=

(k £ 2) a f t e r partial integration and a slight simplification give ι

=

η

5=1 ι Π n-2

as well as I I η n-1

=

n-2 J n-1 n-2

η

n i l η n-1

χ

n-3

=

(n-2) I

ο

=

π

=

2

=

n-2

η

I

l

n-2

l

n-3

n-3

With the elementary integrals I

I

ι

we get the proposition inductively. The t r a n s l a t i o n to the version c o n t a i n ing the r - f u n c t i o n i s an immediate consequence of i t s functional equation.

The d e f i n i t i o n of the volume form as given in theorem 4 . 5 . 1 i s a b i t

incon-

venient because i t r e q u i r e s the s e l e c t i o n of an orthonormal n-Bein.

There-

f o r e a r e p r e s e n t a t i o n r e l a t i v e to an a r b i t r a r y (oriented) frame w i l l be of interest.

Theorem 4 . 5 . 4 : The volume form ω of an n-dimensional oriented Riemann space can be c a l c u l a t e d in any p o s i t i v e l y oriented frame χ ι—» (χ 1

=

/

det gl

dx

Λ . . . Λ dx

χ η ) through

144

4. Riemannian Spaces

Here g is t h e m a t r i x of t h e c o v a r i a n t m e t r i c c o m p o n e n t s

8,J

=

g (

W

Proof: The f o r m u l a in q u e s t i o n is c l e a r l y c o r r e c t a t χ 6 Μ if we u s e n o r m a l c o o r d i n a t e s w i t h o r i g i n χ ( s e c t i o n 4 . 3 ) . The t h e o r e m f o l l o w s f r o m t h e f a c t t h a t b o t h s i d e s t r a n s f o r m in t h e same way if we apply an o r i e n t a t i o n - p r e s e r v i n g c h a r t t r a n s i t i o n to them.

4.6.

Duality

145

4.Β. Duality Aside f r o m the Cartan derivative,

there is still another linear t r a n s f o r m a -

tion on the Graßmann bundle to an oriented Riemann manifold which is independent of the coordinates,

namely the duality.

In contrast

differential

mapping;

to

the

Cartan

the

latter

is

a purely

algebraic

t h e r e f o r e we want to introduce it f i r s t on the Graßmann algebra

A(V) of a suitable vector space V.

Definition 4.6.1: Let V be an n-dimensional real vector space, endowed with the nondegenerate symmetric scalar product < . , . > . We choose a positively oriented orthonormal basis e , . . . , e . l η This means that = 0 holds f o r i * j ,

while all norms

equal ± 1. The volume form is calculable by ω

e

a

ι

...

λ

e

η

The algebraic duality is the linear map »:

Λ( V) α ι—»

Λ( V) *

α

with *

T(k)

n(k+l)

λ

...

λ

e

π(η)

f o r 0 s k a η and π e S . Π

The Graßmann algebra A(V) is the direct sum of its subspaces A k (V) each of

146

4. Riemannian Spaces

which is spanned by the k - f o r m s e

1(1)

Λ.,.Λβ

. Thus there cannot be more n(k) .

than one linear mapping with the required properties. That

it

really

exists

is

a consequence

of

the

anticommutativity

of

the

wedge product of 1 - f o r m s which is compensated by the f a c t o r sgn(n) and of the nonexistence of further linear relations between the generators. It is as simple to see that the definition is independent basis. An alternative formulation is that the product its dual coincides up to sign with the volume form:

[

e

n(l)

Λ.,.Λβ

1 λ n(k)J

[e Λ...ΛΘ 1 [nil) it(k)J

=

of

the

selected

eπ(ΐ) λ. . . Λβ*(k)

with

lie || . . . 11||e 11II ω n(l)" n(k)

11

This property is retained if we move over to any other orthonormal

refer-

ence f r a m e . We compile some t r i v i a l information about the algebraic duality whose proof f o l l o w s directly f r o m the definition.

Theorem 4.6.1: ( a ) The algebraic duality over the (oriented) vector space V = Rn is a l i near b i j e c t i o n between the k - f o r m s and the ( n - k ) - f o r m s . ( b ) Dualizing

twice

reproduces

α e Λ (V)

except

for

a sign f a c t o r

which

arises in some cases. The precise formula is ##

α

=

(-1)

k(n-k)

sgnidet g ) a

Here sgn(det g ) is the sign of the metric determinant.

If the metric is

positive definite and the dimension odd, "*" is an involution.

The duality operation allows us to introduce a scalar product on the Graßmann algebra.

Definition 4.6.2: Every nondegenerate symmetric scalar product < . , . > on the Graßmann algebra A(V) over V = IRn (with orientation) extends to a product (denoted with the same symbol) which is uniquely determined by bilinearity and the f o l l o w i n g conditions:

4.6.

Duality

147

( a ) For α e A k (V) and β 6 A m (V) we have «χ, β> = 0,

whenever k and m are

distinct. (b) If,

however, the ranks of a and β coincide we get

α Λ

*

=

β

ω

where ω means the (positively oriented) Euclidean volume f o r m ,

i.e.

in

standard coordinates ω

One should notice that

=

e

ι

Λ

...

this definition

Λ

e η

is compatible

with

the

identifica-

tion A ^ V ) V. The η - f o r m dual to the C°°-function (= 0 - f o r m ) f is f ω. It is also possible to calculate the scalar product without explicit

refer-

ence to an orthonormal basis:

Theorem 4.6.2: The scalar product of two special k - f o r m s

Α

=

u 1

Λ

...

Λ

u k

Β

=

Ν

Λ

...

Λ Ν

and

(u , ν 1 1

1

k

e V) can be represented as the determinant of the matrix : ι j =

det

The connection with the duality is *

=

# («Α

# Β)

Proof: Follows immediately f r o m the definition.

•

* (βΛ

Α)

148

4. Riemannian Spaces

We c a r r y the a l g e b r a i c duality just introduced over to m a n i f o l d s .

Definition 4.6.3: Assume Μ is an o r i e n t e d n-dimensional Riemann space with m e t r i c tensor g. *

The duality assigns to each k - f o r m ψ an ( n - k ) - f o r m ψ. » * The local value ^rfx) of ψ at the place χ e Μ can be found as the image of k * lir(x) e • Α (Τ Μx ) under the a l g e b r a i c duality over the cotangent space V = Τ Μ by evaluating the scalar X

=

product

g(a, β)

=

g I J at β} ·

* Clearly

ψ(χ) is a smooth f u n c t i o n of x.

r e n t i a l f o r m ( o f rank n-k). well.

Hence

ψ is an a l t e r n a t i n g

diffe-

On A(M) a scalar product can be constructed as

In c o n t r a s t to its a l g e b r a i c counterpart,

however,

it r e q u i r e s an i n -

tegration:

Definition 4.6.4: The scalar product of

two differential

f o r m s α and β of

rank k over

the

With the understanding that A k (M) and Am(M) are supposed to be mutually

or-

oriented Riemann m a n i f o l d Μ reads

κ

=

J α Λ *β Μ

thogonal if k * m, we extend

l i n e a r l y to A(M).

4.7. C l a s s i c a l Vector Analysis

149

4.7. Classical Vector Analysis

The w o r l d we are living in is three-dimensional and to a remarkable degree of accuracy Euclidean. This is the reason why many results concerning mani3

f o l d s have been found f i r s t f o r the p a r t i c u l a r case of R . So,

for

instance,

i n f i n3i t e s i m a l in R

already

calculus

it

shortly was

after

noticed

the

that

discovery

several

e x i s t which have a g e o m e t r i c meaning,

(or

invention?)

differential

of

operators

independent of coordinate

sys-

tems. Most important among them are three invariant d i f f e r e n t i a t i o n procedures of f i r s t order,

namely gradient (grad), curl or rotation (rot), and divergence

(dlv). They are accompanied by the Laplace operator, normally denoted Δ, which of all d i f f e r e n t i a t o r s of

second order is by f a r the simplest and most

widely

used. Since the Cartan differential d is the only invariant d e r i v a t i o n on Riemann spaces known to us, we suspect that the above-mentioned mappings might be expressible

with

its

help.

This

would

immediately

imply

their

invariance

under chart changes. As concerns the g r a d i e n t ,

this is indeed true,

while f o r

the other

f o r m a t i o n s in addition the duality, which is also independent of dinates, To

begin

trans-

the

coor-

operators

under

is needed. with,

we w r i t e

down e x p l i c i t l y

the f i r s t

discussion in Cartesian coordinates ( x , y , z ) . not employ the index notation but f o l l o w

order

For the sake of

c l a r i t y we do

instead the old t r a d i t i o n

press the components of v e c t o r by a d j a c e n t l e t t e r s of the alphabet. 3 The duality over R is c h a r a c t e r i z e d by the f o r m u l a s *

=

1

* *

dx

Λ

dy

dx dy

= =

dy dz

Λ Λ

dz dx

dz

=

dx

Λ

dy

Λ

dz

to

ex-

150

4. Riemanniaxi Spaces

* #

*

The

application

of

Cartesian

(dyAdz)

=

dx

(dZAdx)

=

dy

(dxAdy)

=

dz

(dXAdyAdz)

=

1

(orthonormal)

coordinates

blurs

the

distinc-

tion between c o n t r a - and covariant v e c t o r f i e l d s because the m e t r i c is g i v en by the unit m a t r i x and thus reads in components either g = δ or 1 1 1 1 J g = δ . T h e r e f o r e , one must be a w a r e of the c o r r e c t i n t e r p r e t a t i o n when one wants t o g e n e r a l i z e to a r b i t r a r y charts.

This task is made easier if

we

t r a n s f e r the r e l e v a n t equations into Cartan calculus. In classical vector analysis,

it

tention to (smooth) functions,

i.e.,

variant)

is common p r a c t i s e

vector fields = 1-forms.

to r e s t r i c t

in f a n c i e r language,

Sections

2

3

at(co3

3

in the bundles A (R ) and A (R ) 1 3

0

3

are r e p l a c e d by t h e i r duals which lie in Λ (R ) or Λ (R ), want to f i n d the o p e r a t o r s grad, r o t ,

one's

0 - f o r m s and

respectively.

and div in components.

We

The most c o n -

venient p r e s e n t a t i o n of the intended investigation s t a r t s out f r o m the C a r tan d e r i v a t i v e d. We begin with a 0 - f o r m f . 1-form.

Because of d(A°) £ A1, the d i f f e r e n t i a l of f is a

By section 3 . 5 , df

=

+

+

w h i l e the gradient r e l a t i v e our basis is .. 8rad

The g r a d i e n t

thus

f

is nothing

= else

fäf äf 8fl [ s* · - äi J but

an a l t e r n a t i v e

way

of

writing

the

d i f f e r e n t i a l of a 0 - f o r m . 1 2 Somewhat more complicated is the situation f o r 1 - f o r m s . Now d(A ) S A , and we have to dualize a f t e r having d i f f e r e n t i a t e d in o r d e r to come back t o a 1-form. Let the given f i e l d be (u, v, w ) in c l a s s i c a l , dern notation.

We f i n d

i.e.

u dx + ν dy + w dz in m o -

4.7. Classical Vector Analysis

d (u dx +

ν

dy + w dz) faw

=

[ay

-

=

dvl

.

alj

d y

151

du

dx + dv

Λ

. Λ

fsu

λ

dy + dw

awl

.

, fav

au| , "

Έ9J

faw 3v1 . fau aw") , ^ dx • ^ dy

+

fav au") , ^ - ^ J dz

+

axj

,

[a*

d z

[ΈΈ -

dz

λ

d z

Λ

d x

+

d x

Λ

d y

and, consequently, *_• / ι d (u dx • V dy

... w dz)

=

. , . rot (u,v,w)

=

+

On the other hand,

and we conclude rot =

*

f 3w

3v

au

3w

3v

3u Ί

3

d f o r 1-forms in IR .

Next we consider the Cartan derivatives of 2-forms.

2

3

Since d(A ) S A , we

dualize before and a f t e r the differentiation. In other words: we calculate * · d (u dx + ν dy + w dz). The 2 - f o r m dual to u dx + ν dy + w dz is * (u dx +

ν

dy + w dz)

=

u dy

λ

dz +

ν

dz

λ

dx + w dx

dy

λ

Differentiation yields d

*

(u dx + ν dy + w dz)

=

Γ 3u 3v 3w 1 I + g y + ^ I dx

λ

dy

λ

dz

and a second transition to the dual form provides us with *

, * . d (u dx

+

. ν dy

+

. , w dz)

=

3u 3v _ + _ +

aw _

This can be identified as . . . , div (u,v,w)

=

3u dx

+

dv 3w + Έdy dz

The only remaining case, namely the application of d to 3-forms, does not lead to anything of interest since the three-dimensionality of the underly3

4

ing space implies d(A ) S A

=0.

It is evident that gradient and divergence due to the descriptions with "*" and "d" can be immediately generalized to all IRn (and even to

arbitrary

oriented manifolds). In contrast to this, the given formula f o r the curl is useful merely

in 3-dimensional

spaces because

dtA1) S An

2

if

η denotes

152

4. Riemannian Spaces

the dimension.

If η £ 3, the Cartan derivative of a 1 - f o r m , whose rank is

only 2 would be handier than the ( n - 2 ) - f o r m discussed above. Thus, one n o r 3 mally r e s t r i c t s the notion "curl" to R or - a bit more general - to 3 dimensional manifolds. For reasons of completeness we also give the 2-dimensional counterpart. Now * d(A ι ) £ Λο , and the curl is defined via the explicit expression rot while,

3v öu ax " ay

(u,v)

on the other hand, d (u dx + ν dy)

£ ^ - j dx λ dy

=

and t j ^ j ^ d (u dx 4. V dy)

av au g j - g^

=

*

hold. Hence in the present case "rot" must also be interpreted as " d". We collect the previous deductions and results of this section.

Theorem 4.7.1: ( a ) The gradient of

a C™-function f : Rn —» R is the 1 - f o r m given by

the

differential, grad f

=

df

( b ) The divergence of a 1 - f o r m α in Rn is the O-form div α 2

( c ) In R

=

*

*

d a

3

or R , the curl of a 1 - f o r m a is calculated by rot α

=

#

da

In Euclidean 2-space, rot α is a O-form, in 3-space, on the other hand, a 1-form.

4 . 7 . C l a s s i c a l Vector Analysis

153

If we want to construct chart-independent d i f f e r e n t i a l operators of second order, we must, on account of the Cartan rule dd = 0, insert the transition to the dual between the d i f f e r e n t i a t i o n s . candidates are combinations of d d, These four mappings transform,

Therefore the only

d d, d d , and

in turn,

conceivable

d d .

k - f o r m s to d i f f e r e n t i a l forms of

ranks n-k, k, k and n-k. The second and third of them leave the rank invariant. They are thus of particular interest.

We introduce some more n o t a -

tion.

Definition 4.7.1:

Let Μ be an oriented Riemann space of dimension n. (a) We define by means of the Cartan d i f f e r e n t i a l

d the co-differential

(the co-derivative) δ through

f o r k - f o r m s and extend this linearly to Λ(Μ) = A°(M) ® . . .

Θ Λη(Μ).

(b) T h e Laplace-Beltrami operator i s

Δ

- d δ - δ d

and in p a r t i c u l a r , when applied to a k-form, Δ

=

(-l)nk

d d + (-1)

( c ) A k - f o r m α is closed if da = 0,

and exact if there is a ( k - l ) - f o r m β

f o r which dß = α holds. (d) Dually, α is co-closed whenever δα = 0,

and co-exact provided a (k+1)-

form β with δβ = α e x i s t s . ( e ) Finally, by a harmonic form we mean an a e A(M) with Δα = 0.

The somewhat strange looking sign convention is explained by the f a c t that in Cartesian coordinates of IRn the Laplace operator reduces to the e x p r e s sion

154

4. Riernanntan Spaces

Δ

=

a2 + . .. 1

as is f a m i l i a r f r o m the c l a s s i c a l

theory.

+ a2 η

For compact m a n i f o l d s ,

S is the

a d j o i n t o p e r a t o r o f d. We combine the last statement and a f e w o t h e r mentary p r o p e r t i e s o f d, δ,

ele-

and Δ to

Theorem 4.7.2: L e t Μ be an o r i e n t e d Riemannian space of dimension n. Then L a p l a c e o p e r a t o r and duality map commute:

* Δ

=

Δ *

=

Δ(*α)

i. e. , f o r a l l a e A(M),

*(Δα)

I t e r a t i o n o f d or δ y i e l d s

If Μ is compact,

zero;

d d

=

0

δ δ

=

0

we f u r t h e r m o r e have

( a ) Cartan d e r i v a t i v e

d and c o - d e r i v a t i v e δ are a d j o i n t mappings w i t h

spect t o the s c a l a r product d e f i n e d by

J α Λ

=

Β

Μ ( t h i s is nondegenerate and even p o s i t i v e

=

definite):

( b ) The L a p l a c e - B e l t r a m i o p e r a t o r Δ is s e l f - a d j o i n t :

=

re-

4.7. C l a s s i c a l Vector A n a l y s i s

155

( c ) A Is n e g a t i v e d e f i n i t e :

s

0

with equality only f o r α = 0. (d) A differential

f o r m α e AIM) is harmonic

and c o - c l o s e d .

It

already

s u f f i c e s to

if

and only

require

the

if

it

is

closed

orthogonality

of

α

and Aa:

= 0

( e ) When a is harmonic,

Aa = 0

(da =

0

δα =

Λ

0)

the same is true f o r the dual f o r m

*

a.

Proof: k Consider

*

a e A (M).

calculation.

The equality

Beyond

this,

the

of

* (Aa) and A( a )

Cartan

rule

is v e r i f i e d

(definition

3.5.5)

by

direct

implies

that

dd vanishes i d e n t i c a l l y and thus *

δ δα

=

±

*

*

*

d

*

da

=

*

± d d a

=

0

(the p r e c i s e signs do not m a t t e r ) . We moreover know that the scalar product < . , . > : and p o s i t i v e d e f i n i t e .

A(M)xA(M) —» IR is symmetric

By d e f i n i t i o n ,

-

=

J |da Λ *β - α Λ *ößj Μ

if a and β are f o r m s of

the ranks m-1 and m,

l e f t hand side is obviously equal to 0). * da

Λ

* β

-

Α Λ

* δβ

=

da

Λ

and Stokes' theorem 3. 9. 2 leads to

-

respectively

m-l β + (-1)

=

(otherwise

The integrand is an exact f o r m : * Α Λ

J d(aA*ß) Μ

*

d β

=

0

=

d (ΟΛ

Β)

the

156

4. Riemannian Spaces

This is p r o p o s i t i o n ( a ) ,

which has (b) as an immediate consequence.

State-

ment ( c ) f o l l o w s f r o m

-

=

=

+

since is p o s i t i v e

definite.

The

last

this holds f o r δα and da simultaneously. part of At l a s t ,

+

expression

2

can only

0

be z e r o

if

We thus have proved the n o n t r i v i a l

(d). ( e ) can be deduced easily f r o m

The compactness of

Μ is essential.

(a).

For example,

a harmonic

function

(0-

f o r m ) f by a s s e r t i o n (d) has a vanishing d i f f e r e n t i a l and hence is constant on each connected component of M. Harmonic f u n c t i o n s on compact m a n i f o l d s a r e t h e r e f o r e quite boring. For noncompact

spaces like Rn the situation

is completely

many harmonic f u n c t i o n s e x i s t which are not constant.

different.

Here

We postpone a t h o r -

ough discussion to p a r t F of this book. There we shall g i v e numerous i m p o r tant applications nections with

(spherical

other

harmonics,

problems

as,

Lam£ f u n c t i o n s ) and discover

e. g. ,

numerical

integration

on

con-

spheres,

etc. By the

Cartan

rules

dd = 0 and δδ = 0 every

exact

differential

form

is

closed (and e v e r y c o - e x a c t f o r m c o - c l o s e d ) . The question,

to which extent the r e v e r s a l of this statement is c o r r e c t

a given m a n i f o l d , Tu [1982],

is a topic of cohomology theory ( c f .

Warner [1983]).

Here

case which w i l l be needed l a t e r .

we only

want

to

for

f o r instance Bott &

consider

a very

special

Assume α is a closed 1 - f o r m on the simply

connected m a n i f o l d M. We f i x a basic point X q € Μ and introduce a smooth f u n c t i o n F e Λ°(Μ) through

F(x)

=

J

y

α

where y denotes an a r b i t r a r y path leading f r o m χ to x. The in f a c t is a 0 » meaningful d e f i n i t i o n . If y w e r e another curve connecting X q with x, we could

on account

of

the homotopy

of

y and y

construct *

a surface

(two-

dimensional submanifold) * Γ £ Μ with boundary y - y ( f i r s t y in the f o r w a r d d i r e c t i o n ; t h e r e a f t e r y backward). But in this case the theorem of Stokes

4.7. Classical Vector Analysis

157

would y i e l d

ί α " ί , α = ί * α = Ι α = Jda = ° r

γ

3Γ

γ-γ

Γ

and hence

ί" • J.· y

r

F i x ) thus is independent of the p a r t i c u l a r curve y we use. By a simple calculation we check without d i f f i c u l t y the exactness of a:

dF

=

grad F

=

α

This is the special case k = 1 of the f o l l o w i n g Poincarfe lemma:

Theorem 4 . 7 . 3 : Closed k - f o r m s (k £ 1) on simply connected m a n i f o l d s are e x a c t .

Proof: Thanks to the simple connection we can r e s t r i c t The natural Cartesian coordinates of

Rn

are,

the discussion to Μ = IRn.

as a l w a y s ,

χ1

χη.

The L i e

d e r i v a t i v e r e l a t i v e to the radial v e c t o r f i e l d

ν

=

χ

a

y i e l d s f o r a k - f o r m ω in components * a ω) ν a...a

= ω

l k

a . . . a , m

l

=

m χ + ω

k

xm 9

m

ω

a ... a

1

m m...a

χ k

,a

+ . . . + ω 1

1

+ k ω

a ... a

k

1

k

and vanishes only if

χ

θ

m

ω

a ... a

1

= k

- k ω

a ... a

1

a...m

k

m χ

.a

k

158

4. Riemannian Spaces

holds. The l a t t e r condition is by the Euler homogeneity theorem 2. 5. 1 t a n t amount to ω being homogeneous of degree ( - k ) in the x™. a .. .a 1 k For k > 0 the r e g u l a r i t y of ω at 0 presupposes it to vanish i d e n t i c a l l y , while f o r O - f o r m s we can merely conclude that ω is constant. We now assume k k Then L : Λ (Μ) —> Λ (Μ) is i n j e c t i v e , and there must be a ( l i n e a r ) V map α of Λ (Μ) into i t s e l f which is a l e f t inverse of L : k ν k a 1.

α ο L k ν

=

id

It is not hard to see that the t r a n s f o r m a t i o n given by

α : k

f dx

1

λ

. . .

f o r all ρ e R

c^oL^jfdx

λ

dx

k

(ρ)

J

ι—>

has the desired property.

1λ.

. . Adx

k

j(p)

dx

1

λ

. . .

dx

λ

k

( p )

This is derived f r o m

o^jkf

=

f t t p ) dt

t"-1

+

x'°a1I1f]dx

V..Adx

kJ

(P)

X J

t*"1

[f + x m S m f ]

dt

dx

Λ ...

1

Λ dx

k

(p)

t =0 1 J

at ( t k

)

f ( t p )

dt

dx

1

Λ . . . Λ dx

k

(ρ)

= 0

a

a

[

fdx V . . A d x

One f i n d s equally

k

Cp)

easily

α »d k

=

d ο α k-1

and the i d e n t i t y

d

α « i (ω) k-l ν J

=

d » α

k-l

We next consider theorem 3 . 5 . 5 which,

« i (ω) ν

=

α

k

° d ° i (ω) ν

if ω is closed,

provides us with

4 . 7 . C l a s s i c a l Vector A n a l y s i s

ω

=

α

k

ο L (ω) ν

α

159

ο

k

d ° i (ω) + i «dw ν

ν

J

=

α

k

° d ° i (ω) ν

This leads to d

and t h e

α °i (ω) (_ k-i ν J

=

(k-l)-Form φ

=

a

k-l

«ι (ω) ν

obeys

dip

=

ω

Hence ω is indeed e x a c t .

T h i s would be w r o n g f o r a r b i t r a r y m a n i f o l d s M; t h e e x a c t k - f o r m s a r e e l e m e n t s of a ( n o r m a l l y p r o p e r ) s u b s p a c e of t h e IR-vector s p a c e of t h e c l o s e d k - f o r m s . The f a c t o r g r o u p , t h e s o - c a l l e d k t h ( d e Rham) cohomology g r o u p of M, t h a t is

Η"(Μ)

= { Α 6 AK(M) I Α c l o s e d > / { Α e AK(M) | Α e x a c t >

frequently (e.g.

f o r c o m p a c t M) is f i n i t e - d i m e n s i o n a l .

t o p o l o g i c a l i n v a r i a n t of M.

It is an

important

160

5. Applications to Physics

5. Applications to Physics 5.1. Mechanics In order to make l i f e easier f o r the readers interested in science or engineering, we compile in this short chapter a number of fundamental formulas f r o m theoretical physics. tions w i l l be continued.

The notational conventions of the f o r e g o i n g

sec-

Of course, we do not attempt to substitute a t e x t -

or r e f e r e n c e book of physics. Rather, the considerations to f o l l o w serve as an illustration of our nomenclature system and demonstrate how to apply the d i f f e r e n t i a l geometric calculus. For that reason we deliberately restrict our developments to the most basic physical laws and abstain f r o m a deeper motivation or a detailed discussion of the underlying notions. Nevertheless,

it should be of value f o r many users of this monograph to be

presented some of the main equations in a directly applicable f o r m instead of having to deduce them independently. It is recommended to transform at least a f e w of the formulas listed below to d i f f e r e n t coordinate systems as found, f o r instance, in parts G and H. A comparison of the results with those of a direct evaluation by means of e l ementary calculus will be instructive. Even f o r the simplest examples, we get a considerable reduction of the necessary amount of labour to be invested and a large gain in clarity when we employ the tensor methods developed in chapters 3 and 4. In the sequel, two versions:

some of the physical laws are described in (not less than) Firstly

in the more elegant c o o r d i n a t e - f r e e

language

which

should be p r e f e r r e d during the process of establishing the relevant f o r m u las; and secondly in components (sometimes even in greater detail than demanded by the purpose of The

latter

alternative

is

the d i f f e r e n t i a l equations,

exposition). unavoidable

if

one

seeks

explicit

solutions

since this in general is only possible in

of

coor-

dinates. We start with a f e w ideas f r o m mechanics of point masses. More information can be found in Arnol'd [1978] or Neutsch & Scherer [1992] where f u r t h e r reaching questions are discussed as well.

161

5.1. Mechanics

The state of point

x,

a mechanical system at some instant t is characterized by a

say,

in an n-dimensional

manifold

M,

called

the

configuration

space. The state can and will vary with time; χ thus moves along some curve r in M, parameterized by t: y:

[R

> Μ

t

ι—> x ( t )

We furthermore assume that all forces are conservative, i . e . , ten as the gradient

of

the potential energy V = V ( x ) .

can be w r i t -

This condition

is

met, f o r instance, by the Newtonian law of gravitation. Under these circumstances, the Lagrange equations of motion govern the evolution of the physical system.

We express them most conveniently in an equivalent f o r m via

the Hamiltonian principle, t δ J 2 L ( x ( t ) , x ( t ) ) dt t

=

0

ι

Here the Lagrange function L = Τ - V is calculable f r o m the potential and the kinetic energy T, which depends on the velocity χ = — (and frequently dt also on the position x ) . The variation of the action integral J L dt has to be evaluated

with f i x e d

initial

and final

states

x ( t ) and x ( t ), 1 2

respec-

tively. By the rules of variational calculus the Hamilton principle is tantamount to the Euler-Lagrange equations

(i = 1 χ1

d

aL

aL

dt

dx

ax1

n) if we use an arbitrary chart on Μ to define the

coordinates

of the point χ of M.

As concerns d i f f e r e n t i a l geometry, the solution of the equations of

motion

comes down to a search f o r an orbital curve y: IR —> Μ to the given Lagrange function L:

TM (χ,χ)

» IR ι—>

L(x,x)

5. Applications t o Physics

162

on the tangent bundle T M (= "velocity space") such that the induced curve y:

R

» TM

t

ι—>

(y(t).r'(t))

makes the integral t J 2 L » y ( t ) dt t stationary.

ι

It is important to note that the Euler-Lagrange equations are

of the same shape in any reference system. This is because they are deduced f r o m the variational problem which is coordinate-invariant

itself.

Another formulation of similar importance is obtained when, instead of the velocity coordinates q 1 ,

we introduce the associated canonically conjugate

momenta p^, defined by 3L P'

=

and with their help eliminate the velocities f r o m the Hamilton function

Η (This

is o f t e n possible;

cf.

=

P l q' - L

Neutsch

& Scherer [1992]).

The p^ are

not,

like the velocities, objects in the tangent space TM , but in the dual co« ι tangent space Τ Μ . If we combine them with the position, the (2n)-tuple q ι * (q ,ρ^) is a set of coordinates in the cotangent bundle Φ = Τ Μ. The l a t t e r is in mechanics called the phase space of the system. *

Thus the Hamilton function Η: Τ Μ —» R has to be used in place of the L a grange function L. Then the motion is described by the canonical equations •ι q

3H ap,

an

5.1. Mechanics

163

In comparison to the Lagrangian approach they have the advantage that they f o r m a system of d i f f e r e n t i a l equations of f i r s t order or, what amounts to the same, a vector f i e l d in Φ. We may also view the action integral

S

=

S(q';t)

J L ( q ' ( τ ) , q ' ( τ ) ; τ ) dx

=

with f i x e d initial conditions as a function of time and the coordinates of the f i n a l state.

The variational problem SS = 0 can be used

alternatively

as a foundation of dynamics (Neutsch & Scherer [1992] discuss the details). The partial derivatives of S with respect to the coordinates yield the momenta: 8S 3q and a f t e r

insertion

and a f e w

.

=

simple

Pi

transformations

we

get

the

time-

dependent Hamilton-Jacobi equation as — + Η at

as • · .q ;-

as =

ο

whose solution gives the action function S = S ( q \ . . . , q n ; t ) and at the same time the dynamical evolution of the system. If

the Hamilton function does not contain the time e x p l i c i t l y ,

a further

reduction can be achieved. The expression S

=

Ε t + Wfq1

qn)

solves the variational equation in question whenever the principal function f u l f i l l s the time-independent Hamilton-Jacobi equation aw .q ;aq'

aw

Ε is the mechanical energy of the physical

=

Ε

system.

The l e f t hand side of

this eigenvalue condition f o r W in essence is a quadratic f o r m in the g r a -

5. Applications to Physics

164

dient of

W,

provided the system is "natural".

Similar f o r m u l a s w i l l

occur

again in the sections 5.5 and 5 . 6 as eikonal and Schrödinger equations,

re-

spectively. These techniques are on account of their f l e x i b i l i t y very w e l l

adapted

not

only to describe phenomena of point mass mechanics as e. g. the N-body problem in c e l e s t i a l mechanics. es of

They also play a leading part in other

But we do not intend to dwell tions.

branch-

science.

Instead,

importance.

on a thorough investigation

of

these

we turn over to a r e l a t e d problem which is also of

It is concerned with the c h a r a c t e r i z a t i o n

ρ

=

Β

Jpdx1

the

gravitational

We describe it by a density dis-

action of continuously distributed matter. 3 tribution p: IR —> R whose i n t e g r a l

J

of

ques-

greatest

dx

A

2

Λ

dx

3

Β

(the l a t t e r in Cartesian c o o r d i n a t e s ) states how much mass is contained in some spatial r e g i o n Β which we assume to be regular in the sense of tion 3. 9. 1. In this case one can s t i l l

defini-

describe the g r a v i t y f i e l d caused by

this d i s t r i b u t i o n by a scalar function, the Newtonian potential U. The p o t e n t i a l energy of a "test body" of mass m moving within the f i e l d is then simply given by the product m U. For the sake of letters to

c l a r i t y we use in this and the f o l l o w i n g

characterize

three-dimensional

vectors.

sections

The p o t e n t i a l

is

German calcu-

l a t e d via the f o r m u l a

U(j)

=

U ρ( J )

*p(l>)

= G Β

J-0

which is obtained f r o m its counterpart f o r point masses through a suitable l i m i t process.

Here G is the gravitational constant. 3

The i n t e g r a t i o n domain Β may also coincide with the whole space IR as long as the i n t e g r a l

exists.

From a mathematical point of v i e w ,

the most intriguing p r o p e r t y of the p o -

t e n t i a l is Poisson's equation Δ U

=

- 4 π G ρ

5.1. Mechanics

165

w h i c h r e d u c e s t o t h e potential equation

Δ U

=

*d *d U =

dlv grad U

=

0

if t h e s p a c e i s void of m a t t e r (p = 0 ) . T h e n - d i m e n s i o n a l v e r s i o n of t h i s f o r m u l a a n d i t s s o l u t i o n s , possess

numerous

t h i s book ( c h a p t e r s

6,22,23).

functions,

peculiarities.

They

will

t h e potential

occupy

us

later

in

A d i f f e r e n t , v e r y i m p o r t a n t p r o b l e m of continuum mechanics, n a m e l y t h e m o tion

of

e x t e n d e d rigid bodies ( m o r e p r e c i s e l y :

t h e spinning top) l e a d

to

t h e m a t h e m a t i c a l t h e o r y of t h e rotation groups a n d t h e i r r e p r e s e n t a t i o n s by orthogonal matrices a n d spinors. T h e y a r e a l s o d i s c u s s e d e l s e w h e r e t e r s 8 , 9 , 1 3 ) s i n c e t h i s i s n o t r e a l l y an a p p l i c a t i o n of t e n s o r F o r t h e m o m e n t we w a n t t o l e a v e i t a t

that.

(chap-

calculus.

166

5. Applications to Physics

5.2. Hydrodynamies In contrast to classical mechanics which is concerned with the motion single point masses under the influence of e x t e r i o r f o r c e s , hydrodynamics are based on statistics. ticles

is

regarded

f l u i d are of

as

The individual behaviour of the p a r -

only

the

collective

properties

of

the

interest.

To reach this goal, the general

irrelevant;

of

the methods of

one defines certain

character

of

the flow.

functions which

reflect

This can be done at d i f f e r e n t

averaged

levels.

Here we only discuss the simplest case, namely the mathematical description of

fields

like

(particle

or

mass)

density,

(bulk)

velocity,

and

tempera-

ture. Concerning the relationship to the substantially more general and f l e x i b l e kinetic gas theory which aims at describing the evolution of the system by distribution functions, we r e f e r the reader to the literature ( e . g .

Landau

& L i f s c h i t z [1962-1967]). The definitions of grals

of

the above-mentioned

distribution

quantities

as velocity

functions multiplied by suitable

factors

space is

inte-

assumed

known. The f a m i l i a r mechanical conservation laws of mass, momentum, and energy are r e f l e c t e d in the basic hydrodynamical equations of motion, as there are: ( 1 ) the continuity equation or mass balance; ( 2 ) the momentum conservation in the f o r m of either Euler's equation or, the presence of f r i c t i o n f o r c e s ,

the Navier-Stokes equation;

in

and f i n a l -

ly ( 3 ) the energy balance whose shape depends on the type of the system to be described. The equation of continuity expresses the f a c t that matter (mass) is neither produced nor destroyed,

but only transported f r o m one place to the other.

This means: We consider a f l u i d c e l l Z,

that is some region in space whose boundary

moves with the local velocity u = u ( x , t ) of

the matter

at position J e IR

and time t. Ζ has to be taken so small that the physical parameters like matter density ρ = p(j, t),

particle

density η = n ( j , t )

or

temperature Τ = T ( j , t )

do

not

5.2. Hydrodynamics

167

vary appreciably within Z. On the other hand, the cell must be s u f f i c i e n t l y large,

f o r otherwise the statistical fluctuations would be too strong.

under these circumstances the physical f i e l d functions η,ρ, T,

Only

and so on,

are w e l l - d e f i n e d . Both (contrasting) conditions are compatible by the fundamental of hydrodynamics;

hypothesis

if they were not, we would have to take resort to more

comprehensive theories. We denote the volume of Ζ by dV. The equation of continuity implies that the mass ρ dV contained in Ζ is constant. tion of

the cell,

This is true by the very d e f i n i -

since the f l u i d particles

cannot

(on the average)

move

across the boundary. Hence, the total derivative of the mass contained in Ζ is zero: iL(pdv)

=

0

which we can express also d i f f e r e n t i a l l y as

+ ρ dlv ii

or,

=

0

in components, dp d£

+

Ρ

a

=

u

„ 0

The Einstein summation of course runs f r o m 1 to 3. A more convenient representation is obtained if we replace

by the p a r t i -

al derivative with respect to the time according to the substitution rule dp π

dp

=

sir+

.

dp =

Λ

är

p;a

Although, as we know f r o m the investigations of chapter 3, partial and covariant derivatives of the scalar density function ρ coincide, rable f o r theoretical reasons to introduce the latter.

it is p r e f e -

In this way,

we ob-

tain a more unified representation when we combine the equation of continuity with the other d i f f e r e n t i a l conditions governing the f l u i d motion. We are provided with the f a m i l i a r

d^p + div(ptt)

=

d t p + (pu )

=

0

168

5. Applications t o Physics

Other than the mass, the sum of the momenta of all particles contained in the cell

Z,

the bulk velocity u equals ρ it dV,

which by definition of

not absolutely constant.

it changes under the influence of a force

Rather,

exerted on the f l o w i n g matter.

is

We have to distinguish between interior and

exterior effects. First,

the thermal motion of the atoms or molecules in Ζ produces,

ing on the local temperature Τ = T(j, t ) ,

the gas pressure p.

Its

dependintensity

is determined by the physical structure of the fluid. We express this r e l a tionship by the equation of state

ρ Clearly, External

=

p(p,T)

we can and w i l l not be more specific here. f o r c e s are described

acting on the cell.

by giving

the total

f o r c e per

unit

volume

We describe it by the components f a as r e f e r r e d to our

coordinate system. It is convenient to w r i t e down the gravitation separately

since

its

contribution

to

the

force

density

f

is

proportional

to

the

mass density: ρ grad U

f.g r a v (U is the Newtonian potential,

see section 5.1). The momentum balance thus

is, if we neglect f r i c t i o n e f f e c t s , ί (p u dV) dt

=

(p grad U - grad p) dV

From this we deduce the d i f f e r e n t i a l variant, the Euler equation du dt

grad

υ -

S^LP

which can be changed f u r t h e r into a , a b Q a U ++ u u 3 u U = t ;b

g

ab L ^; b"| U - — L 'b ΡJ

Note the index positioning and the necessity of introducing the metric g ab . The Euler equation may clearly be brought to the covariant f o r m ,

5 . 2 . Hydrodynami es

ο u 9

t a

169

+ u

a;b

b

u

=

a θ u

, gb c u +

a;b uc

t a

itU

=

;a

ρ - —i ®

p

as w e l l . For

reasons of

completeness,

we f u r t h e r m o r e g i v e the extension

which

is

needed to take into account the influence of friction. c a l l e d the dynamic and

Then t w o more f u n c t i o n s , η = η(ρ, Τ ) and ζ = ζ ( ρ , Τ ) , kinematic viscosity, r e s p e c t i v e l y , of

the f l u i d .

must be employed to describe the inertia

They also depend on the chemical

and physical

matter.

It is t h e r e f o r e impossible to give general

der

conceivable

all

circumstances,

although

state of

the

f o r m u l a s applicable

un-

relatively

comprehensive

gas

kinetic t h e o r i e s of v i s c o s i t y have been developed. The momentum conservation f o r a viscous f l o w thus has to obey the NavierStokes equation,

ρ ίϊ-

=

ρ grad U - grad ρ + η Δτι +

which g e n e r a l i z e s E u l e r ' s . t e r are s p a t i a l l y constant;

[ΐΗ Ό

ζ | grad d i v u

+

Here we assumed that the v i s c o s i t i e s of the m a t this is f r e q u e n t l y a good approximation.

Other-

w i s e f u r t h e r terms containing the gradients of η and ζ would have to be inserted. Expressed in an a r b i t r a r y r e f e r e n c e f r a m e , the momentum balance a t t a i n s the f orm

P K

Ö U + PK U

t a

a;b

b

U

.t ;a

= p U

- p

;a

+

fi (3-JJ

+ SC

J

b ;b;a

U

+i)g 6

be

u

a!b;c

When applied to k - f o r m s in 3-dimensional space, the Laplace-Beltrami operator Δ ( D e f i n i t i o n 4 . 7 . 1 ) s i m p l i f i e s to

Δ

and f o r by g r a d ,

=

ν * * U+1 * * (-1Γ d d + (-1Γ d d

the v e c t o r

f i e l d u in p a r t i c u l a r

=

" (-1)"

(k = 1),

* » * *1 d d - d d d d - d d

[

we get

a

representation

div and r o t : *

#

*

*

A u = d d u - d d u =

which is o f t e n p r e f e r r e d .

grad d i v u - r o t r o t u

The N a v i e r - S t o k e s equation now reads

170

5. A p p l i c a t i o n s t o Physics

ρ ^

=

ρ grad U - grad ρ - ij r o t r o t u + [5

If the equation of

s t a t e is independent of

and the same holds true f o r the v i s c o s i t y ,

Sra(ä d i v

71 +

the temperature,

i.e.

11

ρ = p(p),

the continuity and Euler (or N a -

v i e r - S t o k e s ) equations t o g e t h e r f o r m a complete system of conditions which in p r i n c i p l e could be solved f o r given i n i t i a l values of ρ and u a . This assumption is sometimes j u s t i f i e d ;

normally,

however,

it w i l l be n e c -

essary to add an equation which is apt to determine the t e m p e r a t u r e

field.

This is achieved by the energy balance or heat transport equation. E x a c t l y as the t o t a l momentum in Z,

the energy content of the c e l l can be

changed by several transport phenomena. Of the innumerable p o s s i b i l i t i e s we single out only a f e w instructive and t y p i c a l

examples.

The heat equation basically has the shape (Weizel

p

3t

\β

+

h

"

U]

=

[1962-1966])

τ

where the bracket on the l e f t hand side contains in that order kinetic energy, enthalpy

(e = s p e c i f i c energy,

internal

energy,

cf.

Batchelor [1967])

and

the

gravitational

each r e f e r r e d to unit mass.

On the other hand, expansion

or

the f i r s t contribution represents the work done by an

contraction

of

the volume

of

Z,

while τ

is the

sum of

t r a n s p o r t terms whose precise f o r m s depend on the p r e v a i l i n g physical

all con-

ditions. We only mention the heat conduction which is described by

τ

L

=

d i v ( λ grad T )

=

(gab λ Τ

;a

)

;b

=

gab (λ Τ

;a

)

;b

(λ g r a d Τ is c a l l e d the heat flow) and the frictional heat (energy dissipation) τ , which is most easily expressed with the help of the shear tensor

D

ab

We have

=

- \u

2

I a;b

+ u

I

b;al

5.2.

Hydrodynami e s

171

[ w

0

τΗ

or,

=

2

71

[g

bd 8

n

1

n

D »t Dcd

" i 8

ac

bd

g

;e

_

]

g a b Dodj

somewhat simplified, τ

R

=

2 η D ab ίϋ - i g I ab 3

ab

div

tt| J

=

2 η D ab iü

T h e r e a r e many o t h e r heat t r a n s p o r t p r o c e s s e s , etc;

e

u

ab

- i gs 3

ab

uc

like convection,

1 ;cj radiation

but the e x a m p l e s d i s c u s s e d here should s u f f i c e as an i l l u s t r a t i o n .

172

5. A p p l i c a t i o n s t o Physics

5.3. Relativity

Throughout that a l l

the remainder

Greek super-

of

this chapter

we shall

employ

the

convention

and subscripts run f r o m 1 to 3 and the Latin

ones

f r o m 0 to 3; of course, the same agreement applies f o r the Einstein summation. In Newtonian mechanics,

a physical event 8 is completely described by the

c o o r d i n a t e s x " ( f o r the moment supposed to be Cartesian) 1 2

of

the

position

3

3

J = (χ , χ , x ) where it happens,

t o g e t h e r with the time t.

Here j e R

and

t e IR are independent of each other. The theory of relativity combines both quantities

into a f o u r - d i m e n s i o n a l

event vector: x

(t c ^t ; j )Λ

=

=

(/x ο ; x α .)

(x a .)

=

r

=

,

( x 0 , x1 ,2x 3,x. )

e

_4 IR

The f a c t o r c denotes the speed of light in the vacuum, an absolute

invari-

ant in the theory.

system

As an aside,

its value has been f i x e d in the SI

by convention to

c

=

299792458 m sec" 1

the metre (m) becomes a secondary unit depending

With this d e f i n i t i o n ,

on

the p r i m a r y second ( s e c ) . In the

special

theory

with the points of

of

relativity

all

conceivable

a f l a t 4-dimensional manifold,

events

are

associated

the Minkowski space.

distinguishing f e a t u r e is the existence of a metric which in suitable dinates element,

x°

is

expressed

by the Minkowski form of

the

squared

i. e. . 2

ds

In this f o r m u l a ,

-η

ab

=

i)

ab

. a

dx

, b

dx

are the c o e f f i c i e n t s of the constant

matrix

arc

Its

coorlength

5.3.

Relativity

173

' 1 0

0

0

0 - 1 0 0 η

=

0

0 - 1 0

0

0

0 -1

The c o n t r a v a r i a n t m e t r i c components look e x a c t l y the same since η is an involution (η

= 1). The motion of

a f o r c e - f r e e p a r t i c l e then f o l l o w s a g e o -

desic line which in Minkowski space reduces to a s t r a i g h t line.

We are not

going to d w e l l upon a more d e t a i l e d discussion of mechanics in the f r a m e work of

special

relativity

Minkowski f o r m invariant

or i n v e s t i g a t e which chart t r a n s i t i o n s l e a v e the (these s o - c a l l e d Lorentz transformations are

the

main t o p i c of chapter 14). Instead, we immediately go over to the viewpoint of general

relativity.

Now the configuration space is a 4-dimensional, possibly curved m a n i f o l d M. By the correspondence This means,

principle

the m e t r i c g

it

is assumed to

be l o c a l l y

Minkowskian.

like η

has one p o s i t i v e and three n e g a t i v e ab ab 1 3 genvalues and thus the signature ( + , - , - , - ) = ( + , - ) .

ei-

Gravitational fields by the equivalence principle cause f o r all point masses the same a c c e l e r a t i o n . g l e point

or

They can t h e r e f o r e be t r a n s f o r m e d away in a sin-

even on a timelike

curve

in Μ if

we choose

the

coordinates

cleverly. P a r t i c l e s s u b j e c t to no other f o r c e s except g r a v i t y s t i l l

move along

time-

like geodesies: -a

χ

. —a be

*b ·c

+ Γ

x x

(dot = d e r i v a t i v e with respect to the arc length),

or,

if

we introduce

the

dimensionless v e l o c i t y ua = x a , a

u

b

:b

·a

t

u = u + T

_a

b e be

_

u u = 0

The c o o r d i n a t e - f r e e version is simply

g(u,u)

This expresses the o r t h o g o n a l i t y

of

the equation of motion to the f o r m

velocity

and a c c e l e r a t i o n .

If

we

bring

5. Applications t o Physics

174

••a X

—a b c - Γ U U be

=

we discern the meaning of the Christoffel symbols. They describe the g r a v i tational acceleration in the particular reference system we use. In order to determine the metric we moreover need the Einstein f i e l d equations. They generalize the Poisson equation of section 5. 1. Let us suppose that the f i e l d - g e n e r a t i n g matter has density ρ and gas pressure ρ and that

it moves with the velocity

ua.

Then the Energy-momentum

tensor is given by 2

In coordinates,

Τ

=

(pc +p) u β u - ρ g

this reads _ab Τ

=

, 2, . a b ab (pc +p) u u - ρ g

Obviously, Τ is symmetric. We moreover find that energy and momentum are conserved quantities: T ab

;b

=

0

The O-component of the last expression is the r e l a t i v i s t i c equation of continuity, while the three spatial components group to the Euler equation. At the same time we have translated the basic formulas of hydrodynamics to the four-dimensional theory. In the subsequent section we want to consider what changes if we additionally take the electromagnetic

interaction into account.

shall s u f f i c e to formulate the f i e l d equations.

At the moment it

They connect the Ricci ten-

sor R a b with T a b : 8irG R ab - - sg a b R 2 (G = constant of gravitation).

=

4 C

The f a c t (theorem 4 . 4 . 6 ) that the divergence

of the Einstein tensor vanishes leads us back again to energy-momentum conservation.

5.4.

Electromagnetism

175

5.4. Electromagnetism

Especially

beautiful

applications

of

tensor

calculus

are

provided

by

the

mathematical treatment of electromagnetic phenomena. The

evolution

Β = 8(j,t)

of

the

electric

field

& = 5(j,t)

is governed by the Maxwell equations.

and

magnetic

the

They have the

field

following

nonrelativisic f o r m : div 8

=

0

rot 5 + i g

=

0

c at div £

=

4 it ρ

1 dS rot 8 - c- 9t =

4 π . c I'

in Gaussian units (Landau & L i f s c h i t z [1962-1967]). Here, as e a r l i e r ,

c is the vacuum speed of light; ρ denotes the charge den-

sity and j the e l e c t r i c current density. These f i e l d s exert on some particle of charge e,

moving with the velocity

το, the f o r c e g

=

e 6 + e υ χ 8

which is found as a composition of the electrostatic force e Ε and the Lorentz force e τι χ S. If we d i f f e r e n t i a t e the Maxwell equations,

we obtain via the

intermediary

calculation 4tr ^

=

div 6

=

div

=

jj d i v |c rot 8 - 4ir j|

the conservation of charge §£

+

divi

=

ο

=

- 4π d i v j

5. Applications to Physics

176

which s t r o n g l y reminds us of the mechanical continuity

equation.

The r e l a t i o n s h i p is connected with the observation that moving charges g e n erate

electric

currents.

In

this

case,

the

current

density

j

is,

like

section 5.1, found by summation of the products of charge and v e l o c i t y each individual p a r t i c l e within a f l o w

cell.

The t h e o r y becomes much simpler and more elegant if we r e w r i t e i t , ing

to

sion.

the

philosophy

When doing this,

of

relativity

theory,

in

its

l i k e w i s e the last pair of

accord-

four-dimensional

we have to combine the f i r s t t w o Maxwell

(the "homogeneous" ones);

in for

ver-

relations

("inhomogeneous")

for-

mulas. It s u f f i c e s to c a r r y out this process in any coordinate f r a m e since the

re-

sulting tensor equations are valid universally. We s e l e c t an a r b i t r a r y point Ρ in the f o u r - d i m e n s i o n a l space-time

manifold

("world") Μ and determine a Cartesian r e f e r e n c e f r a m e such that the m e t r i c reduces to the Minkowski f o r m η = D i a g t l , - 1 , - 1 , - 1 ) . lence principle

it

is

always

possible

to

find

Because of the equiva-

an inertial 0

1 2

system of

this

3

kind. The 0-component of the position v e c t o r χ = (χ , χ , χ , χ ) corresponds to the time t via the r e l a t i o n x ° = ct. The f o u r - d i m e n s i o n a l

velocity

of

a test

i t s w o r l d line.

By the above argument,

be

naturally

interpreted

as a

particle

is the tangent

vector

of

the e l e c t r i c current density has to

(contravariant)

vector

field.

We

therefore

w r i t e it as Ja

=

(j°,j\j2,j3)

=

(pc,

and charge conservation reduces to the tensor

Ja if

=

j2, j 3 )

relation

0

we at the same time introduce covariant

in place of

partial

derivatives

to g e t invariant f o r m u l a s . T h e r e are several d i f f e r e n t ways to p e r f o r m the required calculations.

One

procedure is to f i r s t w r i t e down all necessary r e l a t i o n s in components

re-

l a t i v e to the given i n e r t i a l f r a m e and then t r y to combine them into tensor equations. The occurrence

of

electromagnetism,

the o p e r a t o r s however,

div

and r o t

in the fundamental

laws

of

suggests to represent the f i e l d s by d i f f e r e n t i a l

f o r m s on M. This leads to a much easier and more elegant f o r m u l a t i o n than

5.4. Electromagnet ism

177

the tensor

approach. 3 we i n t e r p r e t all 3-dimensional vector f i e l d s as 1 - f o r m s in IR which 1 2 3 0 do not only depend on χ , χ , χ but also on the time χ . We view the l a t t e r Hence,

as a p a r a m e t e r and put

β = β(χ°·,χ1,χζ,χ3) · * ! " • > < > '

=

Β dx" α

=

Β dx 1 + Β dx 2 + Β dx 3 1 2 3

τι

=

tj(x°: χ 1 , χ 2 , χ 3 ) '

=

Ε dx" α

=

Ε

l

=

ι (χ ; χ , χ , χ j

,

One can work

0

1

with

2

3

,

· _ • «

=

j dx α

1

·

=

dx 1 + Ε dx 2 + Ε dx 3 2 3 J

1

j dx 1

Α

I

J

2

+ J dx 2

these parameterized d i f f e r e n t i a l

.

-

+ j 3

J

3

dx

forms e x a c t l y

in

the

same manner as with the usual ones. The only d i f f e r e n c e is that d e r i v a t i v e s w i t h respect to the time must be included as w e l l . To prevent misunderstandings,

we denote by β the three-dimensional 1 2 3

f e r s to the components (x , x , x ) alone,

Cartan d i f f e r e n t i a l which

re-

w h i l e d continues to be used

for

the f o u r - d i m e n s i o n a l d i f f e r e n t i a t i o n (including time x ° ) . The technical advantage lies in the e x p l i c i t consideration of of the spatial As concerns marked

the

symmetry

coordinates.

duality,

we » proceed

by an asterisk

similarly:

( ) as b e f o r e ;

the

4-dimensional

the 3-dimensional

mapping

analogue

will

is be

c h a r a c t e r i z e d by a square Γ ) . 3 We c o l l e c t the results. L e t φ, ω be p a r a m e t e r i z e d 1 - f o r m s on IR and f an a r 1 2 3 bitrary O-form, p a r a m e t e r x°.

i.e.

a smooth f u n c t i o n of the coordinates χ , x , x

and the

The rules f o r dualizing f o r m s of ranks 0 , 1 , 2 , 3 , 4 on Μ are in

turn "f

=

#

[f dx° + 0]

[

, 0 ,

dX° Λ ° f

=

°f +

D 1

dx Λψ +

ω

+

dx° Λ Πψ

•, =

-

°f j =

,0

ψ + dx φ

+

λ ι

f dX°

*[dX°ADf] = - f This covers a l l p o s s i b i l i t i e s

(note that det g is negative! ).

By

comparison

178

5. A p p l i c a t i o n s t o P h y s i c s

w e r e g a i n t h e f o r m u l a of t h e o r e m

4.6.1,

**a

(-l)k+1 a

=

f o r k - f o r m s a in M. We e q u a l l y e a s i l y e v a l u a t e t h e r e d u c t i o n f o r m u l a s of t h e 3 C a r t a n d i f f e r e n t i a l s t o t h o s e in R :

d f

=

d | f dx° + tfij = d |dx 0 A^ + ° ω | d [dX0ADV-

n +

(a f ) dx° + 0f ο - dx° λ df + dx° λ d ß + ϋψ - dx° λ ϋψ + dx° λ

=

f]

=

+

- dX° Λ «°φ + dX° Λ

d [dX°A D f]

=

D

(aof)

0

T h e M a x w e l l e q u a t i o n s a r e in t h e t h r e e - d i m e n s i o n a l the

four-dimensional

version equivalent

with

conditions D

a

aDß

=

ο

» y + dQ β

=

l· β π η β

.

d

0

0

=

4 π ρ

„

=

Uϊ C

a n d t h e c o n s e r v a t i o n of c h a r g e i s e x p r e s s e d

d (pc) + ο

D

ö

D

i

t

as

=

0

T h e t r a n s f e r t o 4 - d i m e n s i o n a l f o r m s in s p a c e - t i m e c a n n o w be a c h i e v e d w i t h out e f f o r t . We s h i f t d o w n t h e s u p e r s c r i p t a n d o b t a i n f r o m t h e ( l . O ) - t e n s o r f i e l d J a t h e 1-f orm

5.4. Electromagnet ism

179

J

=

pc dx° - L

whose dual is *

J

N e x t we d i f f e r e n t i a t e this,

d *J

=

dx° Λ j e

Πι

+

•

=

j

pc - dx

ο

•

Λ

ι

getting

DOopc)j

=

dx° Λ

Dc

+ AQ(pc)j

=

0

Thus we have deduced the theorem of charge invariance,

d

*

J

=

0

*

The 3 - f o r m

J is closed and,

nected regions

of

(theorem 4 . 7 . 3 ) . equal to

the w o r l d

if

we r e s t r i c t

manifold M,

Consequently,

our attention to simply

also

exact

by P o i n c a r £ ' s

conlenaa

a 2 - f o r m exists whose Cartan d e r i v a t i v e

is

J.

It is easy to f i n d the desired quantity.

We introduce the field strength as

the 2 - f o r m defined through F

=

- dx° Λ η +

Dß

It connects e l e c t r i c (rj) and magnetic (β) f i e l d s . The dual 2 - f o r m to F is *

F

=

π

ο η + dx

Λ

β

and the Cartan d i f f e r e n t i a l of the l a t t e r yields with the assistance of inhomogeneous Maxwell

d

*

F

=

the

equations

- dx

ο

ββ + dx

Λ

ο

π π η + « η ο

Λ A

4 IT * — J c

=

On the other hand, we have, as a consequence of the two homogeneous laws of electromagnetism,

d F

=

dx°

Λ

«Τ»

+ dx°

Λ

d ° ß

+

e°ß

=

0

This a l l o w s us to base the theory of electromagnetic phenomena on t w o e x -

180

5. A p p l i c a t i o n s t o

c e e d i n g l y simple and e l e g a n t

Physics

formulas:

d

*

F

4 π * — J c

=

d F

P e r h a p s not q u i t e as b e a u t i f u l , sor

but o f t e n expedient a r e the e q u i v a l e n t

ten-

relations -ab

F

F

bc;a

4 π

=

;b

.a

J

C

+ F + F ca;b ab;c

= 0

which a r e accompanied by c h a r g e c o n s e r v a t i o n in the f o r m

Ja

deduced e a r l i e r

;a

=

0

and the equation of motion of

a charged p a r t i c l e

charge e),

[

· a , _a

u + Γ

b be

u u

cΙ

„ab

J

=

eF

u b

The c o v a r i a n t components of the f i e l d s t r e n g t h t e n s o r

0 Ε 1

-E

-E 1 2

0

Β

Ε -Β 2 3 Ε 3 w h i l e the c o n t r a v a r i a n t

read

-E 3 3

0

Β -Β 2 1

-Β

2

Β 1 0

are

(mass m,

5.4. Electromagnet ism

181

We conclude this section by giving the dual tensor,

Β Β 1 2 -Β

1

0

Ε -Ε 3 2

-Β -Ε 2 3 -Β 3 and the

0

Ε -Ε 2 1

Ε 1 0

contravariant

-Β Β 1

-Β 1 2

0

Β -Ε 2 3 Β 3 version.

Β 3

-Β 3

Ε -Ε 3 2 0

Ε -Ε 2 1

Ε 1 0

both in the covariant

5. Applications t o Physics

182

5.S. Optics Although the Maxwell theory of the f o r e g o i n g section is completely general, it

is useful to modify it to a certain degree b e f o r e we apply

it to

the

evolution of e l e c t r i c or magnetic f i e l d s within regions of space which are not void of electrons,

matter.

The reason is that the charges

of

atomic

nuclei

or

react on varying f i e l d conditions.

The f o r c e s exerted by the electromagnetic phenomena cause the particles to change their state of motion;

and this in turn produces e l e c t r i c

currents

discussion would require to consider all particles

individu-

which influence δ and 8. A microscopic ally,

a clearly unworkable task.

Fortunately,

very o f t e n the situation occurs that the f i e l d s change on an

extremely short spatial and temporal scale. Thus a summarizing treatment of the atomic structure (by averaging) becomes feasible. The magnetic induction 8 and the electric field 6 are accompanied by two other t i m e - and position-dependent quantities,

namely the magnetic field 5

and the displacement current 9. In the

limiting

case

of

strong

field

fluctuations

we

are

interested

these variables are connected with 8 and S through linear relations.

in, They

can be expressed with the help of three functions, namely dielectricity ε, (magnetic) permeability μ, and (electric) conductivity σ of the matter. Their exact f o r m is determined by the chemical and physical composition of the medium. We assume that ε, μ, and σ are constant in time,

but possibly

vary with position: ε = ε(,τ), μ = μ ( ί ) , σ = Generally,

ε, μ, and ο- are tensors,

but we r e s t r i c t our discussion to iso-

tropic media f o r which a s u f f i c i e n t l y good approximation by scalar material functions can be achieved.

This excludes an investigation of optical pheno-

mena in crystals which are mostly anisotropic. Basically,

the f o l l o w i n g considerations are unaffected by this

simplifica-

tion. Under the just described conditions, the material equations are 3

=

ε g

8

=

μ 5

5.5.

Optics

183

w h i l e t h e c u r r e n t d e n s i t y in l i n e a r a p p r o x i m a t i o n is given by Ohm's law, j

=

ο- β

The f i e l d e q u a t i o n s div

8 = 0

1 3Ϊ r o t e + - -5V- = c öt div S

=

0

4 π ρ

1 a» r o t S - - 31- = c at

4 π . J c

d i f f e r f r o m t h e Maxwell t h e o r y f o r t h e vacuum m e r e l y by t h e e

substitutions

S, Β -> 5 in b o t h inhomogeneous f o r m u l a s . If we i n s e r t t h e v a l u e s ε = 1,

μ = 1, valid f o r e m p t y s p a c e , we obviously come back t o t h e o r i g i n a l e q u a tions. (or,

However, somewhat

we a r e now mainly i n t e r e s t e d in t h e p r o p a g a t i o n of light more generally,

electromagnetic waves of

sufficiently

high

frequency). In "transparent" m a t e r i a l s t h e p e r m e a b i l i t y n o r m a l l y does n o t d i f f e r a p p r e c i a b l y f r o m 1, hence in w h a t f o l l o w s we s h a l l put μ e q u a l t o u n i t y .

The

r e a s o n l i e s in t h e lack of m a g n e t i z a b i l i t y of t h e s e s u b s t a n c e s . Furthermore,

we d e f i n e t h e refraction index η = n ( t , j ) via t h e Maxwell re-

lation

η

·Ζ~ε

=

which is a p p l i c a b l e w h e n e v e r t h e o p t i c a l c h a r a c t e r i s t i c s of t h e medium v a r y only s l i g h t l y w i t h t h e f r e q u e n c y , i . e . t h e colour of t h e l i g h t . We a s s u m e in a d d i t i o n t h a t we deal w i t h a p o o r c o n d u c t o r (:

- Δ δ

+


0 e t c .

a r e a l w a y s u n d e r s t o o d t o imply t h a t t h e c o n s i -

d e r e d n u m b e r s a r e in R. An i m m e d i a t e c o n s e q u e n c e of t h e above is t h a t G is t h e extension field of R 2 c o n s t r u c t e d by adjunction of i a s a z e r o of t h e p o l y n o m i a l i + 1. I t s d e g r e e o v e r R is t h u s [C:R] = 2. The ( p o s i t i v e ) s q u a r e r o o t of t h e d e n o m i n a t o r of t h e r e c i p r o c a l z" 1 of

z,

t h a t is |z| is c a l l e d t h e modulus of z.

=

•J zz

=

i / x 2 +y 2

It g e n e r a l i z e s t h e a n a l o g o u s q u a n t i t y f o r

n u m b e r s and is multiplicative:

real

6.1. Elementary P r o p e r t i e s of Complex Numbers

191

The (Euclidean) distance of two points w, ζ € C is the modulus |w-z| of the d i f f e r e n c e . With this definition, € is a metric space. The (open) ε-neighbourhood of w is the set

U (w) c

of all points whose distance f r o m w is less than ε (we shall always assume e > 0).

It is also known as the (open circular) disk of radius ε with cen-

tre w. Closed disks are defined similarly. An application of

standard procedures to C yields a Hausdorff space.

We

take the ε-neighbourhoods of arbitrary ζ e C as a basis ( f o r all ε ) .

It a l -

ready s u f f i c e s to include only rational values of ε > 0 and those ζ

which

have rational

real and imaginary parts.

This particular

basis defines the

same topology, but is countable, in contrast to the f o r m e r . 2 The assignment κ: C —> R , which acts by dint of ζ = χ + y i

ι—>

( " f o r g e t t i n g " the algebraic properties),

,2 ( x , y ) e IR'

is a chart and endows C with

the

structure of a 2-dimensional manifold, i . e . an (oriented) surface. We shall have

ample

opportunity

to

apply

this

observation

later;

it

lies

at

the

heart of Riemann's (interpretation o f ) complex analysis. The metric in C is provided by the arc length element ds which is simply the Euclidean one and can be calculated in standard coordinates by the f o r 2 2 2 mula ds

= dx

+ dy . With this,

C even becomes a 2-dimensional

oriented

Riemannian space. The pictorial representation by means of ( x , y ) as Cartesian coordinates of

z = x + yie(Cis

called the Gaussian (number)

plane

or Argand diagram. It is o f t e n advisable to compactify the surface C through the addition

of

an " i n f i n i t e " or "improper" point oo. This results in the 2-manifold y

=

C υ {oo}

It is denoted as the Riemann number sphere, what unhistoric; see section 1.2. This

though this notation is some-

nomenclature, however, has been es-

6. Complex Analysis

192

t a b l i s h e d so s t r o n g l y t h a t a more suitable.

i t is v i r t u a l l y

In any c a s e ,

i m p o s s i b l e t o r e p l a c e i t now by

it is c l e a r w h a t we mean.

In o r d e r t o make if a t o p o l o g i c a l s p a c e ,

we have t o give a c o m p l e t e

listing

of t h e open s u b s e t s in if. They c o n s i s t of t h o s e of C and t h e n e i g h b o u r h o o d s of oo. The l a t t e r ,

by d e f i n i t i o n , a r e t h e c o m p l e m e n t s (in if) of t h e c o m p a c t

C-subsets. One may v e r i f y w i t h o u t any d i f f i c u l t y t h a t if indeed is h o m e o m o r p h i c t o t h e 2 - s p h e r e S . The a l g e b r a i c p e c u l i a r i t i e s

of if, in p a r t i c u l a r

its

interpre-

t a t i o n a s a 1 - d i m e n s i o n a l projective space PC

over t h e c o m p l e x n u m b e r s ,

( i. e.

as

a projective

line)

w i l l be t h e s u b j e c t of o u r s t u d i e s in t h e c h a p -

t e r s t o f o l l o w ; f o r t h e moment we a r e s a t i s f i e d w i t h t h e i n d i c a t e d i n f o r m a tion.

6.2. Convergence of Function Series

193

6.2. Convergence of Function Series For our later (analytical)

investigations,

functions,

we

as f a r as they are concerned with

shall

need several

number of related definitions and theorems. contact with the coordinate concept, quent sections some of

convergence

criteria

complex and a

Since these are only in loose

we merely list in this and the subse-

the most relevant terms and propositions

without

giving detailed p r o o f s or striving f o r the greatest possible degree of

ge-

nerality. Mainly f o r this reason we furthermore avoid to introduce today's t e r m i n o l o gy and p r e f e r the more vivid methods of the 19th and early 20 t h centuries. The arguments leading to the theorems in question usually consist of simple estimates which can either be supplied by the diligent reader or looked up in any textbook

on complex analysis (= theory of complex functions);

the standard literature,

e.g.

Hurwitz & Courant [1964],

Cartan [1966]

cf. or

Peschl [1967], A sequence of points ζ , ζ , ζ , . . .

in a Hausdorff space X, which need not be

mutually d i f f e r e n t , converges to the limit (point) w, if each neighbourhood U of w contains almost a l l members z^ of the sequence ("almost a l l " means all up to at most f i n i t e l y many exceptions). The Hausdorff property guarantees that the limit is unique if it exists. In C, where the topology is determined by the distance relation, we may a l so express this f o r m a l l y as f o l l o w s :

Definition 6.2.1: ( a ) The complex sequence ζ , ζ , z 3> . . . e C has the limit w if to an a r b i t r a ry ε > 0 a positive integer Ν can be found such that f o r all k > Ν the distances

I ζ - w1 I of the sequence members f r o m w are smaller than ε . 1 k If some w e € with this property exists, one writes

and calls the sequence { ζ

| k 6 IN} convergent;

otherwise divergent.

194

6. Complex A n a l y s i s

( b ) The complex numbers ζ , ζ , ζ , . . . f o r m a Cauchy sequence if f o r each g i ven ε > 0 there is an Ν 6 IN such that f o r

a l l m, η > Ν the

inequality

Ι1 ζ - ζ 1Ι < ε holds. m η ( c ) By the series 00

l·-

k = lpartial sums we a l w a y s mean the limit of the

k=l

for Ν

oo, i . e .

Σ

=

lim Ν-*χ>

Σ

If w e want to consider sequences or series on the Riemann sphere !f instead of the complex f i e l d C, a f e w minor m o d i f i c a t i o n s are necessary. is an o r d i n a r y point in

Firstly,

and one t h e r e f o r e has to permit convergence

the l i m i t co. By the very d e f i n i t i o n of

the neighbourhoods of

whenever the moduli of the members of the sequence g r o w

to,

it

a to

occurs

unboundedly.

It is obvious that e v e r y convergent sequence of complex numbers obeys the Cauchy condition.

The r e v e r s e statement is much more i n t e r e s t i n g .

We p r o v e

it here in o r d e r to present at least one example of the estimation methods which are t y p i c a l f o r complex

analysis.

Theorem 6.2.1:

Every r e a l or complex Cauchy sequence converges.

Proof: We again denote the given sequence by {z^ | k 6 IN} and s e l e c t p o s i t i v e tegers Ν ,N ,N , . . .

which f u l f i l l the inequalities

in-

6.2. Convergence of Function Series

ζ

195

-ζ Ν Ν k+1 k


0 t h e r e is some

simultaneously f o r

all

ζ e Γ,

the

sequence is said to converge uniformly. ( c ) Both d e f i n i t i o n s a r e t r a n s f e r r e d to function series in the obvious manner.

Uniform convergence in a domain is a much s t r o n g e r r e s t r i c t i o n than p o i n t wise convergence;

f o r instance,

the limiting function of a uniformly

verging s e r i e s o f continuous functions is i t s e l f continuous.

The

con-

analogous

s t a t e m e n t f o r pointwise convergence would be f a l s e . A t h i r d v a r i a n t of the same concept is also of i n t e r e s t :

Definition 6.2.3: A series GO

a k=0

k

of complex numbers a^ converges absolutely if the s e r i e s formed by t h e i r moduli, 00

converges (and thus has a f i n i t e l i m i t ) .

6.2. Convergence o f Function S e r i e s

197

A useful convergence criterion is

Theorem 6.2.2: L e t f , f , f . . . . be complex functions in the domain Γ and c , c , c . . . . r l 2 3 ι 2 3 s i t i v e r e a l numbers which majorize the f in Γ:

for

all ζ e Γ and k e IN. If

the sum of

the c^ converges,

Kp o -

this implies

the

absolute and u n i f o r m convergence of 00

(and,

consequently,

Absolutely

the continuity of the l i m i t f u n c t i o n ) .

converging

series

are

especially

convenient

to

handle,

because

they do not change t h e i r values if we permute the summands. We are thus a l lowed to "rearrange" them a r b i t r a r i l y . ticular

when dealing

which we turn next.

This is applied f r e q u e n t l y ,

with the HeierstraB version

of

in

par-

complex analysis,

to

198

6. Complex Analysis

6.3. Power Series

Among the f u n c t i o n sequences, the power s e r i e s combine best of a l l the i d e als

generality,

flexibility,

and

convenience

of

application.

This

is

the

main r e a s o n why they form the s t a r t i n g point of all c o n s i d e r a t i o n s in the Weierstraß

version of complex a n a l y s i s ,

the theory

of complex

(analytic)

f unctions.

Definition 6.3.1:

A power series, c e n t r e d a t the point a e C, of the complex v a r i a b l e ζ is a f u n c t i o n s e r i e s of the special f o r m

k k=0 Here,

the coefficients

a r e a r b i t r a r y complex numbers;

likewise f o r

the

centre of expansion a .

The s e t of a l l ζ € C f o r which the sum converges is the domain of convergence o f the s e r i e s .

The domain o f convergence of course depends on the c o e f f i c i e n t s .

I t s gene-

r a l c h a r a c t e r can be described easily:

Theorem 6.3.1:

(a) For

each power

c i r c l e X(p) with

series centre

ρ = p(z|a) a,

E(p) and diverges outside.

such

there that

exists

a uniquely

J) converges

in the

determined interior

is the convergence circle of ip. I t s

of ra-

dius p(p) is c a l l e d the radius of convergence. It may a t t a i n the values 0 (convergence only at a) or oo (convergence everywhere in C;

in this

c a s e the s e r i e s is permanently convergent). ( b ) Inside the c i r c l e of convergence the s e r i e s converges a b s o l u t e l y and on

6.3. Power Series

199

compact subsets even uniformly, ( c ) The convergence radius p(p) can be obtained from the sequence of c o e f ficients

C 1 ' C 2 ' C 3>

·•·

For

instance,

it

is equal

to

the

reciprocal

of

the largest accumulation point of the sequence

This proposition is known as Cauchy's criterion. (An accumulation point is some w ε C all of whose neighbourhoods contain infinitely many terms of the sequence).

Proof: Elementary estimates.

Note that this does not say anything about the behaviour of ρ on the boundary of the convergence circle, i.e. f o r | z-a | = p(p). There are three d i f ferent possibilities:

convergence nowhere,

boundary points.

is not d i f f i c u l t to find examples

It

three conceivable cases.

everywhere,

or only in certain of

each of

these

The absolute convergence in the interior of K(J))

allows to rearrange the series at will.

Two power series which are both

convergent in the domain Γ may be summed or subtracted from each other and multiplied by complex numbers. Furthermore, to multiply the series formally we have to calculate all products of the terms and afterwards reassemble with respect to the degrees, as the formula

indicates. This series represents F ( z ) - G ( z ) if we denote the two f a c t o r s on the l e f t hand side as F ( z ) and G(z). In analogy to this,

it is also possible to form the quotient F(z)/G(z)

if

the denominator function G does not vanish in Γ. We now come to an extremely f r u i t f u l application of these principles. that end we start out from the series

To

200

6. Complex Analysis

00

F(z)

=

y

Ck ( z - a 0 ) "

k=0 centred at a

o

and select a point a

cle of convergence.

which lies in the interior of the

ι

A f t e r suitable rearrangement of the series,

cir-

which con-

verges absolutely in the vicinity of a , we obtain a new representation 00 F(z)

=

Y^

dk ( z - a ^ "

k=0 say. It holds at least in some neighbourhood and thus has a positive radius of convergence.

While the original series converges within a certain c i r c l e

Κ of some radius ρ > 0, the new sum is defined in the interior of some ο ο circle if with radius ρ and centre a . ι Μ ι In the overlapping region η ί , both developments are valid. It may happen that S^ contains points which are not in main of

If so, we enlarge the do-

definition of the function appropriately.

( d i r e c t ) analytic continuation of

F.

This extension is called

We obviously

may i t e r a t e

this

step,

The domain (of definition) of

the

function is the largest connected open subset of C covered by the just

de-

thus coming to an indirect continuation. scribed process.

It is identical with the union of

all

convergence

circles

obtained through repeated analytic continuation. The result is not always independent of the succession of the centres a , a , a , . . . ,

intermediate

which are only subject to the restriction

that

each

of them lies within the circle of convergence of its predecessor. Occasionally, point

without

a f t e r several continuation steps, getting

the

initial

value

of

one returns to the starting

the

function.

This,

should not be seen as a drawback of the method; on the contrary, decisive

advantage.

We merely must interpret

this possibility

however, it is its

in a d i f f e -

rent way. The extended function can attain more than one value at a single point. The underlying idea may be c l a r i f i e d by a suitable example. Let F ( z ) be the square root of ζ and begin the iteration at aQ = 1 (the radius convergence at this place is exactly value of the r o o t ,

equal to 1).

We choose the

thus F ( l ) = 1 and continue analytically in the

of

positive indicated

manner. If,

a f t e r a number of steps, we come back to the initial point 1 a f t e r hav-

ing circled once around the origin ζ = 0, where the function is not d i f f e -

6.3. Power S e r i e s

rentiable,

201

we w i l l f i n d the value F ( l ) = -1. The skeptical readers are ad-

vised to carry out the necessary calculations explicitly. the other branch of

the square root function.

We have reached

A second encircling

of

the

origin leads us back to the initial situation. It is t h e r e f o r e essential to record the intermediate points used during the extension process.

The method sketched above, namely the insertion of

nitely many auxiliary positions, practical

disadvantages.

unfortunately has certain theoretical

For this reason the so-called analytic

fiand

continua-

tion along a curve is normally more appropriate. The idea is to introduce some curve y: [0,1] —> C which starts at the initial point r ( 0 ) = z Q and ends at the place y ( l ) = ζ u p

to which we want to

extend the function. We must interpolate a number of points w , . . . , w 1

the (compact) path y ( [ 0 , l l ) between w to be given in the natural order. the part of the curve f r o m w^

0

= ζ

0

and w

n-l

on

= ζ . They are assumed

η

1

Each of them has to be chosen such that to w^ is completely contained in the inte-

r i o r of both convergence circles with the centres w^ ^ and w^. One may v e r i f y with the help of

elementary

estimates that

Ffz^) is uniquely

deter-

mined by the power series near z q and the path y, independently of the positioning of the w^. A subdivision

with the required properties

can always be constructed

as

long as y is restricted to the domain of F. Two interpretations suggest themselves. The f i r s t , p r e f e r r e d by Weierstraß, defines an analytic function in the indicated way through maximal analytical continuation of a given power s e r i es.

All series developments which are obtainable f r o m each other by this

process (possibly with intermediate steps) are considered as equivalent

lo-

cal descriptions of the same function. The other, founded by Riemann, does not view the domain of definition of an analytical function as a part of the complex Gaussian plane, but as a (Riemannian) surface 5 covering it. In the latter theory, earlier

we have to permit ramifications like the one we met

when discussing the square root.

p r o j e c t i o n ^ —> € need not be invertible;

In special instead,

(isolated) points,

the

d i f f e r e n t "branches"

of

the surface ? may be attached to each other. We can separate them into single sheets by introducing suitable "cuts" ( i . e .

deleting certain lines in C

and their preimages in 5). This, however, is useful only f o r the purpose of localized investigations; In principle,

the Riemann surface itself is a uniform entity.

both views are equivalent.

They supplement each other

since

202

6. Complex Analysis

they emphasize d i f f e r e n t aspects of the theory. We are now able to

divide the boundary points of

of a power series p ( z | a ) into two classes. gular

if

there

is

an

analytic

continuation

p(z|z Q )

which converges in some open set containing z q , Another e x t r e m e l y

the convergence

circle

We call such a point z q reof

the

given

p(z|a),

and singular o t h e r w i s e .

important t r a n s f o r m a t i o n has not been mentioned up t o

now. Every absolutely convergent power series can be d i f f e r e n t i a t e d t e r m by term.

Hence we can associate with the analytical f u n c t i o n 00 F(z)

=

c

V

L·

k

(z-a )k 0

k=0

i t s derivative 00

F'(ζ)

=

V

L·

c

k (z-a )k_1 k

0

k=0

which is also a n a l y t i c . dity of the f o l l o w i n g

At the same time this construction y i e l d s the

vali-

proposition:

Theorem 6.3.2: Every i.e.

analytical

f u n c t i o n is in its domain i n f i n i t e l y

often

differentiable,

00

a C -function.

The concept of complex d i f f e r e n t i a b i l i t y requires a somewhat more d e t a i l e d explanation.

Definition 6.3.2: A f u n c t i o n F: Γ —> C,

which is defined in the domain Γ £ C,

is

(complex)

differentiable at ζ e Γ if the limit R r ' rI z, i)

exists.

F'(z)

-=

n „ F ( z + h ) τ-- F(z). lim

is the derivative of F at the place z.

D i f f e r e n t i a b i l i t y in Γ

6 . 3 . Power S e r i e s

203

of course means that this is true f o r all points in Γ.

Here, the transition to the limit h

0 may be carried out along any path.

The above definition can also be expressed in a different but equivalent way: Each sequence {z+h^} of arguments which tends to ζ yields a convergent set of f r a c t i o n s Ftz+h) - Ftz) h with the limit F ' ( z ) . We consider F ' ( z ) as a new function of z. In marked contrast to the situation in real analysis, complex d i f f e r e n t i a bility in some open region has very far-reaching consequences.

We shall

come back to this aspect as soon as the necessary tools are available (section 6 . 5 ) . Here, we merely add a few remarks concerning power series of analytical functions. It is easy to see that two power series with common centre Z q , which coincide in some open neighbourhood, are equal. They consequently have the same c o e f f i c i e n t s . This implies

T h e o r e m 6.3.3:

Assume F is an analytical function in the domain Γ. The c o e f f i c i e n t s of the power series around a point a e Γ are obtained from the higher order deriv a t i v e s a t a v i a t h e Taylor formula 00

F(z) k=0 where we used the standard notation F

=

(-u

- i v y

+ i u y

- ν ) dx λ d y x

=

0

x

and the p r o p o s i t i o n reduces f o r any regular domain Β to the t w o - d i m e n s i o n a l g e n e r a l i z e d Stokes theorem ( c f . J f ( z ) dz 3B Therefore, is of

the remark a f t e r theorem 3 . 9 . 2 ) : =

J" ω 3B

=

J du

=

0

Β

the v a l i d i t y of Cauchy's theorem is no surprise.

extraordinary

importance f o r the investigation

of

Theorem

6.5.3

holomorphic

func-

tions. In the f o r e g o i n g p r o o f ,

we do not r e a l l y need the continuous

l i t y of the i n t e g r a t i o n path;

differentiabi-

it would be enough to demand that Β admits a

triangulation, i. e. , can be exhausted by t r i a n g l e s .

This is a l r e a d y true

if

the boundary 9B is piecewise C 1 and thus can be built up f r o m f i n i t e l y many C1-sections.

The argument then goes through without

alteration.

6 . 5 . Holomorphic F u n c t i o n s

The l a s t

observation

217

allows

to formulate a slight

variation

of

Cauchy's

t h e o r e m which u s u a l l y is m o r e convenient t h a n t h e old one. It can be s t a t e d as

Theorem 6 . 5 . 4 : Let 3 r 1 .3 r 2 be t w o h o m o t o p i c p a t h s in a domain Γ £ C and f some h o l o m o r p h i c f u n c t i o n in Γ. Then

ι

2

Proof: Apply t h e o r e m 6 . 5 . 3 t o t h e c u r v e y^ - y g ( f i r s t y i in t h e f o r w a r d

direc-

t i o n , t h e r e a f t e r yg b a c k w a r d s ) .

We s h a l l e m p h a s i z e only very f e w of t h e n u m e r o u s i m p o r t a n t c o r o l l a r i e s of t h e Cauchy i n t e g r a l

theorem.

As a f i r s t i n s t a n c e , we c o n s i d e r a c o n n e c t e d open s u b s e t Γ of C and a r e g u l a r domain Β c o n t a i n e d in Γ. The o r i e n t e d b o u n d a r y SB is a c l o s e d

Jordan

c u r v e y, which is a s s u m e d t o e n c i r c l e ζ once in t h e m a t h e m a t i c a l l y p o s i t i v e s e n s e . Then t h e i n n e r p o i n t ζ € Β l i e s t o t h e l e f t of t h e p a t h y. If,

furthermore,

f e 5(Γ) is a h o l o m o r p h i c f u n c t i o n in Γ,

the same

holds

t h o u g h n o t in Γ, b u t only in t h e p u n c t u r e d domain Γ' = Γ \ {z>. T h e r e ,

how-

f o r t h e map d e f i n e d by

e v e r , t h e o r i e n t e d b o u n d a r y c u r v e y = 3B is obviously h o m o t o p i c t o a ( s u f f i c i e n t l y s m a l l ) c i r c l e κ £ Γ w i t h c e n t r e z. We d e n o t e i t s r a d i u s by p. Cauchy's theorem yields

wey

218

6. Complex A n a l y s i s

We p a r a m e t e r i z e κ w i t h t h e a n g u l a r c o o r d i n a t e φ a n d

w

dw

in t h e l a s t i n t e g r a l .

substitute

ζ + ρ e

=

1 ρ e

dp

This leads to 2π

1

of

) dφ

φ=0

wey independently

f(w+pe

t h e p r e c i s e v a l u e of p,

at

least

as

long as κ is

wholly

c o n t a i n e d in Γ. A s i m p l e e s t i m a t e u s i n g t h e c o n t i n u i t y of f a t t h e p o i n t

ζ

p r o v i d e s u s in t h e l i m i t ρ —» 0 w i t h

1 2π1

f(z)

wer Since

by a s s u m p t i o n

ζ has

positive

distance

f u n c t i o n of w i s b o u n d e d a b o v e ( f o r ζ f i x e d ) . absolutely

and

uniformly.

j u s t like t h e i n t e g r a n d ,

It

is

consequently

f r o m y,

the

integrand

Hence t h e i n t e g r a l an

analytic

a n d c a n be d i f f e r e n t i a t e d a r b i t r a r i l y

as

a

converges

function

of

z,

often.

Theorem 6.5.5: ( a ) L e t f b e a h o l o m o r p h i c f u n c t i o n w i t h d o m a i n of d e f i n i t i o n T S C a n d Β a ( s i m p l y c o n n e c t e d ) c l o s e d s u b s e t of Γ. t i o n can be r e c o n s t r u c t e d

In t h e i n t e r i o r of B,

f r o m its values

on t h e

c u r v e y = 3B:

f(z)

1 2nl wey

T h i s r e l a t i o n i s k n o w n a s Cauchy's integral, (b) Moreover,

f is a n a l y t i c ;

i t s n t h d e r i v a t i v e is

(oriented)

the

func-

boundary

6 . 5 . Holomorphic F u n c t i o n s

219

f"»(z)

=

_ η!

Γ f ( w ) dw

2iri

J (w-z)n+1 wey

( c ) Theorem of Liouville: A b o u n d e d a n a l y t i c f u n c t i o n on C i s c o n s t a n t . ( d ) "Fundamental theorem of algebra" a f t e r numbers

is a l g e b r a i c a l l y

closed.

Gauß:

Every

The f i e l d C of

polynomial

of

complex

positive

degree

w i t h c o m p l e x c o e f f i c i e n t s t h u s h a s a t l e a s t one z e r o in C. (It e v e n d e composes completely into linear

factors).

Proof: S t a t e m e n t s (a) a n d (b) h a v e b e e n p r o v e d

already.

If f i s a f u n c t i o n s u b j e c t t o t h e a s s u m p t i o n s t h e e x p l i c i t bound

of L i o u v i l l e ' s

theorem

| f ( z ) | s Μ f o r a l l ζ € C, we f i n d f r o m C a u c h y ' s

with

integral

f ormula

f'n>(0)

e'nl* fiRe")

— R 2π

ί

η! — Ff 2π

y denotes

R

this expression

the

=

Μ — R"

φ=Ο

φ=0

where

Μ dip

circle

of

radius

R > 0

approaches zero,

and

centre

p r o v i d e d η & 1.

0.

In t h e

limit

Consequently,

all

d e r i v a t i v e s of f v a n i s h a t ζ = 0 , a n d t h e f u n c t i o n m u s t b e c o n s t a n t . Now s u p p o s e ,

Ρ is a polynomial over € with η = deg(P) £ 1 w i t h o u t

Then t h e r e c i p r o c a l

f u n c t i o n ^j—- i s a n a l y t i c a l

in a l l of C.

zeroes.

It p o s s e s s e s

a

l i m i t i n g v a l u e f o r ζ —» co ( n a m e l y 0) and t h u s i s c o n t i n u o u s on t h e R i e m a n n s p h e r e . F r o m t h e c o m p a c t n e s s of f we d e d u c e t h e b o u n d e d n e s s of t h e f u n c t i o n ζ —» j^-j- a n d , After all,

by L i o u v i l l e ,

its constancy;

it f o l l o w s t h a t Ρ has a zero ζ

contradiction! e C, s a y ,

t h e f o r m P ( z ) = ( z - z ) - Q ( z ) w i t h Q e Pol (C). 1 n-l t h e p r o o f of G a u ß ' t h e o r e m .

Consequently, existence

(and

even a n a l y t i c .

and c a n be b r o u g h t

By i n d u c t i o n ,

we

c o m p l e x d i f f e r e n t i a b i l i t y of a f u n c t i o n f a l r e a d y i m p l i e s continuity)

of

all

higher-order

derivatives.

to

complete

In f a c t ,

the f

is

H o l o m o r p h y a n d a n a l y t i c i t y in a n o p e n r e g i o n a r e t h u s d i f f e -

rent ways to express the same.

6. Complex

220

T h i s m o s t r e m a r k a b l e r e s u l t i n d i c a t e s how much s t r o n g e r c o m p l e x ability like

i s in c o m p a r i s o n t o i t s r e a l

counterpart

for

Analysis

differenti-

which we have

nothing

this.

The c a l c u l a t i o n

of c e r t a i n i n t e g r a l s is made e a s i e r

by

T h e o r e m 6.5.6: (a) For

any

ζ € Γ,

holomorphic

and e v e r y

function

in

the

punctured

domain

Γ\{ζ),

( p i e c e w i s e ) C 1 - c u r v e y in Γ w h i c h e n c i r c l e s

n a t e d p o i n t ζ o n c e in t h e p o s i t i v e d i r e c t i o n ,

r e s

z

( f ( w ) d w )

=

( 2 i r i )

the

J

_ 1

the

where elimi-

integral

f ( w )

d w

wsj-

is

independent

of

γ.

The

expression

r e s (ω) = r e s ( f d z ) ζ ζ

residue o f t h e 1 - f o r m ω = f ( z ) d z a t t h e p o i n t z .

called

the

It i s a l s o

unaffected

i f w e r e p l a c e t h e v a r i a b l e w by a n o t h e r l o c a l l y u n i f o r m i z i n g

parameter.

(b) The

residue

vanishes

an a n t i d e r i v a t i v e

whenever

near z.

ω is closed,

i.e.

if

the

It t h e r e f o r e c a n a t t a i n v a l u e s

z e r o only if f is not holomorphic at (c)

is

function

f

different

has from

z.

In c a s e t h e r e g u l a r c o m p l e x d o m a i n Β c o n t a i n s o n l y f i n i t e l y m a n y

points

t h e residue theorem

ζ 1 . . . . ,z η in w h i c h f i s n o t h o l o m o r p h i c ,

η J

f ( z )

d z

=

2 π ϊ

9B

^

r e s ^ t f

d z )

=

2 π 1

^

weB

r e s

1=1

( f

z

d z )

1

applies. ( d ) If f h a s in t h e v i c i n i t y of ζ t h e L a u r e n t

f ( w )

=

) )

a\ .

(

series

w ( w -- z )

k

k=-m

w e g e t r e s (f d z ) = a ζ -ι

Proof: Parts

(a)

and

(b)

are

clear;

(c)

follows

from

the

c u r v e y i s in Γ h o m o t o p i c t o t h e u n i o n o f η d i s t i n c t

observation circles,

that

each of

the

which

221

6.5. Holomorphic Functions

surrounds one singularity,

together with theorem 6.5.4.

Finally,

we obtain

statement (d) through a term-by-term integration.

Next,

we turn to two other very important properties of

some functions,

which we f i r s t want to describe abstractly.

Definition 6.5.2: ( a ) We call a continuous real-valued function u in the domain T S C

harmo-

nic if f o r every compact circular disk Κ = K ( p , z ) with centre ζ and r a dius p, which is completely contained in Γ, the condition

1

utz+pe1*1) d(p

Ζπρ

=

u(z)

φ=0

holds. This means that the average of u over the periphery of Κ equals the function value at the centre, ( b ) A potential function in Γ is a twice continuously d i f f e r e n t i a b l e function u: Γ —> R solving the potential equation Au

In the available literature, rant & Hilbert [1968],

=

u

xx

+ u

yy

c f . f o r example Hurwitz & Courant [1964], Cou-

Cartan [19661 etc. , the nomenclature widely

varies.

The compromise used here has been chosen f o r reasons of vividness. U n f o r t u nately,

there is an inconsistency

compared to the notation

(harmonic f o r m s correspond to potential, This, however,

of

chapter

3

not to harmonic f u n c t i o n s ! ) .

does not really matter since we shall see later ( c f .

6 . 5 . 8 ) that both function classes are identical.

theorem

But b e f o r e can do so,

want to find out the interrelation with the holomorphic functions.

we

To this

end, we need

Theorem 6.5.7: ( a ) Let f = u + i v b e a

holomorphic function in the interior

of

a circle

222

6. Complex A n a l y s i s

( c e n t r e z, r a d i u s R), decomposed i n t o r e a l and i m a g i n a r y p a r t s u and v, respectively. the circle.

Assume f u r t h e r t h a t f is c o n t i n u o u s on t h e p e r i p h e r y Then u is uniquely d e t e r m i n e d by i t s b o u n d a r y v a l u e s

of and

c a n b e f o u n d b y Poisson's integral formula, v a l i d f o r r < R,

R2 - r 2

_1 2ir

uiz+re 1 * 1 )

2

R - 2 R r cosΙφ-ι/ι) + r

2

uiz+Re 1 *) dtp

ψ=ο

(b) The conjugate function ν of u is f i x e d e x c e p t f o r an a d d i t i v e

constant,

which we may s p e c i f y t h r o u g h t h e f u n c t i o n value a t t h e c e n t r e z:

R r Ξϊηίφ-ψ) v(z+re

)

v(z) •

u(z+Re 1|S ) dip

-

2

R - 2 R r cos(φ-ψ) + r 2 ψ=ο

(c) F u r t h e r m o r e ,

u and ν a r e h a r m o n i c p o t e n t i a l f u n c t i o n s .

(d) To each p o t e n t i a l f u n c t i o n u in Γ t h e r e is an a d j o i n t f u n c t i o n v, u n i q u e up t o a c o n s t a n t ,

also

such t h a t u + i ν is h o l o m o r p h i c .

Proof:

We w r i t e t h e Cauchy i n t e g r a l in p o l a r c o o r d i n a t e s :

f (z+re

1(l>

)

=

R e

11

Λ

ftz+Re ")

ο

R e ΙΨ

1(1.

- r e l(p

d^i

Φ=ο

T h e r e a f t e r we f o r m t h e a n a l o g o u s e x p r e s s i o n o b t a i n e d f o r t h e mirror point 2

of z + r e'*1, namely ζ + — e 1(P . It r e a d s

I* 0

=

Ή

f (z+Re'1") r e

1|ί

- 1V - R e

dψ

b e c a u s e t h e i n t e g r a n d is h o l o m o r p h i c i n s i d e t h e c i r c l e . p a r a t i o n i n t o f = u + i ν y i e l d s t h e e q u a t i o n s we seek.

S u b t r a c t i o n and s e -

223

6.5. Holomorphic Functions

In both of them, we can ( f o r r < R) interchange the Laplacian with the integral.

This leads immediately

to the validity

of

the potential

Au = 0 and Δν = 0. Setting r = 0, we get the averaging condition: are harmonic.

equations u and ν

If f i n a l l y Δ u

=

u

xx

+u

yy

=

0

we deduce d (u

dy - u dx)

χ

y

and the 1 - f o r m u

χ

connected,

dy - u

y

=

u

xx

dx

Λ

dx is closed.

dy - u

yy

dy

Λ

dx

Since the circular

=

0

disk is simply

the integrability f o l l o w s f r o m Poincar6's lemma, theorem 4.7.3.

Thus there exists an essentially ( i . e .

up to some constant) unique function

ν with dv

=

ν

χ

dx + ν

y

dy

=

- u dx + u y

χ

dy

A comparison of the c o e f f i c i e n t s leads to the Cauchy-Riemann

differential

equations which by theorem 6. 5.1 are equivalent to f = u + i ν being holomorphic. The proof is complete.

The Poisson integral enables us to show that harmonic and potential f u n c tions are the same:

Theorem 6.5.8:

Every potential function in a domain Γ £ C is harmonic,

and vice versa.

Proof: We f i r s t assume that u is harmonic in Γ. Then the maximum principle holds: In every compact domain Κ. £ Γ, the modulus of u attains its maximum at the boundary. We prove this proposition indirectly. inner point ζ of Κ exists with lar,

Let u be a counter-example;

|u(w)| s |u(z)| f o r all w 6 K.

thus an

In particu-

this is true f o r all w on the periphery of any circular disk D which

224

6. Complex A n a l y s i s

is c o n t a i n e d in K, b u t s h a r e s a t l e a s t one b o u n d a r y p o i n t ρ e Rd(K) w i t h K. The d e f i n i t i o n of t h e w o r d " h a r m o n i c " i m p l i e s t h a t u h a s t h e c o n s t a n t v a l u e u(z) on t h e c i r c u m f e r e n c e of D. p o i n t of Κ w i t h t h e r e q u i r e d

In t h i s c a s e ,

however,

ρ is a

An i m m e d i a t e c o r o l l a r y of t h e maximum p r i n c i p l e is:

A function,

h a r m o n i c w i t h i n some domain and v a n i s h e s on t h e b o u n d a r y , zero.

boundary

property.

is

which is identically

This is b e c a u s e t h e maximum of t h e modulus m u s t be equal t o 0.

More g e n e r a l l y ,

a h a r m o n i c f u n c t i o n on a c o m p a c t s e t Κ c C is uniquely d e -

t e r m i n e d by i t s b o u n d a r y v a l u e s (if t h e r e w e r e t w o such f u n c t i o n s ,

their

d i f f e r e n c e would be z e r o on bd K). We a p p l y he l a s t r e s u l t t o t h e s p e c i a l c a s e Κ = c l o s e d c i r c u l a r disk r a d i u s ρ and c e n t r e z.

The r e s t r i c t i o n u |

is c o n t i n u o u s ,

and by

with theo-

K

J 'bd(K) r e m 6 . 5 . 7 t h e P o i s s o n i n t e g r a l gives t h e h a r m o n i c e x t e n s i o n into t h e i n t e r i o r of Κ (which is unique, a s we j u s t s a w ) . The same a r g u m e n t s h o w s t h a t Au = 0 t h e r e .

Next,

we c o n s i d e r t h e r e v e r s e s t a t e m e n t . Let u now be a p o t e n t i a l f u n c t i o n

on Γ. Moreover,

t a k e an a r b i t r a r y p o i n t ζ e Γ and a c o m p a c t c i r c u l a r

disk

in Γ c e n t r e d a t z. It w i l l be enough t o show t h a t t h e z e r o f u n c t i o n is t h e only s o l u t i o n

of

t h e p o t e n t i a l e q u a t i o n in Κ which v a n i s h e s on t h e b o u n d a r y . This is t r u e b e c a u s e P o i s s o n ' s i n t e g r a l p r o v i d e s us w i t h a c o n t i n u a t i o n

in-

t o t h e i n t e r i o r of t h e disk.

We d e n o t e t h i s f u n c t i o n by u . The d i f f e r e n c e

u - u

is

f u n c t i o n and

u - u

=0,

also

a potential

identically

zero

on Rd(K);

thus

and we a r e done.

In o r d e r t o c l o s e t h e gap in t h e f o r e g o i n g p r o o f , we a s s u m e t h a t u is a p o t e n t i a l f u n c t i o n which v a n i s h e s on bd(K). By t h e o r e m 6 . 5 . 7 (d), we can f i n d a h o l o m o r p h i c f u n c t i o n f : Κ —» C whose r e a l p a r t is i d e n t i c a l w i t h u. c o n s t a n t of i n t e g r a t i o n is chosen such t h a t f ( z ) = 0 h o l d s f o r some

The point

ζ 6 Rd(K). Then f = 0 in t h e e n t i r e domain Γ; t h e same c o n c l u s i o n h o l d s f o r u, and t h e t h e o r e m e v i d e n t l y f o l l o w s .

6.6.

Angle-Preserving Transformations

225

6.6. Angle-Preserving Transformations By theorem 6 . 5 . 2 , a n a l y t i c a l (or holomorphic) mappings of the complex numbers are conformal,

except f o r the zeroes of the derivative.

We may ask

which r e g i o n s can be t r a n s f o r m e d b i j e c t i v e l y into each o t h e r by a n a l y t i c a l f unctions. If we r e s t r i c t our a t t e n t i o n to simply connected domains,

the complete a n -

swer is given by the Riemann mapping theorem 6. 6. 3. To prepare i t s p r o o f , we shall need a number of a u x i l i a r y r e s u l t s :

Theorem 6.6.1: Let f j . f g . fg. · · ·

be

a

uniformly bounded sequence of holomorphic

functions

in a complex domain Γ; i. e. |f k Cz)|

< Μ

where the bound Μ does not depend on k e IN and ζ e Γ. Then on each compact subset of Γ a uniformly convergent subsequence of the f

k

e x i s t s (theorem of Hontel).

Proof: Consider an a r b i t r a r y compact subset Β of Γ. We f i r s t claim: To each e > 0 one can find a δ > 0 which f o r all a, b e Β with f i e s the inequality

| a - b | < δ and all k s a t i s -

| f a . J - f ^ i b ) j < ε.

Suppose this is f a l s e .

,

k

Then t h e r e a r e a positive ε and two sequences {a }

and {b } in Β with la - b I < - , but If (a ) - f (b )|1 £ ε. Since Β is comk ' n n ' n ' n n n n p a c t , we can c o n s t r u c t convergent subsequences < a ' } and { b ' > of the a and k k η b , r e s p e c t i v e l y , which tend to the common limit point w e B. π We f i x

ρ > Ο subject

to the condition

that

the

open c i r c l e

Κ = K(2p, w)

around w of radius 2p is contained in Γ. Then obviously a number Η e x i s t s such t h a t f o r all m > Η the inequalities hold.

|1 am ' - w | < ρ as well as

The Cauchy i n t e g r a l f o r m u l a (theorem 6 . 5 . 5 ) moreover yields

|' ibn' - w |1 < ρ

6. Complex Analysis

226

If ( a ' ) - f ( b1 ' ) I

' m m

mm

- Ι Γ f (z) £ ( J m 3K

a

["(z-a' ) _ 1 - ( z - b ' ) _ 1 ] dzl m m I |

2n π

a'-b' m m — 4πρ Μ 2" min{(z-a')(z-b')}

s

s

2pM p" 2 i f 1

=

2M — np

which in the limit η -» , and so on in the indicated manner, n 2 The "diagonal sequence" F = f evidently converges f o r all ζ and consen n,n k quently f o r all points with rational coordinates in Γ. Finally,

we cover the compact set Β by suitable neighbourhoods of the ζ

( e . g . circular disks with f i x e d radius - ) and choose an arbitrary f i n i t e subcover Κ , Κ , . . . ι 2 Each of the circles Κ contains a rational point ζ : hence f o r all ζ e Β at least one ζ

J

J

J

can be found whose distance from ζ is less than δ. The conver-

gence property of the F^ proved above guarantees the existence of a number N(e),

depending on c, with IF ( z J - F 1 m k

(z ) I η k '


Ν (here, of course, we need the finiteness of the covering).

In particular,

the special z^ constructed earlier obey this

restric-

tion, and we have IF

'm

(z)-F (z) I η '

s

IF

'm

(z)-F (ζ ) m j '

I +

(z )-F ( ζ ) ' m j n j '

IF

I +

(z ) - F ( ζ ) 1I ' n j n

IF

which was to be shown.

It is advisable to formulate the next intermediate step separately.

s

3 ε

6.6.

Angle-Preserving Transformations

227

Theorem 6.B.2: Consider

a s e q u e n c e of h o l o m o r p h i c f u n c t i o n s f , f , f , . . .

in t h e o p e n

do-

m a i n Γ S C w h i c h is u n i f o r m l y c o n v e r g e n t in e a c h c l o s e d s u b s e t of Γ. If a l l f are injective or, k f unction

a s one s a y s in c o m p l e x a n a l y s i s ,

ζ

ι—>

univalent, t h e

limit

F(z) = lim f (z) η n-W>

e i t h e r h a s t h e same p r o p e r t y or is c o n s t a n t .

Proof: We k n o w a l r e a d y f r o m t h e o r e m 6 . 2 . 2 and t h e o r e m 6 . 5 . 5 t h a t t h e l i m i t f u n c t i o n F is h o l o m o r p h i c .

Suppose F t o be n o n c o n s t a n t and s e l e c t a point a e Γ

a n d a c i r c l e κ w i t h c e n t r e a on w h i c h F d o e s n o t a t t a i n t h e v a l u e w = F ( a ) ( t h i s i s p o s s i b l e ! ). T h e

expression „ N

_ "

ι Γ F ' ( z ) dz i^T J F ( z ) - w κ

i s by t h e r e s i d u e t h e o r e m e q u a l t o t h e n u m b e r of z e r o e s ( c o u n t e d w i t h m u l tiplicities)

of F ( z ) - w in t h e r e g i o n e n c l o s e d by κ.

In p a r t i c u l a r ,

Ν is

in-

teger. T h i s r e m a i n s t r u e if w e r e p l a c e F by one of t h e f . But t h e i n t e g r a l d e n p e n d s c o n t i n u o u s l y on F; t h e r e f o r e i t d i f f e r s f o r s u f f i c i e n t l y l a r g e η by an a r b i t r a r i l y s m a l l a m o u n t f r o m N. S i n c e b o t h n u m b e r s a r e i n t e g e r s , must coincide.

By c o n s t r u c t i o n ,

they

e a c h f η- w h a s o n l y o n e ( s i m p l e ) z e r o in t h e

d o m a i n b o u n d e d by κ; h e n c e Ν = 1. We n o w a p p l y o u r a r g u m e n t a t i o n

to a circle which does not s u r r o u n d

the

p o i n t a . T h i s l e a d s t o a s i m i l a r i n t e g r a l w h i c h t u r n s o u t t o b e 0, s i n c e o t h e r w i s e a l l f w i t h l a r g e η w o u l d h a v e a s o l u t i o n of f (z) = w i n s i d e t h e π π circle, contradicting the assumption that f is u n i v a l e n t . T h u s t h e r e i s η e x a c t l y o n e ζ w i t h F ( z ) = w, n a m e l y a .

We a r e n o w p r e p a r e d t o w r i t e down t h e m a i n r e s u l t on c o n f o r m a l

equivalence:

228

6. Complex Analysis

Theorem 6.B.3: Every simply connected domain Γ c C is conformally equivalent to the open unit disk

Κ

Thus,

=

j ζ e C

|z| Γ,

is analytic

Γ —> Κ which,

(Riemann mapping theorem,

as w e l l also

as

called

its the

main theorem of conformal mapping). We may even r e q u i r e in addition that a given point w e Γ is t r a n s f o r m e d to f ( w ) = 0 and that the d e r i v a t i v e f ' ( w ) conditions,

is r e a l and positive.

With these

by-

the f u n c t i o n f is unique.

Remarks: ( 1 ) Γ must be simply connected since the desired mapping is a homeomorphism f r o m Γ onto the unit disk. ( 2 ) The r e s t r i c t i o n to proper subsets of C, i. e. is l i k e w i s e essential,

the exclusion of C i t s e l f ,

because every analytic mapping f

f r o m C into Κ

is bounded by 1 and thus constant by L i o u v i l l e ' s theorem 6 . 5 . 5 This is not compatible with the assumptions on f .

(c).

It is the more a s t o n -

ishing that a l l other Γ are admissible. ( 3 ) As a c o r o l l a r y ,

w e obtain: T w o a r b i t r a r y simply connected complex d o -

mains other than C are d i f f e o m o r p h i c to each other,

but not to C ( a l -

though they are homeomorphic with C! ).

Proof: Since the composition of f i n i t e l y many c o n f o r m a l b i j e c t i o n s is of the same type,

we are a l l o w e d to impose certain

additional

restrictions

on the

do-

main Γ. By assumption,

there is a complex number a which is not contained in Γ. We

consider the l o g a r i t h m l n ( z - a ) , cording to the equation

defined via the inverse function of exp

ac-

6.6. Angle-Preserving Transformations

e

Since Γ is simply c o n n e c t e d ,

ln(z-a)

=

the part

above Γ s p l i t s into i s o l a t e d l e a v e s . ln(z-a)

229

z - a

of t h e Riemann s u r f a c e which

We s e l e c t one of t h e m .

lies

This b r a n c h of

is an a n a l y t i c a l f u n c t i o n g: Γ —> C, b e c a u s e i t s only

singularities

( a t co and a) a r e not in Γ. Furthermore,

g is u n i v a l e n t , ζ

ι

=

f o r f r o m g ( z j ) = gtZg).

a + e x p ( g ( z )) ι

=

a + e x p ( g ( z )) 2

=

ζ

2

immediately follows. We now c h o o s e some w e Γ. It pcissesses a c i r c u l a r n e i g h b o u r h o o d which is c o m p l e t e l y c o n t a i n e d in t h e domain of g. A f t e r s h i f t i n g by 2tri, we g e t o t h e r c i r c u l a r disk which,

t h a n k s t o t h e i n j e c t i v i t y of e x p ° g ,

is

an-

disjoint

w i t h g ( D . The a u x i l i a r y f u n c t i o n

g ( z ) - g(w) + 2iri is c o n s e q u e n t l y and a n a l y t i c a l ,

bounded

in Γ.

Except

for

this,

it

is obviously

We a r e t h u s a l l o w e d t o r e p l a c e Γ by i t s image u n d e r transformation, t o be bounded.

univalent

a s i t is composed f r o m f u n c t i o n s e n j o y i n g t h e s e p r o p e r t i e s . the just

described

or - t o e x p r e s s it d i f f e r e n t l y - we m i g h t have s u p p o s e d Γ Without r e s t r i c t i o n ,

we s h a l l a s s u m e f r o m now on Γ t o be

c o n t a i n e d in t h e open u n i t disk

Κ =

| z e c |

By s h i f t i n g and a s u i t a b l e n o r m a l i z a t i o n ,

| z | < 1 j·

we can beyond t h i s a c h i e v e

t h e d i s t i n g u i s h e d p o i n t w is l o c a t e d a t t h e o r i g i n of t h e n a t u r a l nate system:

that

coordi-

w = 0.

After this reduction,

we c o n s t r u c t t h e f u n c t i o n f a m i l y S c o n s i s t i n g of

all

f u n c t i o n s , which a r e h o l o m o r p h i c in Γ, bounded by 1, u n i v a l e n t , and s u b j e c t to the

restrictions

fCO) = 0 and

f ' ( 0 ) > 0.

In any c a s e ,

3 contains

the

i d e n t i t y and is t h u s not empty. We s e l e c t an a r b i t r a r y n o n z e r o p o i n t a e Γ and keep i t t h r o u g h o u t t h e r e s t of t h e p r o o f . Each f e g omorphism).

maps Γ o n t o a domain f ( D £ Κ ( s i n c e it is a h o m e -

230

Our next

6. Complex A n a l y s i s

aim is to determine f

For the sake of s i m p l i c i t y ,

such that

|f(a)| is as l a r g e

as

possible.

we w r i t e

sup I

and build a sequence f ^ f ^ f 3 > . . .

|f(a) I

f e 3 j-

in 3, f o r which

Ilm f ( a ) π

=

μ

holds. As we saw b e f o r e , the identity is a member of 3. and we may conclude μ £ |a| > 0 .

Montel's theorem 6 . 6 . 1 guarantees f o r e v e r y given compact sub-

set of Γ the existence of

a partial

sequence g j . g , g g

say,

which converges u n i f o r m l y to a holomorphic function F.

of

the f ,

On account of

the

conditions F ( 0 ) = 0 and |F(a)| = μ, the limit function is not constant,

and

theorem 6. 6. 2 thus implies that F is univalent,

since the f

are.

It remains to show that the image domain F ( D completely f i l l s out the int e r i o r of the unit disk in which it is obviously

contained.

Suppose not.

Then there is some point w, lying in K, but not in F ( D . We lw e x p r e s s it in polar coordinates: w = ρ e , where 0 < ρ < 1. Furthermore, we introduce the abbreviation 2 w c

2Vp

=

= (l+p)Vp

e

ίφ

1+p

The f u n c t i o n

w ζ ζ

ι—»

h(z)

(z-c)

-ίφ

= ρ (c ζ - 1)

maps Κ b i j e c t i v e l y onto i t s e l f . C, namely

p1/2

e1*

and ρ

1/2

e 1 *.

z

(z-c)

c ζ - 1

The d e r i v a t i v e h' has e x a c t l y t w o z e r o e s in The associated f u n c t i o n values are w and * -2 w = ρ w. (By the way, the

i t s mirror image r e l a t i v e to the unit c i r c l e ,

arguments are also t r a n s f o r m e d into each other by the inversion at 3K). -ι * The inverse function h has r a m i f i c a t i o n points only in w and w . Both •

points are outside F ( D : The t r a n s f o r m a t i o n h

1

connected domain F ( D .

·

w by d e f i n i t i o n and w , because we have thus is unramified, In other words,

i.e.

univalent,

it is a c o n f o r m a l

|w | > 1.

over the simply equivalence.

231

6.6. Angle-Preserving Transformations

But, as one easily v e r i f i e s , to zero than ζ i t s e l f ; distances:

|h

1(z)|

every ζ e Κ has an image h ( z ) which is closer

the inverse function h

> |z| f o r

all

ζ

with

1

consequently

|z| < 1.

This

is a

increases the contradiction

to the defining maximum condition on F. So we do have F ( D = K. The existence part of the Riemann mapping theorem follows. It remains to prove the uniqueness.

This is a consequence of the f a c t that

there is only one angle-preserving mapping of Κ onto i t s e l f , the origin invariant and has a positive derivative there,

which

leaves

namely the iden-

t i t y function F ( z ) = z. The last statement in turn is an immediate corollary of the c l a s s i f i c a t i o n of all ( c o n f o r m a l ) isomorphisms of the unit disk:

Theorem 6.6.4: The conformal automorphisms of the unit disk are just the linear fractional transformations of the special f o r m

τ(ζ)

=

a ζ + b b ζ + a

with complex numbers a, b, f o r which the by-condition a a - b b

=

1

holds.

Proof: These function are clearly

b i j e c t i v e mappings of

the open unit disk

onto

i t s e l f and holomorphic in both directions. We have to demonstrate that there are no further transformations with the required properties. Suppose, τ is an analytical automorphism of K. We may r e s t r i c t the discussion to the special case τ ( 0 ) = 0 and τ ' ( 0 ) > 0, because this normalization can be achieved a f t e r w a r d s by a suitable rational The function f :

Κ —» C defined by

linear

transformation.

232

6. Complex Analysis

' ζ f(ζ)

1

τ(ζ)

( ζ * 0)

= • τ'(0)

( ζ = 0)

is holomorphic in Κ and has everywhere on the boundary of K, points ζ with |z| = 1 ,

This implies

at all

the constant modulus 1. (This can be corroborated by

a simple limit process). theorem 6 . 5 . 8 ,

i. e.

Thanks to the maximum principle in the proof

of

the modulus |f(z)| is bounded above by 1.

|τ(ζ)| s |z|' in all of

K.

For the inverse transformation τ

1

the same argument applies, and we conclude |τ(ζ)| = ζ . Furthermore, in the limit ζ —» 0, we get τ ' ( 0 ) = 1. Theorem 6 . 3 . 4 says that τ ( ζ ) can be developed into a power series τ(ζ)

=

z + a

2

ζ

2

+ a

3

ζ

3

+ ...

which converges everywhere in K. If one of the c o e f f i c i e n t s a

were d i f f e r -

ent f r o m 0,

|τ(ζ)| = |z|.

this would o f f e n d against our previous equation

Hence we come to the desired conclusion: a f o r all ζ with

= 0, or, equivalently, τ ( ζ ) = ζ

Izl < 1.

Remark: The same considerations even yield a bit more, namely the extremely important lemma of Schwarz,

Theorem 6.6.5: A holomorphic function f which transforms the interior of the unit into i t s e l f

is either a rotation,

circle

that is a linear map of the special

f ( z ) = e i a ζ with some real constant a, or f o r all ζ with 0 < |z| < 1 , inequality

|f(z)| < |z| is true (and then also | f ' ( 0 ) | < 1).

form the

6.7. L i e Groups

233

6.7. Lie Groups Many mathematical and physical questions lead naturally to problems in a l gebra or d i f f e r e n t i a l geometry.

So, f o r instance,

composition of mappings is always associative.

it is well-known that the

This is why the symmetries

of any o b j e c t f o r m a group. On the other hand, just those spaces which o c cur in s c i e n t i f i c investigations are usually endowed with some topological, if

not

even

differentiable

structure.

It

suggests

itself

to

combine

both

aspects. We owe the f i r s t systematic considered

smooth

(i.e.

approach in this direction

infinitely

often

to Lie [1970].

differentiable)

which a product exists which defines a group multiplication. especially

its

interrelation

called Lie algebras, ful.

Later,

with a distinguished

on

This idea, and algebras

(today

a f t e r their discoverer) proved to be extremely

fruit-

much more general

class of

he

manifolds,

objects have been investigated.

Here,

we

Though we only need a f e w elementary concepts and results f r o m this

ex-

merely mention b r i e f l y the topological groups. tremely multilayered f i e l d which has continued to be of topical interest up to now, it seems appropriate to include at least a f e w advanced results. For reasons of space limitation we r e f r a i n f r o m discussing explicit p r o o f s ; almost all statements can be v e r i f i e d easily with the means we have at our disposal. The only two exceptions are theorem 6 . 8 . 3 by Ado, which is not inherently difficult,

and,

above all,

the solution of

so-called 5 t h Hilbert problem (see below).

but somewhat complicated,

The latter

demands very

the

deep-

rooted and specialized methods and can t h e r e f o r e not even be sketched in this book. The f u r t h e r developments during the 20 t h century in d i f f e r e n t as there are algebraic geometry, (algebraic) topology,

directions,

and group theory or

the recent applications in physics (dynamical systems, c e l e s t i a l mechanics, quantum theory,

etc. ) can only be mentioned without detailed

explanation.

The literature on these branches is abundant and easily accessible. But let us put these remarks in more concrete terms!

6. Complex A n a l y s i s

234

Definition 6.7.1: ( a ) A topological group is a t o p o l o g i c a l s p a c e G w i t h a p r o d u c t

• :

s u c h t h a t (G, •) i s a g r o u p ,

G χ G

—>

G

(a,b)

ι—»

a-b

and b o t h ,

a

1

and a - b ,

depend

continuously

on a a n d b. CO

( b ) If G, (a,b)

beyond this,

is even a C - m a n i f o l d a n d t h e g r o u p

t—> a · b a s w e l l a s t h e i n v e r s i o n a

a

1

multiplication

are smooth mappings,

we

c a l l G a Lie group.

It i s e a s i l y s e e n t h a t s u b g r o u p s and f a c t o r g r o u p s of G a r e a l s o cal groups.

E v e r y o p e n s u b g r o u p s is a l s o

topologi-

closed.

T h e r e s u l t w e w a n t t o q u o t e n e x t h a s a l r e a d y b e e n f o u n d by Lie h i m s e l f :

Theorem 6.7.1: F o r a n y L i e g r o u p G w i t h C°°-atlas U, one c a n a l w a y s f i n d a n a n a l y t i c a l l a s 11' w h i c h i s c o m p a t i b l e w i t h U s u c h t h a t m u l t i p l i c a t i o n a n d i n v e r s i o n G are (real) analytic

in

mappings.

Lie a l s o s h o w e d t h a t w e a l r e a d y g e t Lie g r o u p s i f ,

in d e f i n i t i o n 6 . 7 . 1 ,

m e r e l y r e q u i r e G a n d a l l g r o u p o p e r a t i o n s t o be t w i c e c o n t i n u o u s l y rentiable.

at-

T h i s r a i s e d t h e q u e s t i o n w h e t h e r t h i s a s s u m p t i o n c a n be

we

differelaxed

f urther. In a t a l k

delivered to the 2nd I n t e r n a t i o n a l

Mathematical

C o n g r e s s in

r i s , H i l b e r t p o s e d a s one of 2 3 q u e s t i o n s t o f u t u r e g e n e r a t i o n s of

Pa-

mathema-

ticians:

Hilbert's 5 t h Problem: To w h a t e x t e n t i s L i e ' s c o n c e p t of c o n t i n u o u s t r a n s f o r m a t i o n g r o u p s sible to our investigations t y of t h e f u n c t i o n s ?

even w i t h o u t t h e a s s u m p t i o n of

acces-

differentiabili-

6 . 7 . L i e Groups

235

A l i t e r a l q u o t a t i o n c a n be f o u n d e . g .

in H i l b e r t [1900] o r ( w i t h a

histori-

c a l e v a l u a t i o n of t h e a t t e m p t s a r i s i n g f r o m i t ) S k l j a r e n k o [1971]. T h e c o m p l e t e a n s w e r i s d e d u c e d f r o m a c o m b i n a t i o n of t h e p a p e r s by P o n t r j a g i n [1934],

G l e a s o n [1952], M o n t g o m e r y & Z i p p i n [1952].

It

reads:

Theorem 6.7.2: E v e r y t o p o l o g i c a l g r o u p d e f i n e d on a C ° - m a n i f o l d i s a Lie g r o u p .

We s h a l l n o t u s e t h i s t h e o r e m ;

in a l l a p p l i c a t i o n s ,

t h e s m o o t h n e s s of

the

r e l e v a n t m a p s will be given f r o m t h e o u t s e t . The n e x t

d e f i n i t i o n is a l r e a d y f a m i l i a r to us (section

of c l a r i t y ,

however,

2.2).

For t h e

w e r e p e a t t h e e x p l a n a t i o n of t h e t e r m s

sake

"commutator"

and "solvability" once more.

Definition 6.7.2: ( a ) T h e conmutator of t w o e l e m e n t s a, b of a g r o u p G is

[ a , b]

=

a

1

b

1

a b

If U a n d V a r e s u b g r o u p s of G, w e d e n o t e by [U,V] t h e s u b g r o u p g e n e r a t ed by a l l c o m m u t a t o r s [ u , v ] w i t h u e U and ν e V. ( b ) T h e lower central series of G is t h e s e q u e n c e G , G^, G 3

recursively

c a l c u l a b l e via G

= G and G = [G,G ]. ι k+i k If t h e r e i s a n a t u r a l n u m b e r k w i t h G^ = 1, t h e g r o u p G i s nilpotent.

(c) Similarly,

= G a n d G 1 " " ' = [ G ^ ' . G 1 " ], w e o b t a i n t h e de-

setting G

rived series of

G.

If

it

eventually

leads

down t o

the

trivial

group,

t h e n G i s solvable ( o r soluble).

One s e e s w i t h o u t d i f f i c u l t y t h a t n i l p o t e n t g r o u p s a r e s o l v a b l e ; s t a t e m e n t is,

in g e n e r a l ,

the

reverse

false.

We r e t u r n t o t h e Lie g r o u p s and i n t r o d u c e s o m e m o r e n o t a t i o n of f u n d a m e n t a l importance.

236

6. Complex A n a l y s i s

D e f i n i t i o n 6.7.3:

( a ) The g r o u p g e n e r a t e d by a l l c o n n e c t e d s o l v a b l e n o r m a l s u b g r o u p s of a Lie g r o u p G is t h e radical of G. We d e n o t e i t by Rad(G). (b) A c o n n e c t e d Lie g r o u p G w i t h Rad(G) is semisimple.

The r a d i c a l

can a l s o be c h a r a c t e r i z e d

as the maximal

connected

solvable

n o r m a l s u b g r o u p of G. The g r o u p G is s e m i s i m p l e if and only if i t is c o n n e c t e d and does n o t c o n t a i n c o n n e c t e d s o l u b l e n o r m a l s u b g r o u p s e x c e p t 1. Both a s s e r t i o n s a r e b a s e d on t h e f a c t t h a t t h e p r o d u c t of s o l u b l e

normal

subgroups has the same property. The s e m i s i m p l e o n e s a r e in a c e r t a i n s e n s e t h e b u i l d i n g b l o c k s f r o m which a l l Lie g r o u p s can be c o n s t r u c t e d .

T h e i r r e l e v a n c e is a l s o d e r i v e d f r o m

T h e o r e m 6.7.3:

( a ) Every Lie g r o u p G c o n t a i n s a t l e a s t one Levi complement of t h e r a d i c a l . By t h i s we m e a n a m a x i m a l s e m i s i m p l e s u b g r o u p S. It f u l f i l l s t h e c o n d i tions Rad(G)-S

= G

Rad(G) η S

=

and 1

(theorem of Levi).

(b) All Levi c o m p l e m e n t s a r e c o n j u g a t e in G (theorem of Mal'tsev). ( c ) The f a c t o r g r o u p G/Rad(G) of a c o n n e c t e d Lie g r o u p G is s e m i s i m p l e .

These p r o p o s i t i o n s r e d u c e t h e i n v e s t i g a t i o n of c o n t i n u o u s Lie g r o u p s t o t h e d i s c u s s i o n of t h e ( s o l u b l e ) r a d i c a l , t y p e of e x t e n s i o n .

To c a r r y t h i s o u t ,

t h o d s of t h e s u b s e q u e n t

section.

its (semisimple) complements,

and

one mainly a p p l i e s t h e a l g e b r a i c

the me-

6 . 8 . Lie A l g e b r a s

237

Β.8. Lie Algebras

The c e n t r a l i d e a of Lie t h e o r y is t o a s s o c i a t e w i t h each Lie g r o u p G an a l g e b r a w h o s e u n d e r l y i n g s p a c e is t h e t a n g e n t s p a c e TG^ a t t h e t r i v i a l ment of G. R e m a r k a b l y ,

t h i s mapping in many c a s e s is i n v e r t i b l e ,

ele-

and t h e

s t r u c t u r e of t h e Lie a l g e b r a t h u s o b t a i n e d in t h e l a r g e d e t e r m i n e s t h a t of G. T h i s Lie correspondence is of d e c i s i v e r e l e v a n c e f o r an u n d e r s t a n d i n g of t h e Lie g r o u p s . With i t s h e l p ,

f o r instance,

Killing [1888-1890) s u c c e e d e d t o c a r r y

a c o m p l e t e c l a s s i f i c a t i o n of a l l s i m p l e complex Lie g r o u p s .

through

In doing so,

p e r f o r m e d a m a j o r p a r t of t h e p r o g r a m d r a w n up in t h e f o r e g o i n g

he

section

(some i n a c c u r a c i e s which s l i p p e d i n t o Killings work w e r e c o r r e c t e d l a t e r by C a r t a n [1952-1953];

c o n c e r n i n g h i s t o r y and r e l e v a n c e of t h e

classification,

c f . Yaglom [1988], Coleman [1989], Hawkins [1982] u . a . ) . It is t h i s p a r a l l e l i s m b e t w e e n d i f f e r e n t i a l g e o m e t r i c and a l g e b r a i c which m a k e s L i e ' s t h e o r y so v e r s a t i l e ,

aspects

and it h a s such a g r e a t many a p p l i -

c a t i o n s in m a t h e m a t i c s and p h y s i c s t h a t we s h a l l not a t t e m p t t o l i s t

them

t o any d e g r e e of c o m p l e t e n e s s . All we can do is t o give some n o n r e p r e s e n t a tive examples. Chevalley's description

of t h e Lie a l g e b r a s in c o m p a r i s o n

to the

original

c o n s t r u c t i o n by Killing h a s t h e g r e a t a d v a n t a g e t h a t a l l t h e structure constants of t h e a l g e b r a (see below) become i n t e g e r and n o t only r a t i o n a l n u m bers. Recently, trary

L i e ' s c o n s t r u c t i o n s could be t r a n s f e r r e d a l o n g t h i s l i n e t o

fields

of

positive

characteristic

(Chevalley [1955];

the

arbi-

investiga-

t i o n s of t h e 19 t h c e n t u r y a r e s t r i c t l y c o n f i n e d t o t h e g r o u n d f i e l d s IR and C).

So,

e.g.,

it

became

possible

to

classify

"almost

a l l " finite simple

groups w i t h i n a u n i f o r m s c h e m e ( a s "groups of Lie type" = Chevalley groups"

or modifications t h e r e o f ,

t h e s o - c a l l e d "twisted types"; m o r e

can be f o u n d in G o r e n s t e i n [1968]).

information

Only t h e alternating groups A (n £ 5) η

and t h e 26 sporadic groups a r e o u t s i d e t h i s c a t e g o r y .

As soon a s we have

developed

once a g a i n ,

some t e c h n i c a l

tools,

we s h a l l

come back

if

only

b r i e f l y , to t h i s important point (section 21.6). We a r e n o t going t o p r e s e n t t h e c l a s s i f i c a t i o n of s i m p l e c o m p l e x Lie a l g e b r a s in t h i s book,

b e c a u s e we have no a p p l i c a t i o n s f o r t h e r e s u l t .

It

is,

238

6. Complex Analysis

however,

closely related to the Coxeter theory of part D. Especially

the

root systems and the Dynkln diagrams described there will play a fundament a l part.

They are slight modifications of concepts which were also i n t r o -

duced by Killing [1888-1890], During the last decades,

many branches of physics utilized Lie groups and

algebras. We cannot do more than j u s t mention (celestial) mechanics and hydrodynamics. In this theory of fluid motion, f o r example, the KdV- (Korteweg-deVries-)equation has aroused great interest.

This is a completely in-

tegrable dynamical system with infinitely many degrees of freedom.

It is

studied with the help of algebraic methods based on certain generalized Lie algebras,

t h e Kac-Moody algebras.

The l a t t e r are also valuable,

if not essential,

f o r a deeper understanding

of f i n i t e groups; furthermore they have important applications in particle physics (key phrase:

superstrings). One can say without exaggeration

Lie groups and Lie algebras are the ideal means to c h a r a c t e r i z e

that

(continu-

ous) symnetries.

But let us now return to our topic proper! We s t a r t with the a b s t r a c t d e f i nition of the term "Lie algebra".

Definition 6.8.1:

An algebra L over the field Κ is a Lie algebra if the product [.,.]:

L χ L

—»

L

(a,b)

ι—>

[a, b]

(aside from the self-evident bilinearity) f u l f i l l s the following two conditions: (a) For every a e L, we have [a,a] (antisymmetry);

=

0

and

(b) f o r all a, b, c e L, [[a,b],c] + [[b,c],a] + [[c,a],b] (Jacobi-identity).

=

0

6.8. L i e Algebras

239

Remarks: ( 1 ) The antisymmetry implies an identity, by polarization,

namely [a,b] + [b,a]

Vice versa,

which is deduced without e f f o r t

=

0

if the characteristic of the ground f i e l d is d i f f e r e n t f r o m

2, the reverse statement is also true,

but f o r char Κ = 2, the version

used in the definition is stronger. (2) The symbol [ . , . ] f o r the Lie product stems f r o m the f a c t that the mat r i x commutator [Α, Β]

=

Α Β - ΒA

endows the vector space K ( n , n > of the (n, n)-matrices with c o e f f i c i e n t s in Κ with a Lie algebra structure. We even have the much more general, but nevertheless elementary

Theorem 6.8.1: ( a ) Every K-algebra L with associative product ( a , b )

i-» a-b becomes a Lie

algebra with the new multiplication rule [a.b]

=

a-b - b-a

f o r all a, b e L. ( b ) The set of the derivations of an algebra A, i . e . the linear maps D of A into itself with D(a-b) = D(a)-b + a-D(b) f o r all a,b e A, is a Lie

al-

gebra Der(A) with the Lie product [D ,D ] 1 2

=

D «D 1 2

- D °D 2 1

( c ) The adjoint transformations to the elements χ of an arbitrary Lie a l g e bra L, ad(x):

y ι—> [ x , y ]

240

6. Complex Analysis

a r e called

inner derivations

of L. They form a (usually proper) s u b a l -

gebra ad(L)

= I ad(x) | χ e L |

of Der(L). The assignment ad:

L —> Der(L) χ ι—> ad(x)

is the adjoint representation of L. Its image ad(L) is isomorphic with the f a c t o r a l g e b r a L / Z ( L ) of L by i t s centre, that is the subalgebra Z(L) (d) The

Killing f o r m

= \ ζ 6 L

[z, L]

g, defined by

g(x,y)

= tr |ad(x)-ad(y)j

is nondegenerate (det g * 0) if and only if L is semisimple (has no nontrivial solvable ideals). (e) Every simple Lie algebra is semisimple. Α semisimple Lie a l g e b r a decomposes into a direct sum of simple ideals.

(Of course, representations of Lie algebras employ in the image space the product provided by the commutator instead of m a t r i x multiplication; with this single exception, the definitions of section 2 . 4 remain valid). Similar propositions as that of the l a s t theorem can be stated f o r the commutator of vector f i e l d s on manifolds:

T h e o r e m 6.8.2:

( a ) The set SUM) of all vector f i e l d s on a smooth manifold Μ is a Lie a l g e bra with the commutator as product.

6 . 8 . Lie Algebras

(b) In p a r t i c u l a r ,

241

the differential operators

d e f i n e an i n f i n i t e - d i m e n s i o n a l r e a l Lie a l g e b r a on t h e s p a c e of a l l gular

vector

fields

in C = C\{0),

c o m m u t a t i o n r e l a t i o n s of t h e L

called

the

Virasoro algebra.

reThe

are

k =

(J-1}

L.

+J

The m a p p i n g L^ —> k y i e l d s a graduation on t h e V i r a s o r o a l g e b r a .

It i s i n -

t i m a t e l y c o n n e c t e d w i t h Kac-Moody a l g e b r a s and of f u n d a m e n t a l r e l e v a n c e f o r p a r t i c l e p h y s i c s (string theory). We now r e t u r n t o o u r g e n e r a l c o n s i d e r a t i o n s .

The a d j o i n t r e p r e s e n t a t i o n

is

n o t a l w a y s f a i t h f u l , b u t we have t h e i m p o r t a n t

Theorem 6.8.3:

Every f i n i t e - d i m e n s i o n a l Lie a l g e b r a over an a r b i t r a r y f i e l d of istic 0 possesses a faithful representation

character-

whose d i m e n s i o n is a l s o

finite

(theorem of Ado).

N e x t we come t o some t e r m i n o l o g y which is e x a c t l y p a r a l l e l t o t h a t of s e c t i o n 6 . 7 . We d e l i b e r a t e l y choose t h e same f o r m u l a t i o n in o r d e r t o e m p h a s i z e t h i s a s c l e a r l y a s p o s s i b l e . The a n a l o g u e of d e f i n i t i o n 6. 7. 2 is

D e f i n i t i o n 6.8.2:

(a) If U and V a r e s u b a l g e b r a s of some Lie a l g e b r a L, we d e n o t e by [U,V] t h e s u b a l g e b r a g e n e r a t e d by a l l c o m m u t a t o r s [ u , v ] w i t h u € U and ν e V. (b)

T h e lower central series o f

c a l c u l a b l e via L

ι

= L and L

L is t h e s e q u e n c e L , L , L , . . . , k+i

recursively

= [L, L ]. If t h e r e i s a n a t u r a l n u m b e r k k

w i t h L^ = 1, t h e a l g e b r a L is nilpotent. (c) Similarly,

setting L

rived series of L.

0

If

t h e n L i s solvable ( o r

= L and L

k+1

it e v e n t u a l l y soluble).

= [L

k

,L

k)

],

we o b t a i n t h e de-

l e a d s down t o t h e

zero

algebra,

242

6. Complex A n a l y s i s

Obviously,

a l l nilpotent L i e algebras are soluble,

but not vice versa.

We want to pursue the analogy between L i e groups and a l g e b r a s a bit f u r t h e r (cf.

definition

6.7.3):

Definition 6.8.3: ( a ) The s o l v a b l e ideals of a L i e algebra L generate the Jacobson radical of L.

We denote it by J a c ( L ) .

( b ) A L i e a l g e b r a L with J a c ( L ) = 0 is semisimple. ( c ) If L is noncommutative,

i.e.

Z ( L ) * L, and, f u r t h e r m o r e , does not c o n -

tain any ideals excepts f o r 0 and L i t s e l f ,

Hence,

we call L simple.

the Jacobson r a d i c a l is the l a r g e s t soluble ideal in L.

The theorems of Levi and Mal'tsev can also be c a r r i e d over to a l g e b r a s ,

but

only with s l i g h t m o d i f i c a t i o n s and the additional r e s t r i c t i o n char Κ = 0:

Theorem 6.8.4: L e t L be a L i e a l g e b r a over a f i e l d of c h a r a c t e r i s t i c 0. (a)

(Theorem of Levi): cobson r a d i c a l ,

L contains at least

i.e.

a subalgebra

S,

one Levi complement of such that

the

Ja-

L s p l i t s into a

direct

All Levi complements are c o n j u g a t e under

auto-

sum of the f o r m

L

( b ) (Theorem of Mal'tsev): morphisms of

=

Jac(L) ® S

L.

( c ) The f a c t o r a l g e b r a L / J a c ( L ) is semisimple.

We are s l o w l y approaching the central topic of shall be able to w r i t e down the main results, notation.

Lie theory,

but b e f o r e

we

we need a f e w more pieces of

6.8.

243

Lie Algebras

Definition 6.8.4: ( a ) T h e left translation w i t h a n e l e m e n t χ of t h e n - d i m e n s i o n a l L i e g r o u p G is t h e map

L : χ

G

—>

G

y

ι—>

L (y) = x - y X

T h e differential dL p r o v i d e s u s w i t h a n i s o m o r p h i s m b e t w e e n t h e t a n g e n t s p a c e s TG a n d TG . It a l l o w s u s t o c a r r y o v e r an a r b i t r a r y b a s i s y xy of t h e t a n g e n t s p a c e TG a t t h e u n i t e l e m e n t t o TG . i n ι χ (b) A v e c t o r

field

ν ε B(G)

left invariant if

is

the j u s t defined bases or,

is

to say it o t h e r w i s e ,

constant

relative

to

if f o r a l l x , y e G t h e

equation (dL ) v ( y ) X

=

v(L ( y ) ) X

=

v(x-y)

holds.

T h e s p a c e of l e f t i n v a r i a n t v e c t o r f i e l d s on G i s c l o s e d u n d e r f o r m i n g c o m mutators each

and thus

v(l) e TGj

a

a subalgebra well-defined

of

B(G).

vector

Is it

field

also

with

can be f o u n d . This makes t h e f o l l o w i n g c o n s t r u c t i o n

easy to

the

see t h a t

given

initial

for value

possible:

Definition 6.8.5: ( a ) L e t G b e a L i e g r o u p o v e r DC ε {R,C}. T h e associated Lie algebra of G i s the

space

ν e 8(G)

2(G) •

ν is l e f t

invariant

{

of a l l l e f t i n v a r i a n t v e c t o r f i e l d s w i t h t h e c o m m u t a t o r

as product.

It

h a s t h e g r o u n d f i e l d K, a n d i t s d i m e n s i o n e q u a l s t h a t of G. ( b ) A s s u m e L i s a Lie a l g e b r a o v e r Κ e . By A d o ' s t h e o r e m , (k k) b e d L in a s u i t a b l e m a t r i x a l g e b r a Κ ' is the

set

we c a n e m -

. T h e universal Lie group of L

244

6. Complex A n a l y s i s

S(L)

=

< expU)

I λ e L )

g e n e r a t e d by t h e images of a l l L - m a t r i c e s u n d e r t h e e x p o n e n t i a l map.

We a r e now p r e p a r e d t o f o r m u l a t e t h e main p r o p o s i t i o n of t h e w h o l e t h e o r y . The v a r i e t y of i t s a p p l i c a t i o n s is immense.

T h e o r e m 6.8.5:

Let L be a r e a l o r c o m p l e x Lie a l g e b r a . Every Lie g r o u p w i t h £(G) = L is a h o m o m o r p h i c image of i?(L). (This j u s t i f i e s t h e t e r m " u n i v e r s a l Lie g r o u p " ) .

T h e Lie correspondence

G


L = .2(G)

L


G = S(L)

t h u s e s t a b l i s h e d c o n n e c t s t h e o b j e c t s b e a r i n g t h e same n a m e s (e. g. ple, nilpotent,

solvable,

semisim-

. . . Lie g r o u p s and a l g e b r a s ) w i t h each o t h e r .

s u b g r o u p s of G a r e b i j e c t i v e l y a s s o c i a t e d w i t h t h e s u b a l g e b r a s of L, l a r l y f o r n o r m a l s u b g r o u p s and i d e a l s ,

The simi-

etc.

The a u t o m o r p h i s m g r o u p s of G and L a r e c a n o n i c a l l y i s o m o r p h i c , and t h e r a d icals are mutual

counterparts: Rad G
Jac L

relationship

mainly c o n s i s t s

in t h e

reduction

of t h e d i f f i c u l t i e s one e n c o u n t e r s in c l a s s i f i c a t i o n p r o b l e m s of Lie g r o u p s with specified properties.

The c o r r e s p o n d i n g

Lie a l g e b r a s

are

technically

much e a s i e r . The s o - c a l l e d classical Lie groups a r e t h e s p e c i a l l i n e a r , g o n a l o r s y m p l e c t i c ones over t h e r e a l or c o m p l e x n u m b e r s ,

unitary,

ortho-

i. e. t h o s e d e -

s c r i b e d by t h e s y m b o l s SL (C), SL (IR), Sp (C), Sp (R), SU (C) and SO (R). η η 2n 2n η η It is c u s t o m a r y t o d e n o t e t h e a s s o c i a t e d Lie a l g e b r a s by G e r m a n (or G o t h i c ) minuscules,

a s will be a p p a r e n t f r o m t a b l e 6 . 8 . 1 .

Concerning the

theoreti-

6.8.

cal

Lie

Algebras

impact

of

245

the classical

Lie

theory (part D of this book),

algebras

and t h e i r

connection

with

t h e r e a d e r may l i k e t o c o n s u l t t h e

Coxeter

monographs

by H u m p h r e y s [1975] o r J a c o b s o n [1962] o r t h e v o l u m i n o u s o r i g i n a l p a p e r s o f Killing

[1888-1890].

The

latter

source

also

contains

the

classification

the s e m i s i m p l e L i e g r o u p s o v e r C.

Tab.

L i e group

complex:

real:

6.8.1

Lie

algebra

SL ( C ) η

$r co

Sp

sp

2n

(C)

su

SL

sr

Sp

CR)

2n

CR)

SO ( R ) π

n2-l

η

SU C O η

η

dimension

sp

so

2n

η

π

CO

I

CR)

(R)

(2n-1)

Η

CN-1)

n2-l

CR)

2n

η

η

CO

η

- η 2

C2n-1)

Cn-1)

of

246

7. P r o j e c t i v e

Geometry

7. Projective Geometry 7.1. Affine and Projective Coordinates

The h i s t o r i c a l

development towards projective geometry

a t l e n g t h a l r e a d y in s e c t i o n 1 . 2 ; ther

has been

discussed

s o w e s h a l l go in m e d i a s r e s w i t h o u t

fur-

delay.

T h e affine plane o v e r t h e f i e l d Κ is t h e t w o - d i m e n s i o n a l

vector

space Κ ,

w h o s e e l e m e n t s a r e c a l l e d points. A (straight) line g i s t h e s e t of a l l l u t i o n s of a n a l g e b r a i c e q u a t i o n of d e g r e e 1, l i k e u q + u ^

+

so-

= 0.

T h e c o e f f i c i e n t s u 0 , u 1, u2 e Κ a r e t h e Plücker coordinates of g. β All t r i pκ l e s (u . u , u ) a r e p e r m i t t e d , e x c e p t t h o s e w i t h u = u = 0 ; t h e s e w o u l d n o t 0 1 2 1 2 describe

a nontrivial

straight

line

are

not

condition. unique,

On t h e since

other

hand,

multiplication

the of

nonzero f a c t o r λ € Κ merely r e p l a c e s the above polynomial l i n e by a n e q u i v a l e n t •> ^ with λ e Κ

will

one.

The t r a n s i t i o n

be denoted

Plücker

inhomogeneous o r affine coordinates Xj.Xg of

other,

or

m a k e u s e of

we say,

an i l l u s t r a t i n g

l i e s on g" o r "g p a s s e s t h r o u g h P",

a

of

e q u a t i o n of

coordinates

geometric

and so on.

terminology

with

like

"P

T h i s m a n n e r of s p e a k i n g

is

it. t h e r e is

a

are

system

u + u x + u x 0 1 1 2

2

u + u yJ + u y ο 1 1 2 '2

the normalization.

are

a p o i n t Ρ = ( χ , χ ) in

u n i q u e l y d e t e r m i n e d connecting line w h o s e P l ü c k e r c o o r d i n a t e s u o> u ^ u ^

The P l ü c k e r c o o r d i n a t e s uq,

the

(Xu , Au , Xu ) ο 1 2

( d i f f e r e n t ) points Ρ = ( x ^ x ) and Q = ( y ^ y ^ ·

f o u n d by s o l v i n g t h e l i n e a r

a

fixed

Ρ a n d g a r e incident

so w e l l e s t a b l i s h e d t h a t w e n e e d n o t c o m m e n t u p o n F o r e a c h p a i r of

by

rescaling.

t h e p l a n e o b e y t h e l i n e e q u a t i o n of g , each

u(

(u , u , u ) t o o' l ' 2

a s rescaling. H e n c e ,

only d e t e r m i n e d up t o an a r b i t r a r y If t h e

from

coordinates

the

u t

.u2

= 0 = 0

of t h e s t r a i g h t

They s p e c i f y the r a t i o

u : ua>

line g a r e independent

of

w h i c h d e s c r i b e s t h e slope

of g. Two lines g and h w i t h c o o r d i n a t e s u , u , u 0 1 2

a n d ν , ν , ν , r e s p e c t i v e l yJ , ο ι 2 ^

are

247

7 . 1 . A f f i n e and P r o j e c t i v e Coordinates

parallel to each other

(we express this by the symbolic notation g||h),

they have the same slope,

if

i. e.

u : u 1 2

=

ν

: ν 1 2

It makes a principal d i f f e r e n c e whether even

u : u : u 0 1 2 holds or not. are d i s j o i n t .

=

ν

0

: ν

: V 1 2

In the f i r s t case the lines are identical;

tion point in common.

Its coordinates

x

in the second they

they have e x a c t l y one intersec-

If g and h are not p a r a l l e l , j>x2

can

be obtained f r o m the linear

conditions

Whenever t w o s t r a i g h t also

mutually

u + u x + u x 0 1 1 2 2

= 0

v + v x + v x 0 1 1 2 2

= 0

lines h, k are p a r a l l e l

parallel:

to a third one ( g ) ,

g||h and g||k imply h||k.

Parallelism

they

is an

are

equiva-

lence r e l a t i o n . This

and 2

plane IR The

similar

observations

are

very

familiar

to

us;

the

real

affine

of

lines

brings

is the standard model of Euclidean geometry.

distinction

between

parallel

and

nonparallel

along numerous complications and inconveniences.

pairs

Mainly f o r this reason the

t r a n s i t i o n to projective geometry was c a r r i e d out ( f o r a more d e t a i l e d

ac-

count, see section 1.2). The points and lines of the projective plane IP2 Κ over the basic f i e l d DC are 3 the o n e - and two-dimensional subspaces of DC . Incidence of a point Ρ and a line g occurs if and only if Ρ c g. 3 T w o points g e n e r a t e a two-dimensional subspace of DC and thus possess e x T w o points g e n e r a t e a two-dime a c t l y one line connecting them. Since lines are of codimension

1, they

are maximal

3

in DC .

Two

different

by the homomorphism

theorem,

3

lines t h e r e f o r e span the whole of Κ , and, their plane,

intersection

is a subspace of

dimension 2+2-3 = 1: In the

projective

t w o lines a l w a y s have a (unique) point in common. The a f f i n e notion

of p a r a l l e l i s m thus loses its meaning in p r o j e c t i v e g e o m e t r y . A point Ρ is generated (as a subspace) by any nonzero v e c t o r contained in

248

7. P r o j e c t i v e Geometry

it,

f o r i n s t a n c e by χ = (χ , χ , χ ) e Ρ\. I t s c o o r d i n a t e s (which a r e f i x 0 1 2 ed only up t o r e s c a l i n g ) a r e c a l l e d t h e homogeneous o r projective coordinates of

P.

This

leads to the following interpretation

of

the

projective

plane: P2K

P 2 (K)

=

=

(K3\0)/(K\0)

3 As a s u b s p a c e of codimension 1 in Κ , a s t r a i g h t line g can be d e s c r i b e d by a linear equation,

s i m i l a r t o t h e s i t u a t i o n in a f f i n e g e o m e t r y : u x + u x + u x 0

0

1 1

2

2

= 0

It i s s e l f - e v i d e n t t h a t we may c h o o s e t h e s c a l i n g of t h e so d e f i n e d line coordinates a t w i l l ,

b u t t h i s is t h e only f r e e d o m we have.

the a f f i n e coordinates,

now a l l t r i p l e s a r e a d m i s s i b l e ,

In c o n t r a s t

with the sole

to ex-

c e p t i o n of (0, 0, 0). Beyond

this,

it

will

prove

useful

to

interpret

coordinates a s r o w v e c t o r s (= ( 1 , 3 ) - m a t r i c e s ) column v e c t o r s ,

i.e.

the

homogeneous

point

and t h e line c o o r d i n a t e s

as

(3, l ) - m a t r i c e s .

F u r t h e r m o r e , f r o m now on we s h a l l i d e n t i f y p o i n t s and l i n e s in

with the

a s s o c i a t e d coordinate vectors and t h u s speak of t h e "point x '= (XQ, x ^ . x ^ ) " and t h e "line u =

(Uq, u , u )".

The i n v a r i a n c e u n d e r

rescaling has

always

t o be t a k e n i n t o a c c o u n t . I n c i d e n c e of some p o i n t χ and some line u c o n s e q u e n t l y is e x p r e s s e d by t h e linear

condition x-u

=

x u 1 1

=

x u + x u + x u 0 0 1 1 2 2

=

The f i r s t v e r s i o n h a s t o be r e a d as a m a t r i x p r o d u c t , Einstein convention is a p p l i e d ,

which

0 in t h e second

(only in t h e p r e s e n t

e x t e n d e d t o p a i r s of i d e n t i c a l l o w e r i n d i c e s .

chapter

7)

In t h e p a r t i c u l a r c a s e of

the is the

p r o j e c t i v e p l a n e , t h e s u b s c r i p t s r u n f r o m 0 t h r o u g h 2, f o r g e n e r a l projective spaces of dimension η t h e y v a r y b e t w e e n 0 and n. This s i m p l e and e l e g a n t f o r m of t h e incidence r e l a t i o n i n d i c a t e s t h a t t h e p r o j e c t i v e c o o r d i n a t e s a r e highly s u p e r i o r t o t h e a f f i n e ones. T h i s a d v a n t a g e b e c o m e s even m o r e e v i d e n t if we t a k e t h e c o m p l e t e s y m m e t r y b e t w e e n 2

p o i n t s and l i n e s in IP Κ i n t o c o n s i d e r a t i o n , t h e duality principle of Poncelet:

which i s given e x p r e s s i o n t o by

7. 1. A f f i n e and P r o j e c t i v e

Coordinates

249

Theorem 7.1.1: 2 The duality of the p r o j e c t i v e plane IP (K) is the map which associates

with

e v e r yJ point line

(χ , χ , χ ) its dual line * ( x , x , x ) and, r e v e r s e l y , with each 0 1 2 0 1 2 (u , u , u ) i t s dual point (u ,u ,u ). In m a t r i x notation, this simply 0

1

2

0

1

2

corresponds to transposing. Any true

statement

concerning

points and s t r a i g h t

plane and incidence between them yields we dualize

a likewise

lines

in the

correct

projective

proposition

if

it.

Proof: Incidence of the point χ and the line u means that x-u = 0 holds. is ^ - ' x

The Poncelet p r i n c i p l e reduces the amount of most",

The dual

= 0 and thus equivalent.

because

certain

self-dual

interesting). 2 The a f f i n e plane Κ is naturally

labour by almost 50 % ( " a l -

configurations

and theorems

exist;

they

are e s p e c i a l l y

2 embedded in Ρ Κ.

In o r d e r

to v e r i f y

this

assertion,

we only have to map the a f f i n e point (x , x ) to ( l , x ,x ) e p \ . 1 2 1 2 This mapping is c l e a r l y i n j e c t i v e . Its image consists of all points of the p r o j e c t i v e plane whose 0 t h coordinate does not vanish. They are c a l l e d pro2

per or finite points;

the remaining χ 6 IP K, namely those with x q = 0,

are

the improper, infinite or ideal points, sometimes also r e f e r r e d to as the points at infinity. The a f f i n e s t r a i g h t ordinate

vectors

lines correspond with those p r o j e c t i v e lines whose

co-

,u ,u ) are l i n e a r l y independent of l ( l , 0 , 0 ) . We d e 2 note them as proper lines. In Ρ Κ, only one improper line e x i s t s . This line at infinity is

l(u

t(l,0,0).

The t r a n s i t i o n f r o m the a f f i n e to the p r o j e c t i v e point of

view can be d e -

scribed as f o l l o w s : If we are given the a f f i n e plane Κ , we call its points and lines p r o p e r ; the incidence r e l a t i o n s between these o b j e c t s are e x a c t l y 2 the same as in Κ . To each set of

parallel

with a l l of them,

lines we associate an ideal point which is incident

but with no other proper line.

Finally,

w e complete

system thus constructed by adding a single line at i n f i n i t y .

The l a t t e r

the has

250

7. P r o j e c t i v e Geometry

to be incident

with all

ideal,

but no f i n i t e points.

All

points

and

lines

together f o r m the (obviously unique) p r o j e c t i v e plane over the f i e l d K. The reversal

is even simpler.

We delete an arbitrary

incident with it f r o m the p r o j e c t i v e plane

line and all

points

What remains is the a f f i n e

plane. The points Ρ ,P , Ρ , . . . are said to be collinear if a line exists which 1 2 3 passes through all P ^ analogously we call the lines g , g 2> g ,... copunctal if there is a point lying on each of them. All points on a line f o r m a point row; the t o t a l i t y

of

all

straight

lines

through a f i x e d point is a line bundle. The connecting line of two points A, Β has the line coordinates [AB]

=

' ( Α Β -Α Β , Α Β - Α Β , Α Β - Α Β ) 1 2

2 1 2 0

0 2

0 1

1 0

and all its points are of the f o r m α A + β Β with (α, β) e K 2 \{(0,0)>.

The

intersection point of the line a, b is [ab]

=

(a b - a b ,a b - a b ,a b - a b ) 1 2

2

1 2 0

0 2

0 1

1 0

and the lines passing through it have a representation α a + β b with the same r e s t r i c t i o n on α and β as before. A triangle Δ = Δ(Α, Β, C; a, b, c)

consists

of

three

points

A,B,C

and

three

lines a, b, c, such that A is incident with b and c, Β with c and a, C with a and b, while there is no incidence between A and a, Β and b, or C and c. It is clear that a triangle is determined by either

its vertices A,B,C

or

its sides a, b, c. T h e r e f o r e we sometimes p r e f e r to w r i t e simply Δ = A ( A , B , C ) or Δ = A ( a , b , c ) .

The vertices or sides can be chosen with the single

striction that they must not be collinear or copunctal, The r o l e

of

the coordinate axes in a f f i n e geometry

re-

respectively. is in the

projective

plane taken over by the coordinate or fundamental triangle. It has the v e r tices

(1,0,0);(0,1,0);(0,0,1)

and

the

sides

t(l,

0, 0); 1 ( 0 , 1 , 0); l ( 0 , 0,1).

shall see in the next section how to construct point and line

We

coordinates

by using it. If we w r i t e down the coordinate vectors of three points A,B,C one underneath the other, we get the vertex matrix

7 . 1 . A f f i n e and P r o j e c t i v e

251

Coordinates

E(A, B,C)

In p e r f e c t a n a l o g y ,

w e d e f i n e f o r t h r e e l i n e s a , b , c by j u x t a p o s i t i o n of

the

c o o r d i n a t e v e c t o r s t h e side matrix

S(a,b,c)

T h e f r e e d o m t o r e s c a l e i s e x p r e s s e d by t h e f a c t t h a t i t i s a l l o w e d t o tiply E(A,B,C) f r o m the l e f t and S ( a , b , c ) vertible diagonal The c o l l i n e a r i t y

mul-

f r o m t h e r i g h t by a r b i t r a r y

in-

matrices. of

three

p o i n t s A , B , C in t h e p r o j e c t i v e

l e n t t o t h e v a n i s h i n g of d e t E ( A , B , C ) ;

plane is

t h e t h r e e l i n e s a, b, c a r e

equiva-

copunctal

if a n d o n l y if d e t S ( a , b , c ) is z e r o . W h e n e v e r E(A, B, C)· S ( a , b, c) is a r e g u l a r vertices valid.

a n d a , b, c t h e

s i d e s of

diagonal matrix,

a triangle;

the reverse

A, B, C f o r m statement

is

This gives us a simple means t o c o n s t r u c t t h e sides when t h e

ces a r e given or vice versa.

also verti-

We o n l y n e e d t o c h o o s e t h e m a t r i c e s E ( A , B , C)

and S ( a , b , c ) inverse to each o t h e r .

The c o o r d i n a t e s c a n t h e n i m m e d i a t e l y be

r e a d o f f . I n s t e a d of t h e r e c i p r o c a l m a t r i x , adjoint,

the

we could use, f o r i n s t a n c e ,

w h i c h d i f f e r s f r o m i t o n l y by a n u m e r i c a l f a c t o r

T h e f o l l o w i n g theorem of Desargues is of e m i n e n t

the

(determinant).

importance to

projective

geometry:

Theorem 7.1.2: L e t Δ = Δ(Α, Β, C; a, b, c) and Δ' = Δ( A ' , Β ' , C ' ; a ' , b ' , c ' ) h a v i n g no V e r t e x e s o r s i d e s in c o m m o n . below are

triangles

in

stated

equivalent:

( a ) The l i n e s [AA'], triangles are (b) The

be t w o

Then t h e t w o a s s e r t i o n s

intersection

a n d Δ' a r e

[BB'],

ICC'J c o n n e c t i n g c o r r e s p o n d i n g v e r t i c e s of

both

copunctal. points

collinear.

[aa'l,

[bb'l,

[cc']

of

associated

sides

in

Δ

252

7. P r o j e c t i v e Geometry

Proof: We a r e allowed to choose the normalization such t h a t each of the two v e r t e x m a t r i c e s Ε = E(A, Β, C) and E ' = E ( A ' , B ' , C ' ) S = S ( a , b , c ) or S ' = S ( a ' , b ' , c ' ) , E-S

is inverse to i t s side

matrix

respectively, =

E'-S'

=

1

which is possible by the preliminary remark. The p r o p o s i t i o n s

(a)

and

(b)

are

mutually

dual;

it

will

thus

suffice

to

prove the implication (a) =» (b). We f i r s t consider the special c a s e in which the common point Ρ of the t h r e e connecting lines [AA' ], [BB' ], [ C C ] coincides with one of the s i x For symmetry r e a s o n s , If t h i s condition

vertices.

we may assume Ρ = C.

holds,

lie on the

straight

line [PA] = [CA] = b. But A' is incident with the side b ' as well;

the points P,

thus we

have A' = [bb' ]. S i m i l a r l y , is t h e r e f o r e reason,

[A'B'J = c ' .

B' = [aa'].

A,

and A'

all

The line connecting [ a a ' l and [ b b ' ]

It evidently also passes through [ c c ' ] .

For

we may c o n c e n t r a t e on the generic case t h a t Ρ is d i f f e r e n t

this from

each o f the Vertexes A, Β, C; a, b . c . As we saw b e f o r e , t h i s implies the v a l i d ity of r e l a t i o n s of the form Ρ

=

a A + α' A'

=

β Β + β' Β'

=

y C + y ' C'

with a p p r o p r i a t e nonzero c o e f f i c i e n t s α, α ' ; β, β ' ; y, y ' in the ground f i e l d K. I f we w r i t e t h e s e t h r e e r e p r e s e n t a t i o n s as the rows of a ( 3 , 3 ) - m a t r i x ,

we

get E(P,P,P)

=

D-E + D ' - E '

with D = Diagta, ß, y) and D' = D i a g l a ' , β', y ' ) . The rank of E ( P , P , P ) is equal to 1, as is the rank o f DE + D ' E ' . If,

on the other hand,

one can find i n v e r t i b l e diagonal m a t r i c e s D, D'

with

r a n k ( D E + D ' E ' ) = 1, then all rows of DE + D ' E ' are proportional and d e t e r mine a common point on [AA'],

[ B B ' ] and [CC'].

P r o p o s i t i o n (a) thus is tantamount to the e x i s t e n c e of r e g u l a r diagonal m a t r i c e s D , D ' f o r which Rank(DE+D'E')

=

1

7 . 1 . A f f i n e and P r o j e c t i v e C o o r d i n a t e s

is t r u e .

A s s e r t i o n (b) is dual t o t h i s ,

253

and we c o n c l u d e w i t h o u t

difficulty

t h a t it e q u i v a l e n t t o Rank(SM+S'M')

=

1

f o r some d i a g o n a l m a t r i c e s Μ and M' w i t h n o n z e r o d e t e r m i n a n t s . S D"1 + S' D'" 1 This immediately

=

E - 1 D"1 + E'

1

D'" 1

implies the theorem,

=

Moreover,

E"1 D"1 (D'E'+DE) E ' " 1 D'" 1

since t h e m u l t i p l i c a t i o n

by

regular

m a t r i c e s d o e s not c h a n g e t h e r a n k .

We d e l i b e r a t e l y r e s t r i c t e d our i n v e s t i g a t i o n t o t h e c o n s t r u c t i o n of t h e f i n e and p r o j e c t i v e p l a n e s over that,

a given

basic

f i e l d K.

af-

The a d v a n t a g e

in c o r r e s p o n d e n c e w i t h t h e t o p i c of t h e p r e s e n t book,

is

we have f r o m

t h e o u t s e t c o o r d i n a t e s a t our d i s p o s a l . The c o m m u t a t i v i t y of K, h o w e v e r , not essential;

w i t h a pinch of s a l t ,

all our previous findings remain

is

valid

if Κ is a skew f i e l d . In t h i s m o r e g e n e r a l c a s e , we only have t o be c a r e f u l w i t h t h e o r d e r of t h e f a c t o r s in p r o d u c t s and, e q u a t i o n s which c o n t a i n d e t e r m i n a n t s

in p a r t i c u l a r ,

modify the

slightly.

As o u r a p p l i c a t i o n s w i l l mainly c o n c e n t r a t e on t h e f i e l d s Κ = 18 o r t h e m o r e i m p o r t a n t c a s e Κ = €, we r e f r a i n f r o m d i s c u s s i n g t h e i n d i c a t e d tion explicitly. tigated.

generaliza-

In g e o m e t r y , much more a b s t r a c t t y p e s of p l a n e s a r e i n v e s -

It seems a p p r o p r i a t e t o say a t l e a s t a f e w w o r d s a b o u t t h e m .

One s t a r t s o u t f r o m a d e f i n i t i o n which is not t i e d t o t h e e x i s t e n c e of g r o u n d (skew) f i e l d . I n s t e a d , void

sets

f

and

a

a projective plane is b u i l t up f r o m t w o n o n -

Ϊ. ("points"

and

"lines") b e t w e e n

which

a

relation

("incidence") is d e f i n e d , such t h a t t h e f o l l o w i n g f o u r a x i o m s h o l d s : (1) F o r each t w o p o i n t s P, Q e f t h e r e is a uniquely d e t e r m i n e d connecting line [PQ] e ΐ which is i n c i d e n t w i t h both of t h e m . (2) F o r each t w o l i n e s g, h e if t h e r e is a uniquely d e t e r m i n e d intersection point [gh] e Τ which is i n c i d e n t w i t h b o t h of t h e m . (3) The p l a n e c o n t a i n s a t l e a s t f o u r p o i n t s , no t h r e e of which a r e c o l linear. (4) The p l a n e c o n t a i n s a t l e a s t f o u r l i n e s , punctal.

no t h r e e of which a r e

co-

254

7. P r o j e c t i v e Geometry

The f o r m u l a t i o n chosen h e r e l e a d s d i r e c t l y back t o P o n c e l e t ' s · p r i n c i p l e of duality,

b e c a u s e t h e c o n d i t i o n s (1) and (2) a r e dual t o each o t h e r ,

and t h e

couple (3) and (4) h a s t h e same p r o p e r t y . With r e g a r d t o economy,

t h i s s y s t e m of a x i o m s is n o t o p t i m i z e d ,

however.

The f o u r p o s t u l a t e s a r e not i n d e p e n d e n t . We could, f o r e x a m p l e , o m i t one of the last

two,

since

(3) is a c o n s e q u e n c e

of

(1), (2), (4),

while dually

(4)

can be deduced f r o m (1), (2), (3). A f f i n e p l a n e s a r e c o n s t r u c t e d in t h e same way as we did b e f o r e , namely by d e c l a r i n g one line a s " i m p r o p e r " and d e l e t i n g it f r o m t h e p r o j e c t i v e p l a n e . It is t h e r e f o r e s u p e r f l u o u s t o r e p e a t o u r e a r l i e r

discussion.

As a l a s t r e m a r k t o t h i s t o p i c , we m e n t i o n w i t h o u t p r o o f t h a t a p r o j e c t i v e p l a n e is i s o m o r p h i c t o some IP (DO over a s u i t a b l e skew f i e l d Κ if and only if i s D e s a r g u i a n . T h i s of c o u r s e m e a n s t h a t it obeys t h e o r e m 7. 1. 2. We do n o t i n t e n d t o p u r s u e t h i s any f u r t h e r and r e f e r t h e i n t e r e s t e d to the l i t e r a t u r e ,

a s , e. g . ,

P i c k e r t [1955],

reader

7.2.

Projectivities

255

7.2. Projectivities

Many a g e o m e t r i c p r o b l e m c a n be s i m p l i f i e d c o n s i d e r a b l y by a p r o p e r of c o o r d i n a t e s . are admissible, Let us give an

choice

Which r e f e r e n c e f r a m e s a n d w h i c h t r a n s i t i o n s b e t w e e n d e p e n d s on t h e s t r u c t u r e one w a n t s t o k e e p

them

invariant.

example: 2

T h e E u c l i d e a n p l a n e IR i s e n d o w e d w i t h a n a t u r a l m e t r i c w h i c h a l l o w s u s t o d e f i n e t h e distance of t w o p o i n t s a n d angles b e t w e e n t w o s t r a i g h t l i n e s .

If

w e d e m a n d t h e s e m e a s u r e s t o be i n v a r i a n t d u r i n g t h e t r a n s f o r m a t i o n , w e a r e r e s t r i c t e d t o m a p p i n g s of t h e f o r m χ

ι—>

χ D + a

2 (2 2) translation vector a e IR and a m a t r i x D 6 IR ' which

with an a r b i t r a r y

i s s u b j e c t t o t h e s u p p l e m e n t a r y c o n d i t i o n D l D = 1 (orthogonality). Such t r a n s f o r m a t i o n s a r e called Euclidean; contains

the

set

of

t h e y f o r m a g r o u p g(IR ),

translations (D = 1) a s a n o r m a l

all

which

subgroup.

It

2

i s o m o r p h i c w i t h IR , w h i l e t h e f a c t o r g r o u p c o r r e s p o n d s t o t h e g r o u p of orthogonal (2,2)-matrices,

is

real

2

i.e.

GO(IR ):

g(IR2)

£

IRZ : G0(IR2) 2

(The

extension

splits;

a

complement

of

IR

is

provided

by

the

Euclidean

t r a n s f o r m a t i o n s with a = 0). An s eor be lt ya i nreedq uifi r e wet h ado c o n sael rt ve er nd ,a t i bv ue t i m t tnhoet iwnacni dt e nt hc ee lr ee nl agtt ihosn a nind IRa2n gilse s i nt ov a rbe i2 a n t u n d e r t h e t r a n s f o r m a t i o n (in o t h e r w o r d s : w e i n t e r p r e t IR a s t h e a f f i n e real plane). includes

T h e g r o u p of p e r m i t t e d c h a r t c h a n g e s t h e n b e c o m e s l a r g e r

and

all χ (2 2)

w i t h a r e g u l a r m a t r i x Μ e IR '

ι—»

, or,

χ Μ+ a

more precisely,

2

Μ e GL(IR ). 2

T h e l a s t c o n s i d e r a t i o n c a n be e x t e n d e d i m m e d i a t e l y t o a n y a f f i n e p l a n e Κ . 2

An affine transformation of Κ

2

i s s i m p l y a b i j e c t i o n of Κ

onto

itself

by

256

7. P r o j e c t i v e Geometry

2 which χ e Κ g o e s o v e r

to

χ

where a e Κ

2

2 and Μ e GL(K ).

*

=

χ Μ+ a

The set of these maps is the affine group

A{K2)

K 2 : GLCK2)

st

2 E s p e c i a l l y f o r the j u s t discussed case Κ = IR, we f i n d that ) is a m a n i 2 f o l d of dimension 6, w h i l e the Euclidean group £(IR ) is only 5 - d i m e n s i o n a l . The r e n u n c i a t i o n of

2

length and angular measurements in IR

is a l s o

combined

w i t h a c o n s i d e r a b l e gain of f r e e d o m in the choice of c o o r d i n a t e s . This even 2 becomes m o r e e v i d e n t when w e i n v e s t i g a t e the p r o j e c t i v e plane Ρ (Κ). As w e saw in s e c t i o n 7. 1, the point r o w termined

by

a

linear

condition;

incident w i t h

therefore

in

the

a fixed

present

p o l y n o m i a l t r a n s f o r m a t i o n s of d e g r e e 1 a r e admissible.

line is

situation,

Beyond t h i s ,

It i m p l i e s t h a t t w o

ar

same

coordinate

vectors,

describing

the

mapped t o v e c t o r s which a r e l i k e w i s e p r o p o r t i o n a l stricts

the p o s s i b i l i t i e s

for

point,

have

t o each o t h e r .

the point coordinates of

the

image

only

we also

have t o t a k e the s c a l i n g i n v a r i a n c e into account. dependent

de-

lineto

This to

be re-

linear

f unctions: χ

w h e r e Κ is an a r b i t r a r y w i t h det Κ * 0,

or,

ι—>

regular

*

χ

= χ Κ

(3,3)-matrix

more c o n c i s e l y ,

w i t h e n t r i e s in Κ:

Κ e GL (Κ). 3

( 3 3)

Κ 6 IK

'

2

On account of the s y m m e t r y between points and lines in IP Κ imposed by d u a l ity,

the

line coordinates a r e s u b j e c t to an analogous t r a n s f o r m a t i o n ,

for

example, u

Here L e GL 3 (K), cidence

condition *

equation χ

but this m a t r i x χ u = 0 for

*

ι—»

u

= L u

is not independent of

a point

χ and a line

K.

Instead,

the

u must have the

indual

* u

= 0 as a consequence and v i c e v e r s a . From χ u = 0 we t h e r e -

f o r e deduce 0

=

*

x u

*

= x K L u

7.2. P r o j e c t 1vi t i e s

This is only true if

257

Κ and L are,

up to a scalar f a c t o r ,

inverse to

each

other:

Κ L

=

λ·1

with λ e K* = DC\{0>. On the other hand, we may r e s c a l e the coordinates.

This permits us to

im-

pose the n o r m a l i z a t i o n λ = 1, which w i l l be done f r o m now on. We conclude:

Definition 7.2.1:

2

The projectivity of IP (K) given by the regular in Κ is the t r a n s f o r m a t i o n of

(3,3)-matrix

Κ with

the point and line coordinates

entries

according

to

the rules χ

ι—>

χ

*

= χ Κ

* u

ι—»

u

-1 = Κ

u

A l l p r o j e c t i v i t i e s f o r m a group isomorphic with

GL (K)/Z(GL (HO) 3

=

PGL CK)

3

3

the projective group on Ρ 2DC. The last a s s e r t i o n r e f l e c t s the f a c t that the scalar matrices, ers, map a l l v e c t o r s to multiples and consequently 2 the sets of points or lines in IP OC.

and no o t h -

induce the identity

The group PGL3(D0 of pro j e c t i v i t i e s is 8-dimensional and thus even

on

strictly

l a r g e r than the a f f i n e group. Aside f r o m the i n t e r p r e t a t i o n chart t r a n s i t i o n s ,

of

a f f i n e and p r o j e c t i v e t r a n s f o r m a t i o n s

as

which we p r e f e r here, we could equally w e l l have viewed

the maps in question as automorphisms of IK

and

If we want to do so, however, we have to observe that not every t r a n s f o r m a tion which respects the incidence r e l a t i o n in the plane is contained group

*i(K ) or

PGL (HO.

For

example,

for

certain

ground

fields

in the

(like

C:

258

7. P r o j e c t i v e Geometry

complex conjugation) there might exist additional symmetries produced by nontrivial field automorphisms of DC. Such a τ € Aut OC simply has to be a p plied to all coordinates of points and lines. If we are only interested in the incidence structure,

but not in the c a t e -

gorization of the geometric o b j e c t s as points or lines, we can also enlarge the automorphism group by the duality operation,

etc.

The term "automor-

phism of an a f f i n e or projective plane" f o r this reason requires a precise definition. We are not going to discuss these problems to any extent and leave the subj e c t as it stands. In what follows, the only automorphisms we need are the ( a f f i n i t i e s and) p r o j e c t i v i t i e s .

2

Having fixed a r e f e r e n c e frame, in the a f f i n e plane Κ the point Ο = ( 0 , 0 ) , called the origin of the coordinates, is distinguished, together with two s t r a i g h t lines a j and a.^, the axes, which intersect in 0. The Plilcker c o o r dinates of a and a are ( 0 , 0 , 1 ) and ( 0 , 1 , 0 ) : the associated point rows 1 2 contain all ( x ^ O ) and (0, x^l,

respectively.

The configuration j u s t described is c h a r a c t e r i s t i c f o r the r e f e r e n c e system in use and is known as its coordinate cross. The transition to a new set of coordinates is simply e f f e2c t e d by the definition of another pair of axes. In the p r o j e c t i v e plane Ρ DC, this construction is replaced by the fundamental triangle introduced in section 7.1, points

(1, 0, 0); ( 0 , 1 , 0); (0, 0 , 1 )

of

whose vertices are situated at the

our coordinate

system,

while the

sides

are the lines t ( l , 0. 0); ι ( 0 , 1 , 0); *·(0, 0 , 1 ) .

2 It is obtained from the basic cross in Κ by adding the improper line as the third side,

together with the three points of intersection.

the situation in the a f f i n e plane,

Similar to

a projectivity is tantamount to

select-

ing an ( a r b i t r a r y ) triangle as fundamental. The p r o j e c t i v e group t h e r e f o r e a c t s s t r i c t l y transitive on the set of all 2 2 t r i a n g l e s in IP (DC), j u s t as the affine group does on the crosses in DC . This implies the relation i4(DC2)


χ

= χ Μ + a

2 where Μ e GL (DC) and a 6 Κ amounts to the projectivity with the matrix 1 -

0 0

in point coordinates;

a

a 1 2 Μ Μ 11 12 Μ Μ 21 22

the corresponding (inverse) matrix for the line coor-

dinates is 1

-b

0

Ν 11 Ν 21

0

-b

1

2 Ν 12 Ν 22

-1

and

Ν = Μ

That all

matrices of the form 1 * * ' 0 * « 0 » * in GL3

τ

τ

Α

β+δτ = α,+γτ

The a r i t h m e t i c of the point at i n f i n i t y (oo) hasto be obtained f r o m the proj e c t i v e version:

If the denominator in the last f o r m u l a vanishes,

r a t o r is a l w a y s nonzero;

in this case we have τ

the nume-

= oo. The image of the ide-

al point is in the same way found f r o m

(0, 1) · A

=

(y,6)

as α

CO

Since

it

is permitted

to

rescale

δ — 7

=

the homogeneous

coordinates

of

the

ele-

ments of p ' k ,

t w o m a t r i c e s which d i f f e r only by a nonzero constant f a c t o r 2 λ e K, give r i s e to the same t r a n s f o r m a t i o n of the subspaces of Κ , i . e . , the same permutation of the points in P I . The set of all projectivities of the line IP1(K) t h e r e f o r e is a f a c t o r of GL 2 (K),

group

namely GL (DC) / K* 2

=

GL (Κ) / Z(GL ( Κ ) ) 2

=

2

PGL (DC) 2

the projective group. This p a r a l l e l s the t h e o r y of the f o r e g o i n g section;

the most important

pro-

p e r t y of PGL (K) which has no equivalent in higher dimensions is

Theorem 7.3.1: The

group

PGL 2 (K)

of

projectivities

is

strictly

3-fold

transitive

on

the

line p V k ) .

Proof: We f i r s t

show that every t r i p l e

a,b,c

of

pairwise

different

is c o n j u g a t e under the action of P G L ^ K ) to a p a r t i c u l a r one,

points

in p'lK

say oo, 0, 1 (in

this o r d e r ) . If a 6 K, we apply the t r a n s f o r m a t i o n τ ty.

We are

therefore

a l l o w e d to

ι-* ( τ - a )

assume without

1

which moves a to i n f i n i restriction

of

generality

262

that τ

7. P r o j e c t i v e Geometry

a =

In

the

next

step,

we

reduce

the

ι-> τ - b t o t h e s p e c i a l c a s e a = oo, b = 0.

field element p r o o f of t h e

other

than

zero,

and

task

by

the

translation

The t h i r d p o i n t , c, t h e n i s a -1 by c completes the

a multiplication

3-transitivity.

It r e m a i n s t o s h o w t h a t e v e r y

invertible α

β

7

δ

in GL 2 (K) w h i c h h a s oo, 0 , 1 a s f i x e d p o i n t s i s a s c a l a r m a t r i x and c o r r e s p o n d s t o t h e i d e n t i t y in t h e f a c t o r g r o u p P G L ^ K ) .

(0,1) A

implying γ = 0, while 0 = 0 namely Diagta, δ),

(r,3)

=

therefore

The i n v a r i a n c e

of

(0,*)

l e a d s t o β = 0. T h u s ,

A is a diagonal

a n d f r o m 1* = 1 we d e d u c e α = δ,

matrix,

a s s t a t e d in t h e

theo-

rem.

Doubly

transitive

groups

are

not

very

frequent;

threefold

transitivity

an e x t r e m e l y r a r e o c c u r r e n c e .

More i n f o r m a t i o n c o n c e r n i n g m u l t i p l y

tive permutation

be p r o v i d e d

groups

will

in s e c t i o n

20.6

is

transiHathieu

on t h e

groups. T h e o r e m 7 . 3 . 1 h a s q u i t e a n u m b e r of c o n s e q u e n c e s w h i c h a r e of f u n d a m e n t a l importance. prescribed

While w e c a n a l w a y s f i n d a p r o j e c t i v i t y of P1K w i t h images

of

three

points,

this

is

impossible

for

arbitrarily

four

or

more

points. In o r d e r

to give an i l l u s t r a t i o n ,

a , b, c, d 6 Ku{to>.

The

first

we s e l e c t f o u r m u t u a l l y d i f f e r e n t

three

of

them

fix

a

unique

points

projectivity

σ e PGL 2 (K) w h i c h t r a n s p o r t s t h e m in t u r n t o oo, 0, 1. T h e n t h e i m a g e d0^ of d is

determined

as

well.

It

only

depends

on t h e

write

CR(a, b, c , d)

In p a r t i c u l a r ,

=

d0"

given

points

a, b , c , d .

We

7.3. The Cross Ratio

263

CR(co, 0,1, d)

=

d

f o r all d € K\{0,1}. "CR" is an abbreviation of "cross ratio", a term whose meaning w i l l become apparent soon. By its definition,

CR(a,b, c , d ) is an invariant,

i.e.

it does

not change if we subject the points a , b , c , d to any p r o j e c t i v e t r a n s f o r m a tion. Up to now, the cross ratio is only defined f o r the case that no two of the arguments coincide.

Its value is also a point on the p r o j e c t i v e line P 1 (K).

The direct calculation of CR(a, b,c, d) does not present any d i f f i c u l t y .

Let

us assume f o r the moment that a , b , c , d are all contained in K. The p r o j e c tive transformation which brings the quadruple (a,b, c , d ) in the correct o r der to (co, 0, 1, * ) is represented,

among others, by the matrix a(c-b)

b(c-a)

Μ = b-c

a-c

and turns out to be Μ_

(c-a)b+(a-c)T (c-b)a+tb-c)r

as can be seen by inserting τ = a, b.c. We thus obtain the cross ratio if we put τ = d, CR(a,b,c,d)

=

(c-a)b+(a-c)d (c-b)a+(b-c)d

(a-c)(b-d) (a-d)(b-c)

In f o r m e r times, one p r e f e r r e d to write the last expression as CR(a, b, c, d) This alternative

=

-c b-c a-d ' b-d

has its origin in the theory of proportions and

explains

the name "cross ratio". The formula f o r C R ( a , b , c , d) requires a f e w supplementary conventions in the case that one of the points lies at infinity. stead of τ

ι-» τ

K

the transformation τ-b c-b

If,

for

instance,

a = co,

in-

264

7. P r o j e c t i v e Geometry

s u f f i c e s , f r o m which we deduce CR(n>,b,c, d) = ( b - d ) / ( b - c ) . The o t h e r s p e c i a lizations are similar. Furthermore,

it

is a b s o l u t e l y

evident

that

the cross

ratio

t h e only projective invariant of f o u r p o i n t s in p'tK).

is

essentially

This is t o say

that

a l l o t h e r i n v a r i a n t s can be e x p r e s s e d in t e r m s of CR. Each of t h e a r g u m e n t s a , b , c , d can be r e g a i n e d f r o m t h e o t h e r t h r e e and C R ( a , b , c , d). We s u m m a r i z e t h e p r o p e r t i e s f o u n d up t o now:

Theorem 7.3.2:

(a) The c r o s s r a t i o of a, b, c, d e IP1 CK) is c a l c u l a t e d

w i t h t h e h e l p of

the

It is d i r e c t l y a p p l i c a b l e if none of t h e a r g u m e n t s is equal t o t h e

in-

f ormula ,

f i n i t e point.

Otherwise,

(a-c)(b-d) (a-d)(b-c)

t h e f o l l o w i n g r e l a t i o n s hold: CR(a>, b, c, d)

b-d b-c

CR(a, oo, c, d)

a-c a-d

CR(a, b, oo, d)

b-d a-d

CR(a, b, c, to)

a-c tPi

They a r e f o r m a l l y d e r i v e d f r o m t h e g e n e r i c ones by o m i t t i n g t h e

two

f a c t o r s c o n t a i n i n g "co" o r f r o m t h e s u b s t i t u t i o n r u l e oo-x oo-y

_ y-oc

if x , y e Κ and t h e e q u a t i o n

a s long a s χ * 0. The same a g r e e m e n t a l s o a l l o w s t o d e f i n e t h e r a t i o w h e n e v e r only t h r e e of t h e f o u r p o i n t s a r e d i s t i n c t , (b) The c r o s s r a t i o is i n v a r i a n t u n d e r p r o j e c t i v i t i e s :

cross

7 . 3 . The C r o s s R a t i o

265

CRta", b " , c11, d " )

=

CR(a,b,c,d)

f o r all π ε PGL (DO. 2

( c ) Two o r d e r e d q u a d r u p l e s ( a , b , c , d ) and ( a ' , b ' , c ' , d ' ) f r o m Ρ (Κ) a r e e q u i valent r e l a t i v e to the group PGL^OO if and only if CR(a,b,c,d)

=

CR(a', b ' , c ' , d ' )

holds.

We now r e t u r n to the p r o j e c t i v e plane.

Every s t r a i g h t

2 line g in IP (DC) is

c a n o n i c a l l y isomorphic to p'flK), a s can be e a s i l y d e m o n s t r a t e d d i r e c t l y in homogeneous

coordinates

(χ , χ , χ ) of

the

points

χ incident

with g.

At

l e a s t one of the t h r e e components is redundant when we r e s t r i c t to g; if we omit it, we a r e l e f t with p r o j e c t i v e c o o r d i n a t e s on g = P1K. The c r o s s r a t i o n of f o u r points on g is then defined a s well.

It does not

depend on the r e f e r e n c e f r a m e , b e c a u s e a c h a r t change is e f f e c t e d by a l i n e a r c o o r d i n a t e t r a n s f o r m a t i o n which induces a p r o j e c t i v i t y on g. The i n v a r iance g r o u p of g in PGL^tDC) c o n s i s t s of the maps which have (the line c o o r dinates of) g as fixed vectors.

They all t o g e t h e r f o r m a subgroup

isomor-

phic with PGL 2(DC). At the s a m e time we have c o n s t r u c t e d the c r o s s r a t i o n of f o u r p o i n t s in the plane.

Dually,

collinear

we a l s o have the p o s s i b i l i t y to i n t r o d u c e the

c r o s s r a t i o of f o u r copunctal lines g , g , g , g . This is done by choosing a line h which is not contained in the same bundle a s the given l i n e s .

The

c r o s s r a t i o of the g^ then is understood a s the a n a l o g o u s quantity of

the

i n t e r s e c t i o n points with h, CR(g1,g2,g3,gt) CR(a,b, c,d)

depends

on

the

=

CR([hgiJ,[hg2],[hg3],[hg4])

order

of

its

arguments.

Nevertheless, 24

4! = 24 p e r m u t a t i o n s of the 4 v a r i a b l e s yield only at most — = 6 4

the

different

v a l u e s since the c r o s s r a t i o does not change if we apply an element of the f o u r group

266

7. Projective Geometry

V4

=

I 1, ( a b M c d ) , ( a c ) ( b d ) , ( a d ) ( b c )

J

as can be seen by d i r e c t evaluation of the r e l e v a n t expression.

In g e n e r a l ,

the 6 remaining quotients are mutually d i f f e r e n t . We compile them here e x plicitly

f o r the convenience

of

the r e a d e r .

The cross r a t i o

of

a, b, c, d is

denoted by λ. Then we f i n d

CR(a,b, c , d )

=

CR(b, a, d, c )

=

CR(c, d, a, b )

=

CR(d,c,, b, a )

CR(a, b, d, c )

=

CR(b, a, c , d )

=

CR(c, d, b , a )

=

CR(d, c,, a, b )

CR(a, c, b, d )

=

CR(b,d, a, c )

=

CR(c,a,d,b)

=

CR(d,b,, c, a )

CR(a,c, d , b )

=

CR(b, d, c, a )

=

CR(c,a,b,d)

=

CR(d,b,, a, c )

CR(a, d, b, c )

=

CR(b,c, a, d )

=

CR(c,b,d,a)

=

CR(d, a,, c, b )

CR(a,d, c, b )

=

CRCb.c, d, a )

=

CR(c, b, a, d )

=

CR(d, a,, b , c )

Though,

the cross r a t i o

exceptional

λ·

1_λ·

b

are usually

configurations

projective ( 1 ) If

1 λ =

1-λ 1 1-λ λ-1 λ λ

Τ=λ· V " ' λ^Τ

different,

which

for

deserve

our

}

certain

special

attention

λ we

because

obtain

of

their

invariance:

λ e {oo, 0 , 1 } ,

each of

these three

we speak of a singular quadruple.

values occurs

(2) Similarly, again

if

λ attains one of all

of

them

are

twice.

In this

case

The condition is equivalent with the

coincidence of t w o of the f o u r points a, b , c , d ;

ther,

λ

as stated above, the six values

{

of

=

hence the name.

the numbers - 1 , 2 , i found

twice.

Then

which the

belong

set

toge-

{ a , b , c , d)

is

c a l l e d harmonic. Whenever the points are ordered such that even C R ( a , b , c , d) = -1, we say that the quadruple ( a , b , c , d ) is harmonic. the fourth harmonic point to (3) Finally, ratic

If this holds,

d is known as

a,b.c.

it is possible that λ equals one of the solutions of

equation λ2 - λ + 1

=

0

the quad-

7 . 3 . The C r o s s R a t i o

267

Then a, b , c , d a r e equianharmonic,

2 and b o t h z e r o e s of λ - λ + l a r e

s e n t e d t h r i c e by p e r m u t a t i o n s of t h e a r g u m e n t s . condition

c a n n o t be f u l f i l l e d in every f i e l d .

However,

repre-

the relevant

If t h e c h a r a c t e r i s t i c

of

Κ is e q u a l t o 2, i t can never be t r u e ; w h i l e f o r c h a r Κ = 3, we would g e t λ = -1, i. e. t h e h a r m o n i c c a s e .

In a l l c h a r a c t e r i s t i c s o t h e r t h a n 2

and 3, we have λ * -1, and b e c a u s e of

(λ+l)

(λ 2 -λ+1)

( - λ ) m u s t be a p r i m i t i v e t h i r d r o o t of u n i t y .

These (and hence e q u i a n -

h a r m o n i c s e t s ) e x i s t w h e n e v e r - 3 is a s q u a r e in K, f o r t h e s o l u t i o n of the quadratic equation f o r λ reads

λ

2

2

Of t h e s e t h r e e s p e c i a l c a s e s , t h e h a r m o n i c is by f a r t h e m o s t i m p o r t a n t in projective geometry.

7. P r o j e c t i v e Geometry

268

7.4. B6zout's Theorem We have learnt to characterize

the straight

lines in the p r o j e c t i v e

plane

IP (IK) over the ground f i e l d Κ by linear equations in point coordinates. al to this representation

is the description

of

the points in line

Du-

coordi-

nates. Both are particular examples of algebraic curves, which are defined as the sets of zeroes of homogeneous polynomials in three variables (χ , χ , χ ).

In

doing so, it is immaterial whether we interpret the X ( as point or line coordinates. A f u r t h e r generalization suggests i t s e l f .

The investigation can be extended

to higher dimensional spaces, f o r instance to IPn(K). We only have to consider vectors over Κ with (n+1) components (χ

χ ), η

0

and algebraic

equa-

tions now define submanifolds of codimension 1. The r e s t r i c t i o n

to

homogeneous polynomials

is essential,

since

only

then

the invariance under rescaling is retained which has proved to be so useful in the f o r e g o i n g sections. On the other hand, the choice of the underlying f i e l d Κ is much less important;

to avoid unwanted complications,

however, we shall always assume it

to be algebraically closed. As a particular example, the reader is urged to keep in mind the set Κ = C of the complex numbers (and r e f e r to Gauß' fundamental theorem of algebra; theorem 6 . 5 . 5 ) .

Definition 7.4.1: ( a ) An n-ary form over a f i e l d Κ is a homogeneous polynomial in η variables taken f r o m K. I f ,

in particular,

ternary, quaternary forms,

η = 2,3,4

one speaks of

binary,

etc.

( b ) All zeroes of a f o r m F: Kn —> K, i . e .

the vectors χ = (χ

ι

χ ) e Kn η

with F(x)

=

F(x

ι

χ ) π

=

0

are the points of the hypersurface defined by F; or 3 this is also called a point,

in the cases η = 1, 2

(algebraic) curve or surface,

respec-

7.4.

E t e z o u t ' s Theorem

269

tively. ( c ) If χ * 0 i s a n o n t r i v i a l z e r o of F, on a c c o u n t of t h e h o m o g e n e i t y of F a l l m u l t i p l e s of χ a r e l i k e w i s e s o l u t i o n s of t h e

FUx)

=

F(Ax

λχ ) π

1

equation

=

0

=

κ

The s e t

DC-x

is

called

=

j λ χ

λ € κ |

ray (of solutions) of

the

F.

We u s u a l l y

interpret

it

as

a

p o i n t in s o m e p r o j e c t i v e s p a c e . T h e n i t s c o m p o n e n t s x t a r e t h e h o m o g e n e o u s c o o r d i n a t e s of x. (d) We do n o t m a k e a d i f f e r e n c e b e t w e e n v e c t o r s of t h e s a m e r a y .

Therefore,

t h e number of s o l u t i o n s of an a r b i t r a r y s y s t e m of f i n i t e l y many F^: Kn —> Κ is a l w a y s t h e c a r d i n a l i t y of t h e s e t of t h e

F

forms

simulta-

n e o u s s o l u t i o n r a y s of t h e F . In o t h e r w o r d s ,

i t i s t h e n u m b e r of intersection points of t h e

hyper-

s u r f a c e s belonging to t h e given f o r m s .

Any i n f o r m a t i o n on t h e c o m m o n p o i n t s of h y p e r s u r f a c e s i s of g r e a t to geometry.

Under very general

yields their number.

conditions,

the theorem

of

interest

B^zout

We s h a l l f o r m u l a t e it in a v e r s i o n s u i t i n g o u r

e s a t t h e e n d of t h i s

[1769]

purpos-

section.

But b e f o r e , we p r e s e n t s o m e h e u r i s t i c a r g u m e n t s w h i c h c a n b e t u r n e d i n t o a n e x a c t m a t h e m a t i c a l t h e o r y w i t h t h e m e a n s of m o d e r n algebraic geometry ( s e e Hartshorne

[1977]).

Generically,

t h e z e r o s e t of a f o r m o v e r a n a l g e b r a i c a l l y c l o s e d f i e l d

has

codimension

1; w e t h e r e f o r e e x p e c t

the

i n t e r s e c t i o n of t w o , t h r e e , ...

under

"normal" circumstances

h y p e r s u r f a c e s s h o u l d h a v e c o d i m e n s i o n 2,

If s o m e of t h e f o r m s p o s s e s s common f a c t o r s of p o s i t i v e d e g r e e ,

conclusion

i s of

b o u n d of t h e In

...

that

the

nates,

course

invalid;

it

nonetheless

provides

us with

a

3, this

lower

dimension.

n-dimensional

projective

space

P n (K),

endowed

with

(n+1)

coordi-

we t h u s need a t l e a s t η a l g e b r a i c a l l y i n d e p e n d e n t f o r m s t o g e t

l a t e d p o i n t s = r a y s of

solutions.

iso-

270

7. P r o j e c t i v e Geometry

To get acquainted with Bezout's theorem,

we f i r s t consider several

simpler

special cases which are important f o r themselves.

Theorem 7.4.1: Every f o r m in 2 variables over

an algebraically

closed f i e l d splits

com-

pletely into linear f a c t o r s .

Proof: We w r i t e down the f o r m in question - let it be of degree η - e x p l i c i t l y , F(x ,χ )

=

0 1

f

xn + f

0 0

1

χ"" 1 χ 0

1

+ ...

+ f

n-l

χ

x n_1 + f

0 1

x"

n l

If f

vanishes, then χ is a f a c t o r of F; otherwise a f t e r having t r a n s f e r n ο red to the inhomogeneous coordinate χ ζ

=

1 X 0

we obtain a polynomial of degree n, namely

f(z)

=

F(l.z)

=

f

0

+f

1

z + . . . + f

n-l

z"" 1 + f

η

zn

which, on account of the assumption that the underlying f i e l d Κ is a l g e b r a ically (ζ χ0 -

closed,

has a zero ζ ,

say.

Then F ( l , ζ ) = 0,

and F is divisible

by

V ·

Thus, we may separate a linear f a c t o r f r o m F. By induction with respect to the degree, the theorem f o l l o w s .

Already f o r ternary forms, the above statement is f a l s e , as simple examples show. Up to now we only considered one f o r m at a time; now we ask f o r s i multaneous solutions of two binary forms.

Theorem 7.4.2: Let

7 . 4 . B 6 z o u t ' s Theorem

A(x , x ) ο ι

271

a

0

x

k k-1 + a χ χ 0 1 0 1

+ a

k-1

k-1 . k x x + a χ 0 1 k 1

and TW \ B(x , x ) 0 1

1_ no— 1 . IB—1 . m b x + b x x + . . . + b x x + b x 0 0 1 0 1 , m-101 m l m

=

be t w o b i n a r y f o r m s of d e g r e e s k and m o v e r t h e a l g e b r a i c a l l y c l o s e d f i e l d K. Then t h e f o l l o w i n g a s s e r t i o n s a r e e q u i v a l e n t : ( a ) The s y s t e m A(x) = 0; B(x) = 0 h a s a n o n t r i v i a l

solution;

(b) Both f o r m s have a n o n c o n s t a n t common d i v i s o r ; ( c ) T h e r e a r e f o r m s F and Η of d e g r e e s m-1 and k-1, obeying t h e i d e n t i t y A(x) F(x) + B(x) H(x)

=

0

f o r a l l χ e K2; (d)

T h e resultant o f

A and

B,

that

i s Sylvester's determinant

constructed

f r o m t h e c o e f f i c i e n t s of t h e f o r m s a c c o r d i n g t o t h e s c h e m e

Res(A,B)

=

det

(m r o w s w i t h a, k r o w s w i t h b), is i d e n t i c a l l y z e r o . The r e s u l t a n t is homogeneous of d e g r e e m in t h e a ' s and of d e g r e e k in the b's.

272

7. P r o j e c t i v e Geometry

Proof: ( a ) => ( b ) :

If

( ξ , ξ ) * ( 0 , 0 ) is a common z e r o of both f o r m s A and B, then 0

1

A ( x , x ) and B(x , x ) are multiples of ( ξ χ - ξ χ ). r ο ι ο ι η ο ^o ι ( b ) => ( c ) :

Let Κ be a f a c t o r of A and Β with d e g ( K ) > 0. By theorem 7.4.1,

we can f i n d a linear f a c t o r L ( c ) =* ( a ) :

We decompose,

again with the help of theorem 7.4.1,

A , B , F , H into linear f a c t o r s . which

does

either

| K. Now set F = 5- and Η = -

divide

Η not

at

all

or

with

lower

multiplicity

(consider the degrees! ). We may thus, e . g . , assume that Then L "

| BH and L

(c) β (d):

=

forms

L*1

than

| A, but i f

A

f H.

| B.

If the f o r m s F and Η are,

F ( x ,χ ) 0 1

all

Among those of A, there must be at least one

f

0

explicitly,

x™"1 + f x m " 2 χ + . . . 0 1 0 1

+ f

m-2

χ

χ*" 2 + f x"" 1 1 m-1 1

0

and

H(x , x ) = h x 0 1 0

k

+ h x k 1 0

l

0

x + . . . + h χ xk 2 + h xkl 1 k-2 0 1 k-1 1

2

a comparison of the c o e f f i c i e n t s of x k+m ral numbers i = 0 , 1 , 2 , . . .

a

0

f

1 1

x j in AF +BH = 0 f o r all

natu-

in turn yields the conditions

+ b

0

0

h

= 0

0

a f + a f 1 0 0 1

+ b

+ b h 0 0 1

= 0

a f + a f + a f 2 0 1 1 0 2

+ b h + b h + b h 2 0 1 1 0 2

= 0

a f + a f + a f + k m-3 k-1 m-2 k-2 m-1 a

k

f

m-2

+ a

k-1 a

k

f f

m-1 m-1

h 1

b h +b h +b h m k-3 m-1 k-2 m-2 k-1

= 0

b h + m k-2

= 0

+

b

+

h m-1 b

k-1

h m k-1

=

0

which a l l t o g e t h e r are a system of m+n linear equations in just as many unknowns,

namelyJ

;h m-1 0

h k-1

algebra,

this has a nontrivial

solution

zero.

f

0

f

But the c o e f f i c i e n t m a t r i x

in the d e f i n i t i o n of the resultant.

According if

to

and only

is the transpose of

the if

the

rules

of

linear

determinant

the m a t r i x

is

occurring

7.4.

B 6 z o u t ' s Theorem

Next,

we

multiple

give

273

a necessary

and

sufficient

criterion

for

the

existence

of

zeros:

Theorem 7 . 4 . 3 : The b i n a r y f o r m F = F ( x q , X j ) o f d e g r e e η o v e r an a l g e b r a i c a l l y c l o s e d f i e l d possesses

a solution

of

multiplicity

at

least

2 if

its

discri-

2 n - 2 in the

coeffi-

and only

if

minant,

Dsc(F)

vanishes.

The d i s c r i m i n a n t

c i e n t s of

F.

itself

=

Res(9 F,a F ) ο ι

is a f o r m

of

degree

Proof: If of As

F has the double or ( sξ χ - ξ χ i ο ^o ι an e l e m e n t a r y

shows,

higher

order

zero

ξ^) * ( 0 , 0 ) ,

it

is a

multiple

differential

calculus

)2.

in t h i s

d i v i s i b l e by

application

case

both

of

partial K

the product derivatives

of

are

(^χ^ξ^^.

On the o t h e r hand,

on account of the

identity

χ 9 F ( x , x ) + χ 3 F ( x ,x ) 0 0 0 1 1 1 0 1 which i m m e d i a t e l y

=

η F(x ,x ) 0 1

f o l l o w s f r o m E u l e r ' s homogeneity theorem,

s o l u t i o n of 9 q F ( 5 o , ζ ^ ) =

F ( x ,χ ) ο ι

ξ ^ ) = 0 is also a z e r o of F.

=

every

common

We w r i t e

( ξ χ - ξ χ ) G(x ,χ ) η ο ^ο ι ο ι

w i t h some f o r m G of d e g r e e n-1 and f i n d , stitution of

rule

d F ( x , x ) and d F ( x , x ) 0 0 1 1 0 1

again by d i f f e r e n t i a t i o n and sub-

ξ,

0 = 9/CVV = 0 = V^o-V =

+

G

depending on the parameter κ. Using a slightly

different

notation than in theorem 7. 4. 2, we develop A and Β into f i n i t e power series of the variables χ

ο

and χ : ι

A(x , χ ) ο ι

=

V

F (χ ,χ , κ·χ ) ι ο ι ο

L i =0 g„

a

r ϊ 1 i(K)xo

X

i

Σ,V9 b (κ) χ 1 0

χ 1

where we expressly emphasize the dependence of the c o e f f i c i e n t s on κ. cording to the relation x g = κ·Χ 0> degrees

s i.

i s m holds,

The

(i,j)-entry

of

the a j and b Sylvester's

a κ-polynomial of degree s j - i ;

one of degree s j-i+m.

are polynomials

determinant

is,

as

Ac-

in κ of long

as

f o r i > m, on the contrary,

(A reduction of the degree could be caused by can-

cellation of the highest order contributions). All terms the determinant is composed of get f r o m each row and each column one f a c t o r . This leads to an upper bound f o r the degree of the resultant as a polynomial in κ, deg

κ

Res(A,B)

s

m-n

In f a c t , we have equality, since the summands of maximum degree combine to the resultant of G (χ , ι ι

F (0,

V

and G (x ,x ) 2 1 2 gotten by setting x q = 0 in F

=

F (0,χ , χ ) 2 1 2

and F .

Clearly R e s f G ^ G ^ cannot be identically 0, because the G^ by assumption do not have nontrivial solutions in common.

7 . 4 . B 6 z o u t ' s Theorem

277

The d e g r e e of Rest A, B) as a f u n c t i o n of κ t h u s is g -g , and t h e r e a r e e x a c t l y g j ' g 2 v a l u e s of κ f o r which Res(A,B) v a n i s h e s , to our initial problem:

Ν = g · g^.

We can c o n t i n u e t h i s a r g u m e n t a t i o n well.

and a s many s o l u t i o n s

in t h e i n d i c a t e d

m a n n e r f o r η £ 3 as

The d e t a i l e d e l a b o r a t i o n can be l e f t c o n f i d e n t i a l l y t o t h e

reader.

interested

The r e s u l t of our e f f o r t s is

Theorem 7.4.5:

The n u m b e r of s i m u l t a n e o u s s o l u t i o n r a y s of η f o r m s F , . . . , F in (n+1) i n 1 π d e t e r m i n a t e s w i t h c o e f f i c i e n t s in an a l g e b r a i c a l l y c l o s e d f i e l d and r e s p e c tive degrees gI

n g

is e i t h e r i n f i n i t e or equal to t h e p r o d u c t of t h e g1

(theorem of B6zout).

Remark:

Of c o u r s e ,

h e r e as a l w a y s ,

we have t o t a k e t h e algebraic multiplicities of

the linear f a c t o r s into account;

some or all of t h e s o l u t i o n s m i g h t

coin-

cide. In e a c h p a r t i c u l a r c a s e t h e c o r r e c t c o u n t i n g can be f o u n d f r o m a s i m p l e c o n s i d e r a t i o n of t h e individual f a c t o r s . Bezout [1769] h i m s e l f

p r e f e r s t h e a f f i n e i n t e r p r e t a t i o n and s t u d i e s t h e

r o s e t s of p o s s i b l y inhomogeneous p o l y n o m i a l s in η v a r i a b l e s χ 1

ze-

χ η in-

s t e a d of f o r m s in homogeneous c o o r d i n a t e s , a s we did. This l e a d s t o t h e f a m i l i a r c o m p l i c a t i o n s which a r e c o n n e c t e d w i t h t h e ( p r o j e c t i v e ) s o l u t i o n s on t h e hyperplane at infinity.

F o r t h i s r e a s o n t h e o r i g i n a l v e r s i o n of B e z o u t ' s t h e o r e m is much m o r e a w k w a r d t h a n t h e one given in t h i s s e c t i o n :

there are numerous exceptions

to

be c o n s i d e r e d which u n d e r c e r t a i n c i r c u m s t a n c e s may r e d u c e t h e n u m b e r of solutions

in t h e

s e n s e of

a f f i n e geometry.

We a r e

not

going t o

discuss

t h e s e q u e s t i o n s any f u r t h e r ; t h e s i t u a t i o n is much like t h e one e n c o u n t e r e d in s e c t i o n 7. 1. The above f o r m u l a t i o n of B e z o u t ' s t h e o r e m is by no m e a n s t h e m o s t g e n e r a l . In m o d e r n a l g e b r a i c g e o m e t r y , cepts are investigated.

f a r - r e a c h i n g e x t e n s i o n s of t h e r e l e v a n t

For e x a m p l e ,

t h e p r o j e c t i v e p o i n t s obeying

con-

certain

s y s t e m s of a l g e b r a i c e q u a t i o n s a r e r e p l a c e d by t h e irreducible components of

the

solution

s e t s whose

d i m e n s i o n s may be p o s i t i v e .

If

one

evaluates

278

7. P r o j e c t i v e Geometry

s u m s o v e r s o - c a l l e d i n t e r s e c t i o n n u m b e r s of t h e t h u s c o n s t r u c t e d algebraic varieties, one g e t s s i m i l a r , b u t even m o r e f u n d a m e n t a l r e s u l t s t h a n B£zout; c f . t h e book of H a r t s h o r n e [1977], For our purposes,

t h e t h e o r y developed h e r e is s u f f i c i e n t . We s h a l l

f o r e n o t e x t e n d t h e d i s c u s s i o n beyond t h i s p o i n t . priate

as the presentation

This is the more

is only i n t e n d e d a s an i l l u s t r a t i o n

of

thereapprothe

d e r l y i n g i d e a s (and t o s e r v e as a p r e p a r a t i o n f o r t h e i n v e s t i g a t i o n of nar complex algebraic curves).

unpla-

7.5. Planar Algebraic Curves

279

7.5. Planar Algebraic Curves

The realm of applications of the two types of projective coordinates in the plane

extends

far

beyond

the

description

of

points

or

straight

lines

as

discussed in section 7.1. Among other things,

they also f a c i l i t a t e the study of the zeroes of

higher

degree ternary forms, the so-called algebraic curves. In this book, we usually r e s t r i c t to curves in IP (C), which, f r o m a practical viewpoint,

are an

uttermost important special case. All definitions and the m a j o r i t y of the theorems can be t r a n s f e r r e d without d i f f i c u l t y to more general p r o j e c t i v e planes, braically the

closed

principal

scription

fields.

properties

in point

and

This of line

s u f f i c e s to algebraic

curves

coordinates.

at least to those over

serve At

our

intentions

to

algesketch

and their

mathematical

any

it

time,

will

be

declear

which generalizations are admissible.

Definition 7.5.1: By a curve we always mean a complex planar algebraic curve if not e x p l i c i t ly stated otherwise. Expressed in point coordinates,

a curve of order η e IN is the set of all 2 solution rays of a not identically vanishing f o r m F: P C —> C of degree n. If

2

the point χ with homogeneous coordinates (X q , x j . x 2 ) e P C

satisfies

the

equation F(x , χ , χ ) = 0, we say that χ lies on the curve F = 0 (and, 0 1 2

the

other way round, that the curve passes through the point). Dual to this,

we have a representation in line coordinates. The

solutions

of a f o r m G * 0 of degree k define a curve of class k. When G(u 0, u ,u 1 2) = 0 holds, the straight line u = (u ,u ,u ) is called a tangent of the curve 0 1 2 G = 0.

Remarks: ( 1 ) The necessity from

the

wish

(section 7. 1).

to consider two mutually to

utilize

the

extremely

dual

kinds of

fruitful

curves

Poncelet

derives

principle

280

7. P r o j e c t i v e Geometry

( 2 ) It is common practise not to distinguish systematically between a curve and its

equations

or

the associated

solution

sets.

We have

followed st

this custom already when we investigated points = curves of

1

class

st

and straight lines = curves of 1 The defining f o r m s of sense.

order.

course have to be understood in the p r o j e c t i v e

A curve thus is l e f t invariant if we multiply its equation

with

a nonzero complex number. ( 3 ) It is very important to c l a s s i f y the curves in decomposable = reducible ones, i . e .

those which can be f a c t o r i z e d into curves of lower order or

class,

and those which do not admit such a split-up.

curve,

be it given in point or line coordinates,

Evidently,

each

is representable in a

unique way as the product of irreducible components. In doing so, we have to acknowledge c a r e f u l l y the multiplicities with which the single components divide the equation of the curve. For example, we must distinguish between the straight line u χ +u χ +u χ = 0 0

r

and the curve of 2

0

1

order with the equation (u χ +u χ +u χ ) 0 0 1 1 2 2

Though the latter has the same zeroes as the f o r m e r ,

1

2

2

=0.

they have to be

counted twice. The quadratic f o r m t h e r e f o r e describes a so-called double line. ( 4 ) Whenever we speak of

a curve of

order η or class k,

this

implicitly

contains a statement on the type of coordinates (point or line) we have to use.

Before we shall discuss the interrelations and connections between the two categories of curves, which greatly surpass the duality proper, we r e f o r m u late a special case of B6zout's theorem in order to adapt it to the present situation:

Theorem 7.5.1: ( a ) Two curves of orders m and η intersect in m-n points. ( b ) Each two curves of classes k and 1 have k-1 tangents in common.

Proof: Theorem 7. 4. 5.

7.5.

281

P l a n a r A l g e b r a i c Curves

We remind the r e a d e r f o r the last time that the c o r r e c t a l g e b r a i c cities

of

all

factors

must

be

taken

into

account;

otherwise

multipli-

the

theorem

would be f a l s e . Under c e r t a i n c i r c u m s t a n c e s , tion

or

ficance,

tangents

coincide.

it occurs t h a t t w o or more p o i n t s o f

This

because the c o n d i t i o n

w e choose.

possibility

is

of

great

does not depend on the special

As a f i r s t a p p l i c a t i o n ,

intersec-

geometrical

signi-

coordinates

we shall use it to go o v e r f r o m one t y p e

of c u r v e s t o the dual. Assume

a curve

straight

line g

line

shall

of

(we

F ( x ) = 0 of

order

in η · 1 = η points. soon g i v e

η to With

an e x a m p l e ) ,

d e g r e e n. M u l t i p l e s o l u t i o n s e x i s t s ,

and only i f gree n(n-l).

i t s d i s c r i m i n a n t vanishes. T h i s i m p l i e s (one half

be given.

It

is

intersected

a suitable

parametrization

t h i s leads to

an a l g e b r a i c

by

of

equation

and g is a tangent t o the c u r v e , The l a t t e r

a

the

by t h e o r e m 7. 4 . 3 has

if de-

of)

Theorem 7 . 5 . 2 : A curve o f o r d e r η has c l a s s at most n ( n - l ) ; s

curves of c l a s s k a r e o f

order

k(k-l).

Here,

a r e d u c t i o n f r e q u e n t l y occurs,

a cubic

(curve),

η = 3.

Its

as is seen a l r e a d y f r o m the e x a m p l e of

equation

in line

coordinates

j u s t d e s c r i b e d method has the d e g r e e 3 - 2 = 6. r e p r e s e n t a t i o n w i t h point c o o r d i n a t e s , It o b v i o u s l y cannot be i d e n t i c a l

If

as found by

we r e t u r n to the

the

original

we obtain a curve o f o r d e r 6 - 5 = 30.

w i t h the curve w e s t a r t e d f r o m ,

but

con-

t a i n s it as a f a c t o r . The t r a n s i t i o n f r o m one curve v e r s i o n to the dual and back again thus p r o duces a d d i t i o n a l branches,

caused by the f o r m a l i n c r e a s e of the d e g r e e .

The

r e a s o n is t h a t a c u r v e which is d e f i n e d in point c o o r d i n a t e s a t t a i n s a v e r y special f o r m i f t r a n s f e r r e d to line c o o r d i n a t e s , We do not ther,

intend to

and v i c e v e r s a .

discuss the questions r a i s e d by this o b s e r v a t i o n

but o n l y consider a f e w types of curves (Clebsch

L e t us begin w i t h the simplest and f r o m a p r a c t i c a l point o f v i e w most p o r t a n t nonlinear

curves.

fur-

[1968]). im-

7. P r o j e c t i v e Geometry

282

Definition 7 . S . 2 : A conic section is a curve of 2 nd order and 2 n d class.

Since 2-1 = 2;

the degree

in this case does not g e t

larger

if

we

switch

over f r o m one version to the other. To i l l u s t r a t e the methods,

we c a r r y through the calculation e x p l i c i t l y .

The

general equation of a conic section in point coordinates is

Κ χ χ 1J ι J if we apply the Einstein convention.

=

0

The subscripts run f r o m 0 through 2.

We may demand in addition that K ^ = K ^ .

Κ

00

Of

the

χ® + Κ 0

six

11

x2 + K 1

22

χ2 + 2 Κ

coefficients

2

x x + 2 K

12

Κ

00

Thus,

1

Κ

12

,

2

only

20

the

explicitly,

x x + 2 K 2

0

five

01

ratios

multiplying the f o r m u l a as a whole by some constant

x x

= 0

are

essential;

0 1

f a c t o r * 0 does

not

change anything. When considering conic sections, tion of

section 7.1.

class than 2).

it is useful to remember the m a t r i x

nota-

(This is no longer true f o r curves of higher o r d e r

According to our e a r l i e r

agreement,

nates are represented by r o w and column v e c t o r s ,

point and line respectively.

or

coordi-

The

above

The rank of Κ can be equal to 1, 2 or 3; Rank(K) = 0 is not a l l o w e d

since

r e l a t i o n then reads =

with the symmetrical

0

matrix Κ Κ Κ 00 Ol 02 Κ Κ Κ 10 11 12 Κ Κ Κ 20 21 22

it corresponds to the z e r o f o r m . sentially d i f f e r e n t ,

i.e.,

simple c o o r d i n a t e t r a n s f o r m a t i o n , der i n v e r t i b l e linear maps.

The three remaining p o s s i b i l i t i e s are

es-

they cannot be t r a n s f e r r e d into each other by a as the rank of a m a t r i x is invariant un-

7.5. Planar Algebraic Curves

283

The case rank(K) = 3 is equivalent with det Κ * 0 (irreducible conic section).

We shall later treat these curves explicitly.

or proper

If Κ is not

regular, we obtain a degenerate conic section. For rank 2, the kernel of Κ is one-dimensional. We assume it to be generated by a vector χ * 0; thus χ · Κ = 0. It corresponds to a certain point on the conic section.

If y is a d i f f e r e n t point on the curve,

the same holds

true f o r all λ χ + μ y with arbitrary complex numbers λ, μ. The straight line connecting χ and y then is contained as a whole in the conic section which therefore decomposes into two straight lines. The same conclusion r e mains valid if rank(K) = 1, but then the two f a c t o r s of degree 1 are identical. In conclusion,

a degenerate conic section in point coordinates is simply a

(straight) line pair (intersecting at x),

if

the rank of Κ is equal to 2,

and a double line f o r rank 1. Going

over

to

line coordinates,

we get

the dual configurations,

namely

point pairs and double points, depending on the rank. From now on we restrict

our investigations to irreducible conic

x-K-tx

sections

= 0 and consider f i r s t the line coordinates u = ,u ,u ) of the 0 1 2 tangent at some curve point χ = (XQ, Χ , χ ). TO find it, we select a second point

l(u

y * χ on u.

Then χ + Xy is

incident

with

the

line u = 0 f o r

all

λ € C. The point of tangency of u is x; therefore the quadratic equation obtained by insertion, 0 or,

=

(x+Xy) Κ ^x+Xy)

=

λ |x Κ *y + y Κ Scj + λ 2 y Κ ' y

simplified, 2 λ y Κ 4x + λ 2 y Κ V

=

0

must have the twice counting solution λ = 0. Consequently,

the c o e f f i c i e n t

of λ vanishes: y Κ *x

=

0

This linear relation in y must be equivalent with y-u = 0. Up to an i r r e l e vant numerical f a c t o r it is allowed to set

284

7. P r o j e c t i v e Geometry

u

=

Κ 'χ

By assumption, Κ is symmetric and invertible, χ

t

=

so we can solve f o r x:

„-1 u Κ

and insert the result into the basic equation.

This yields the formula f o r

our conic section as a curve of class 2, namely 4u

K _1 u

=

0

It is not d i f f i c u l t either to find the two tangents which can be drawn f r o m an ( e x t e r n a l ) point χ e Ρ C to the curve.

Let y be the contact points

of

one of them. We get the condition

χ Κ V

=

y Κ Sc

=

0

A comparison with the equation of the conic section itself provides us with two solutions: y and the point of contact of the other tangent. We c o l l e c t the results just found,

together with their dual

counterparts,

and take the opportunity to introduce at the same time some more important notions.

Theorem 7 . 5 . 3 : ( a ) Let X be an irreducible

conic

section.

Its equation

in point

coordi-

nates Χ = (XQ, X I> X 2 ) can be brought to the f o r m χ Κ Sc

=

0

while in line coordinates u = Su , u , u ) it reads 0 1 2 tu

L u

=

0

Here Κ and L are regular symmetric matrices which, except f o r an unimportant constant multiplier,

are inverse (or,

by the nonzero determinant,

a d j o i n t ) to each another.

if we p r e f e r to

multiply

( b ) If the point χ and the line u (up to some f a c t o r ) f u l f i l l the relation

7.5. Planar Algebraic Curves

285

u

=

Κ lx

χ

=

t . u L

or t h e e q u i v a l e n t dual one,

w e c a l l u t h e polar (line) of x , and χ t h e pole of u w i t h r e s p e c t t o X. (c) Pole and polar determine each other uniquely.

Furthermore,

a l l p o l e s of

l i n e s p a s s i n g t h r o u g h χ l i e on t h e p o l a r of u, and v i c e v e r s a . A pole-polar

pair

x , u is i n c i d e n t

a n d χ i t s p o i n t of ( d ) More g e n e r a l l y ,

if

and

o n l y if

u is a t a n g e n t

to X

contact.

t h e p o l a r of s o m e p o i n t χ i s t h e c o n n e c t i n g l i n e of

p o i n t s of t a n g e n c y of b o t h t a n g e n t s f r o m χ t o X. A n a l o g o u s l y ,

the

the pole

of a s t r a i g h t l i n e u is t h e i n t e r s e c t i n g p o i n t of t h e t w o c o m m o n

points

of u a n d X.

T h e r e a r e n u m e r o u s f u r t h e r p r o p e r t i e s of p o l e s and p o l a r s , be d e f i n e d f o r o n e of t h e m ,

higher

degree

curves,

cf.

C l e b s c h [19681.

which can

also

We d i s c u s s

only

s i n c e i t p r o v i d e s d e e p e r i n s i g h t in t h e s t r u c t u r e of t h e

s e c t i o n s and t h e i r In t h i s r e s p e c t ,

conic

equations.

p a r t (d) of t h e f o r e g o i n g t h e o r e m p l a y s a m a j o r r o l e .

We

c o n s t r u c t w i t h i t s h e l p t o a g i v e n p o i n t E, w h i c h i s s u p p o s e d n o t t o l i e on X but is otherwise a r b i t r a r y , p o i n t F on e,

the polar

b u t n o t on X. I t s p o l a r ,

l i n e e. f,

T h e r e a f t e r we choose

any

i n t e r s e c t s e in a t h i r d p o i n t G.

In t h e s a m e vein w e d e n o t e t h e l i n e c o n n e c t i n g Ε and F by g. We t h u s o b t a i n a polar triangle w i t h r e s p e c t t o t h e i r r e d u c i b l e c o n i c

sec-

t i o n X. E a c h of e,f,g.

its vertices E,F,G

is t h e p o l e of t h e c o r r e s p o n d i n g o p p o s i t e

It i s t o be e x p e c t e d t h a t

the equation

of X a t t a i n s

a

side

particularly

s i m p l e f o r m if w e t a k e Δ(Ε, F, G; e, f , g) a s t h e fundamental triangle of a n e w coordinate frame. This is indeed the case. we

deduce

proposition:

(after

having

From the f a m i l i a r r e l a t i o n between pole and defined

the

multipliers

suitably)

the

polar

following

7. P r o j e c t i v e Geometry

286

Theorem 7.S.4: Relative to a polar triangle,

the equation of a proper conic section is in

point coordinates 1

x x

2

^

2

^

x +

2

=

x + 0

x 1 2

=

2 ^ 2 ^ 2 u + u + u 0 1 2

=

η

0

and in line coordinates 1

Hence,

uu

η = 0

all irreducible conic section can be transformed into each other by

pro j e c t i v i t i e s ,

i. e. chart changes.

For curves of higher order (or class) than 2 the analogous statement dently is f a l s e ,

since a f o r m of degree η contains exactly i(n+l)(n+2)

evico-

e f f i c i e n t s , one of which can be divided out. The remaining -n(n+3) parameters cannot be transformed away by an application of the 8-dimensional p r o j e c t i v e group PGL^C). This is already the case f o r η = 3, where we have 9 essential

parameters.

For f o r m s of

higher

transformations,

degree (n £ 4),

the possible simplification by

namely elimination of (at most) 8 c o e f f i c i e n t s ,

linear

is compa-

rably i n e f f i c i e n t , and we shall not pursue these approaches here. On the other hand, we can t r y to reduce the cubic curves (3 r d order or 3 r d class) to a one-dimensional family of standard curves.

This is indeed f e a -

sible. We shall come back to this question later the e l l i p t i c functions.

(chapter 18) in connection

with

7.6.

Stereographic

Projection

287

7.6. Stereographic P r o j e c t i o n

2

I t is known t h a t the 2 - s p h e r e S

2

the Euclidean plane IR . F o r , 2

of IR

existed,

consequently (χ

then Μ would be, isomorphic

= 1), h o w e v e r ,

Thus t h e r e

disk.

The Euler

number

of

and

the

disk

d i f f e r s f r o m that of the sphere (χ = 2).

2

S

which consists of

only

one c h a r t .

e v e r y c h a r t of the sphere has a s i n g u l a r i t y .

To

This f a c t ,

c o m p l i c a t e s the p o s i t i o n i n g

considerably.

On the o t h e r hand, face

into

onto a subset Μ

2

t o c a r t o g r a p h e r s f o r many c e n t u r i e s , on the sphere

2

like S , compact and simply connected

to the c l o s e d

is no a t l a s of

another way,

cannot be embedded h o m e o m o r p h i c a l l y

if a t o p o l o g i c a l mapping of S

of

put

known points

2

i f we d e l e t e only one point f r o m S , the r e m a i n i n g

is homeomorphic

and even

(analytically)

plane. We want to e s t a b l i s h such a b i j e c t i o n , model S2 by the set

{

it

the s i m p l e s t in f a c t .

3

(x,y,z)

diffeomorphic

2

e IR

2

2

x +y +z

to

the

surentire

To do so,

we

I

= 1 1-

3

and choose the south pole

of a l l unit v e c t o r s in IR

Σ as the e x c e p t i o n a l p o i n t . Σ with

the

south

pole

=

(0,0,-1)

€

Ω

=

3

S2

We connect an a r b i t r a r y v e c t o r ( x , y , z ) o t h e r than

by a s t r a i g h t

line

in space.

It

hits

the

plane ζ = Ο in a w e l l d e f i n e d point w i t h c o o r d i n a t e s ( u , v , 0 ) , the i r r e l e v a n t t h i r d component, The

assignment

Clearly,

other

described

normalizations

is

are

standard f o r m .

known

as

possible,

the

Omitting

stereographic

mapping.

but

we

+ (1-λ)

shall

Our next task w i l l

citly. 2 The l i n e through Σ and ( x , y , z ) e S Μ Σ ) c o n s i s t s of

λ (x,y,z)

say.

w e g e t ( u . v ) e IR .

( x , y , z ) «=» (u, v )

the j u s t

equatorial

(0,0,-1)

=

always

stick

be to f i n d i t

all

(λχ,\y,λζ+λ-1)

to

expli-

288

7. Projective Geometry

w i t h r e a l λ.

One of them must be ( u , v , 0 ) .

U

=

V

=

This i m p l i e s λ = (1+z)" 1 and thus

χ 1+z JL 1+z

The i n v e r s e map is obtained s i m i l a r l y .

The line through Σ and ( u , v , 0 )

con-

t a i n s the p o i n t s

λ (u,ν,0)

The second

+ (1-λ)

intersection with 2 2 2

f r o m the c o n d i t i o n χ +y +z

f o r λ,

one of

(0,0,-1)

the

sphere,,

=

(Au,λν,λ-1)

aside f r o m Σ,

= 1. It y i e l d s the q u a d r a t i c

can be

calculated

equation

λ2 (uZ+v2+l) - 2 λ + 1 = 1 whose s o l u t i o n s (λ = 0 ) has to be d i s c a r d e d since it 2

ponds to the south p o l e ,

2

corres-

—1

w h i l e the o t h e r leads to λ = 2(u +v +1)

and

the

(x,y, z)

are

point 2 u χ

=

2

1+u +v

2

2 ν y

Ζ

The p r o j e c t i o n it:

(x,y,z)

~

=

i-» ( u , v ) ,

~

2

1+u +v

2

2 2 1-U -V 1+U2+V2 and its inverse

w = u+iv 6 C

c a n o n i c a l l y w i t h the f i e l d C of the complex numbers.

This procedure

results

in a c o n s i d e r a b l e gain of mathematical s t r u c t u r e . F u r t h e r m o r e w e a r e p r o v i ded w i t h a v e r y n a t u r a l projective

space a l r e a d y f a m i l i a r to us,

namely the

complex

line

C υ { 3 is only u s e f u l f o r

representation.

description

numerical

of

the

calculations.

investiga-

rotation

groups

The e v a l u a t i o n

of

e x p ( S ) is m o s t e f f i c i e n t l y done n o t w i t h t h e h e l p of t h e d e f i n i t i o n f o r m u la,

b u t by a t r a n s f o r m a t i o n of t h e skew s y m m e t r i c a l m a t r i x S t o

principal

a x e s ( t h i s i s a l w a y s p o s s i b l e ! ). To t h i s end we have t o seek a s u i t a b l e τ e GL(n,R) such t h a t t h e t r a n s f o r m ed m a t r i x S T = τ 1 Sx is d i a g o n a l , S'

=

ST

=

e.g.

τ" 1 S τ

=

DiagU

1

λ ) π

The λ^ t h e n a r e t h e e i g e n v a l u e s of S, and t h e e x p o n e n t i a l mapping p r o v i d e s us w i t h

D'

=

exp(S')

=

, λ Diagle 1

λ •> e "

and t h u s

D =

τ D' τ

=

λ f λ 1 τ Diag e , . . . , e n τ "

N u m e r i c a l p r o c e d u r e s f o r c a r r y i n g out t h e t r a n s f o r m a t i o n t o p r i n c i p a l

axes

o r t h e d e t e r m i n a t i o n of t h e e i g e n v a l u e s can be f o u n d in n u m e r o u s c o m p u t e r libraries;

we a r e n o t going t o give any advice.

At t h e end of our i n v e s t i g a t i o n i n t o t h e e x p o n e n t i a l r e p r e s e n t a t i o n of rotation groups,

the

we should come back once m o r e t o t h e a m b i g u i t y which h a s

310

8. Orthogonal Groups

been already discussed f o r the plane. Since all of

(see chapter

2) the identity

is the only

whose eigenvalues are equal to unity,

= - S yields

D = 1 if

its

eigenvalues

orthogonal

transformation

every matrix S € R t n , n ) are

integer

multiples

with

of

2ni.

Thanks to theorem 8.2. 1, this means in particular that all skew symmetric (3, 3 ) - m a t r i c e s S(v) with

ρ

=

jν I

e

2nl

(and only these) describe the identical rotation. T h e r e f o r e all elements of SO(IR ) are already included in the formula with

I ν I < 2ir, which gives each of them, except f o r D = 1, exactly

D(v)

twice.

This observation would allow us to determine the topological structure

of

the Lie group SO(3,IR), but we shall postpone the necessary considerations. In the next section, the desired result w i l l be handed us on a plate.

8.3.

Rational

Parametrization

311

8.3. Rational Parametrization 1 The r e p r e s e n t a t i o n

of

the

standard

1 - s p h e r e Ω^ = S

2 in R

as

the

set

of

c o m p l e x n u m b e r s w i t h m o d u l u s 1 c a n be e x p l o i t e d f o r a n o t h e r d e s c r i p t i o n of t h e r o t a t i o n m a t r i c e s w h i c h c a n be c a r r i e d o v e r t o o r t h o g o n a l

transforma-

n

t i o n s in IR . All w e n e e d i s t o g e n e r a l i z e t h e s i m p l e s t c o n c e i v a b l e c o n s t r u c t i o n of

num-

b e r s of m o d u l u s 1 in C, n a m e l y f o r m i n g t h e q u o t i e n t of t w o c o n j u g a t e c o m plex

quantities + b i

and

a - b i

(a, b € IR). T h u s , b e g i n n i n g w i t h t h e c a s e of t h e E u c l i d e a n p l a n e , we h a v e t h e n e w c h a r a c t e r i z a t i o n of t h e s p h e r e S 1 ,

s1

It h a s

to

be t a k e n

for instance,

η

s

into

2

=

/

account

j

(

I a-bi

e

r2n{0}

that

Ί J

mutually proportional vectors 2 ( a , b ) a n d A - ( a , b ) = (Aa.Xb) in IR \{0> l e a d t o t h e s a m e

like, point

in Ωζ- T h i s y i e l d s a n i s o m o r p h i s m of Sl^ and t h e o n e - d i m e n s i o n a l real projective space, i. e. t h e projective line:

(compare t h i s with section 7.1). F o r m a n y a n a p p l i c a t i o n i t w i l l be c o n v e n i e n t t o r e p l a c e t h e c o u p l e of projective coordinates a a n d b by t h e i r

t

=

a

€

ratio

IP1 R

=

IR υ {•*>}

312

8. Orthogonal Groups

with the r e s u l t

•

{ PIT I " " · ' » }

As we explained a l r e a d y in chapter 7 about p r o j e c t i v e g e o m e t r y , the " n o r m a l " f r a c t i o n s

(for

which a * 0 ) ,

aside f r o m

we also must admit the

special

Thus we are f o r c e d to include the infinite or

case a = 0 (implying b * 0).

improper point t = oo, subject to the rule 1+ooi 1-ooi It

corresponds

real

values of

to

the point

the parameter

+1 -1

-1

( - 1 , 0 ) e Ώ^ c R t.

which

The t o p o l o g y

of

is not

IP1(IR),

obtainable

d e f i n e d in

with

section

8. 1, goes through the assignment

1+tl 1-ti

over into the natural t o p o l o g y of the c i r c u l a r line S . We return to the m a t r i x notation and apply the f a m i l i a r 2-dimensional gebra)

(al-

representation 1

0 =

0

0

1

-1

0

1

=

1

I

of C. We have discussed it already at length in the last section. ber t i e C ( w i t h t € R) is associated with the skew symmetric

t

0

t

•t

0

The num-

(2,2)-matrix

I

For the g e n e r a l case of a r b i t r a r y ( f i n i t e ) dimension η a 2 we are in an abs o l u t e l y analogous situation.

In conclusion,

we g e t

Theorem 8.3.1: Suppose S € R ( n , n > to be skew symmetric:

lS

= -S.

Then

8.3. Rational

Parametrization

D =

D(S)

is an n - d i m e n s i o n a l

=

313

=

rotation

1

(1+S) · (1-S)

matrix.

=

Moreover,

1

(1-S)

· (1+S)

every e l e m e n t

of

SO(n,IR)

which d o e s n o t p o s s e s s t h e e i g e n v a l u e -1 i s uniquely r e p r e s e n t a b l e in t h i s m a n n e r . The i n v e r s i o n f o r m u l a r e a d s D-l D+l

Proof: The e i g e n v a l u e s of S a r e p u r e l y i m a g i n a r y ,

such t h a t in t h e given e q u a t i o n

t h e d i v i s i o n i s well d e f i n e d . Beyond t h i s , auxiliary

D is an e l e m e n t of SO(n,IR),

as can be seen f r o m t h e

simple

calculation

*D

=

His l^S

and t h e o b s e r v a t i o n

that

=

l·^

=

D -1

of

1+S and

1+S

the determinants

l+lS are

identical

with each other. The t r i v i a l p r o o f of t h e l a s t e q u a t i o n f o l l o w s by solving f o r S.

Remark: ( a ) E x a c t l y like f o r η = 2,

in t h r e e - d i m e n s i o n a l

s p a c e it is a l s o

possible

t o avoid t h e e x c e p t i o n caused by t h e r o t a t i o n s w i t h e i g e n v a l u e -1. We only have t o s e t , s l i g h t l y m o r e g e n e r a l l y t h a n b e f o r e ,

w i t h d e IR and 1,S e£ R ( 3 , 3 ) ; S + l S = 0. C l e a r l y , t h e p a i r s (Xd.XS), w h e r e λ e IR , give t h e same t r a n s f o r m a t i o n , (b) T h i s a r g u m e n t a t i o n e s t a b l i s h e s , going section,

the isomorphism

(d,S)

p a r a l l e l t o t h a t s k e t c h e d in t h e

and

fore-

314

8. Orthogonal Groups

S0(R 3 ) (as topological

IP3 IR

s

spaces).

E s p e c i a l l y f o r η = 2 we g e t t h e r e p r e s e n t a t i o n

D =

™

-

1-t

2t

-2t

1-t

(l+tV>

By c o m p a r i s o n w i t h t h e e x p o n e n t i a l f o r m of D, cos a

sin a

-sin α

cos α

D =

we o b t a i n

cos a

=

sin α

=

1-t 1+t 2 2t 1+t 2

and u n d e r s t a n d t h e g e o m e t r i c a l meaning of t h e p a r a m e t e r t , t As an a s i d e ,

=

namely

tg

e x c e p t f o r t h e t r i g o n o m e t r i c f u n c t i o n s of a,

the d i f f e r e n t i a l

of t h e a n g l e a,

doc

2 dt l+t2

is a l s o r a t i o n a l

in t and d t .

This is o f t e n used t o r e d u c e

i n t e g r a t i o n s t o e a s i e r ones w i t h r a t i o n a l At l a s t ,

trigonometric

integrands.

we give t h e e x p l i c i t f o r m u l a s f o r t h e r a t i o n a l r e p r e s e n t a t i o n of a

t y p i c a l D e SO(3,IR). It is n o t a p p r o p r i a t e rectly.

Instead,

to c a r r y

out t h e i n v e r s i o n of t h e m a t r i x

we f i r s t e x c l u d e ,

(1-S)

as in s e c t i o n 8 . 2 in t h e d i s c u s s i o n

diof

8 . 3 . Rational Parameterization

315

the exponential mapping, the trivial case S = 0, D = 1 and note that D com2 mutes with S and thus is a linear combination of the matrices 1 , S , S :

D =

1+S 1-S

=

A·1 + B-S + C-S

Multiplication by the common denominator yields 1+S

=

A·1 + (B-Α) S + (C-B) S 2 - C S 3

We insert 0 S

=

S(o,ß,r)

-r

jr ο

β -a

and compare the c o e f f i c i e n t s ( 1 , S , S

=

/

aZ+ß2+rZ

and the minimal equation S3

=

- p2 S

A =

1

getting the solution

Β

= C 1+P*

and consequently D =

Hence we have proven:

a 0

are linearly independent!),

the shorthand notation

ρ

-β

1 + — (S + S 2 ) 1+P2

again with

316

8. Orthogonal Groups

Theorem 8.3.2: 3

( a ) The rotation D £ SO(R ) belonging to the parameter matrix

Sict.ß.y)

0

r

-y

0

1

β -a

D

=

1 +

α 0

(s + sn i+a 2 +ß 2 +r 2

( b ) All rotations in IR , except those which possess the eigenvalue -1,

can

be (uniquely) written in this way.

In order to include the hitherto neglected orthogonal transformations with eigenvalue -1, i. e. rotation angle 180°, we go by setting

α

=

a d

. :

=

ß

b d

c :

7

=

d

3 over to p r o j e c t i v e coordinates ( a , b , c , d ) in the parameter space P R .

Theorem 8.3.3: ( a ) Each rotation matrix D e SO (IR) is of the f o r m 3

' a 2 -b 2 -c 2 +d 2 D(a, b , c , d )

2

,2

a +b +c +d

2(ac-bd)

2(ba-cd)

2(bc+ad)

2(ca+bd)

2(cb-ad)

c 2 -a 2 -b 2 +d 2

( b ) If a , b , c are not simultaneously zero, f r o m the identity,

2(ab+cd) b 2 -c 2 -a 2 +d 2

or,

in other words,

'

if D d i f f e r s

the axis of D ( a , b , c , d ) is parallel to ( a , b , c ) e R3.

( c ) The rotation angle φ is determined by . 2 (VI tg W

-

a 2 +b 2 +c 2

—p-

8.3. Rational

Parametrization

317

Proof: The Statement under ( a ) has been confirmed e a r l i e r .

Furthermore,

one f i n d s

f r o m theorem 8 . 3 . 2 and

(a,b,c)-S

=

0

or by d i r e c t calculation that

(a,b,c)-D

and thus ( b ) holds true.

=

(a, b, c )

The eigenvalues of

D are l . e ' ^ . e

of their sum

1 + e 1 * + e" 1(P

with the t r a c e of D g i v e s

(c).

=

1+2

cos φ

a comparison

318

9. Linear Transformations of Complex Spaces

9. Linear Transformations of Complex Spaces 9.1. Pauli Matrices

We a s s o c i a t e with an a r b i t r a r y complex ( n . n ) - m a t r i x Μ via

a pair Η, Κ of Hermitian matrices, f r o m which we can g e t Μ back by the f o r mula

Μ =

Η + i Κ

For η = 1 this s p l i t - u p reduces to the decomposition of into i t s r e a l and imaginary parts.

a complex

number

We analogously denote Η as the Hermitian

and iK as the skew-Hermitian part of M. Real linear combinations of Hermitian m a t r i c e s e n j o y the same p r o p e r t y well.

as

T h e r e f o r e all Hermitian (n, n ) - m a t r i c e s f o r m an IR-vector space (of 2, mension η ).

di-

The simplest

of

the

importance;

for

nontrivial

case

(n = 2) concerns

the

linear

mappings

2

complex plane C

into i t s e l f .

It is of p a r t i c u l a r p r a c t i c a l

this reason we are going to discuss it here at some length. The m a t r i x a

b

c

d

Η

( a , b , c , d e C) is Hermitian if

and only if

are complex c o n j u g a t e to each other.

a and d are r e a l ,

ansatz

b

=

χ - 1 y

w h i l e b and c

It is t h e r e f o r e reasonable to make the

c

χ + 1 y

9.1. Pauli Matrices

319

with real numbers x, y. We can restore the symmetry of the formulas by substituting d

with likewise real w, z,

Η

=

w - z

getting

=

H(w,x,y,z)

w+z

x-iy

x+iy

w-z

=

If we drop the condition that the parameters lie in IR and allow

arbitrary

(2 2)

complex numbers instead, we obtain all elements of C '

; but the s i g n i f i -

cance of the representation in question of course is founded on the especially simple and elegant description of the Hermitian matrices.

We conse-

quently r e s t r i c t our considerations to ( w , x , y , z ) € IR . The invariants of Η are easy to evaluate: the trace is Tr Η =

2 w

while the determinant det Η

=

(w+z) (w-z) - ( x + i y ) ( x - i y )

=

w2 - χ 2 - y 2 - z 2

represents the Lorentz form in the (Cartesian) coordinates ( w , x , y , z ) . indicates an intimate relationship

This

with the special theory of relativity,

which we shall attend to in chapter 14. But b e f o r e , we consider the spectrum of H. It is immediately found f r o m the last two equations:

EW(H)

=

I w ± / x 2 +y Z +z 2

The Hermitian ( 2 , 2 ) - m a t r i x Η can alternatively be written as Η

=

wo- + x

twenty

conjugation

M(a,-b,-c,-d)

discussed.

But G a u ß w a s n o t t h e f i r s t m a t h e m a t i c i a n e i t h e r w h o c a m e u p o n t h i s a l g e b r a ic

concept;

E u l e r [1911-

it

had

been

discovered

and

investigated

much

earlier

by

].

Aside f r o m t h i s i n s t r u c t i v e c o m p a r i s o n ,

we w a n t t o e m p h a s i z e a l s o a s o m e -

338

9. L i n e a r T r a n s f o r m a t i o n s of Complex Spaces

what different parameterization

of t h e m u t a t i o n g r o u p which h a s

likewise

been f o u n d in G a u ß ' s n o t e s . The a n s a t z +a +d - c Κ =

- d +a +b

+ f

(b,c,d)

(b,c,d)

+c - b +a

+a+b f Κ =

leads,

+d+bcf 2

-c+bdf

-d+bcf

+a+c f

+b+cdf

+c+bdf

-b+cdf

+a+d f

a s t h e m u l t i p l i c a t i o n by t h e t r a n s p o s e d m a t r i x s h o w s , t o a m u t a t i o n

if we s u b s t i t u t e f

=

(n+a)" 1

Here η a g a i n h a s t h e a b o v e - m e n t i o n e d i n t e r p r e t a t i o n a s a s c a l i n g f a c t o r . We t h u s have g o t t e n a new r e p r e s e n t a t i o n of t h e g r o u p GM^IR) of m u t a t i o n s ; t y p i c a l e l e m e n t is

Κ =

By d i r e c t c a l c u l a t i o n ,

b +a(n+a)

bc+d(n+a)

bd-c(n+a)

bc-d(n+a)

c +a(n+a)

cd+b(n+a)

bd+c(n+a)

cd-b(n+a)

d2+a(n+a)

one indeed f i n d s Κ lK

=

Κ =

η2· 1

which i m p l i e s t h e m u t a t i o n p r o p e r t y (with s c a l e n).

its

9 . 5 . Euler Angles

339

9.5. Euler Angles 3 The r o t a t i o n s of IR f o r m a three-dimensional seek a p a r a m e t e r i z a t i o n

of

L i e group.

SO^ilR) by three angles,

description of SC>2(IR) with the r o t a t i o n

natural

similar to the f a m i l i a r

To understand

we f i r s t need the special r o t a t i o n s around the three axes of

Cartesian r e f e r e n c e f r a m e .

For s i m p l i c i t y ,

we want to avoid

f i n d the m a t r i c e s

of

the r o t a t i o n s

with

p o s i t i v e d i r e c t i o n ( " l e f t t u r n " ) about the x - ,

D (?)

=

1

0

0

cos

0

cos

sin

D (?)

φ

0

=

φ

the

we

mathematically

and z - a x e s .

They are

0 sin

φ

-sin φ

Dy ( φ )

angle φ in

y-,

his

sub-

s c r i p t s and c a l l the coordinates x, y, z. With the results of section 8 . 2 , easily

to

angle.

A v i a b l e method of this kind goes back to Euler [1758]. approach,

This suggests

φ

cos φ

0 -sin 1

0

0

cos

φ

φ

cos φ

sin φ

0

-sin φ

cos φ

0

0

0

1

How can w e build up the general r o t a t i o n of IR f r o m these? Following

Euler,

we f i r s t c a r r y out a z - r o t a t i o n with the angle a,

y i e l d s new coordinates x ' , y ' , z ' , tion;

say.

T h e r e a f t e r we apply a second

which rota-

this time about the x ' - a x i s with angle β. This d e f i n e s a new f r a m e ,

x",y",z".

Finally,

t a t i o n angle y ) .

we only have to r o t a t e once more around the z " - a x i s

The result of the procedure is the orthogonal

tion with the m a t r i x

E(a,ß,y)

When multiplied out,

it reads

=

D (a) D (ß) D ( y ) ζ χ ζ

(ro-

transforma-

340

9. Linear Transformations of Complex Spaces

sina cosß siny + cosa cosy

sina cosß cosy + cosa siny

sina sinß

cosa cosß siny - sina cosy

cosa cosß cosy - sina siny

cosa sinß

sinß siny

- sinß cosy

cosß

In spite of the unsymmetrical way in which the partial operations are applied, the Euler angles α, ß, y are in widespread use, mainly in the physical literature. The reason is that the "symmetry breaking" by the decomposition into "elementary rotations" is thoroughly welcome and meaningful,

particu-

larly f o r the investigation of the motion of rigid bodies (spinning tops). An extremely, and perhaps overly comprehensive evaluation of the many mathematical

facets of

the Euler angles and their

utilization

for

physical

problems can be found in the handbook of Klein & Sommerfeld [1897-1910]. A presentation of the theory of the gyroscope, which is more apt to practical

aspects,

is

the book

Schneider [1992-1993]

by Whittaker [1937],

mainly

consider

while

astronomical

Bucerius [1967]

applications

and

(rotation

of the Earth, precession, nutation, etc. ). As concerns history and the impact of the theory of the spinning top,

especially the work of Euler him-

self and about later developments in the prize-winning paper of Kowalewskaja [1890] and other achievements, cf. Cooke [1984], The orthogonality of the product matrix given above is manifest, is composed of three rotations. tirely

On the other hand,

since it

it is perhaps not en-

obvious that the Euler matrices Ε(α, β, y) form a group

isomorphic

with SO (IR). 3

In order to corroborate this, we want to have a closer look at the transformations in question. Trivially,

the identical map is equal to E ( 0 , 0 , 0 ) ,

while the inverse matrix

of E(a, ß,y) is

[E(a,ß,y)j

=

D (-y) D (-ß) D (-a)

E(-y.-ß.-a)

Note the order of the angular parameters! The product of two Ε-matrices is by no means easy to calculate; we theref o r e shall not attempt to do so,

but instead v e r i f y the assertion by di-

rectly comparing the Euler approach with the Cayley-Klein formulas of section 9. 2. To begin with, we find without much e f f o r t the (2,2)-matrices belonging to the rotations D («>), D (a>) and D («ρ), χ y ζ

9.5. Euler Angles

341

Δ (φ)

=

Δy (φ)

= -sin 2

Δ (φ)

cos,

ΗΝ

=

exp

by

explicitly

solving

the

2

correspondence

(?)

equations.

The

(2,2)-version

of

Ε ( α , β , γ ) turns out to be

=

Α ( α , β , γ )

e

Δ

i(+a+y)/2

(α)

cos

Δ

(β)

ρ I

Δ

(y)

i e1(+"-*)/2

sinl^

Δ(α,β,τ) ι

s i n

l ( - a -

ι

5

Because of that, the Cayley-Klein parameters of A

eK+a+jr)/2

Β

=

, i (+a-y)/2 , \β i e sIn I

c

=

ι

=

e1 e ' * w w i t h φ e R, and the inversion

Since the above substitutions g e n e r a t e the e n t i r e Möbius g r o u p P G L ^ C ) , proof

the

o f the t h e o r e m is c o m p l e t e .

We a r e now capable of t r e a t i n g our f i r s t g e o m e t r i c

problem:

Theorem 9 . 6 . 3 : The antipode o f

a e P 1 (C) is

Proof: By d e f i n i t i o n ,

a

*

#

=

π(ο- ( a ) )

*

=

π(α )

=

π(-α)

A f t e r a f e w t r i v i a l i n t e r m e d i a t e steps, the t r a n s f o r m a t i o n rules o f

theorem

7. 6. 1 y i e l d the c o r r e c t n e s s of the claim.

The most i m p o r t a n t s u r f a c e of

curves on the sphere a r e the c i r c l e s . 2

second o r d e r ,

three

different

them.

Its

points

intersection

S

algebraic

does not contain c o l l i n e a r point t r i p l e s ;

a, 6, c uniquely with S

t i o n can be f o u n d e a s i l y .

As an

define

a

plane

is the ( o n l y ) c i r c l e

Ε

passing

Κ 3 α, b, c,

We s t a t e it in a s y m m e t r i c a l

i.e.,

through

whose

equa-

f o r m as a c r i t e r i o n

f o r the c o c i r c u l a r i t y of ex, b, c and a f o u r t h point 6.

Theorem 9 . 6 . 4 : The f o u r

2 p o i n t s a, b,c,6 e S r e s p .

their

images

a,b,c,d

under

the

stereo-

348

9. Linear Transformations of Complex Spaces

graphic p r o j e c t i o n l i e on a common c i r c l e if and only if the cross r a t i o

is

r e a l or i n f i n i t e :

CR(a, b, c, 6)

=

CR(a,b,c,d)

e

R υ {»}

=

IP1 (IR)

Proof: The points a, b, c, d have the required p r o p e r t y whenever the same condition holds f o r

M(a), μ(1>), μ(ο), μ( q is an a n t i a u t o m o r p h i s m :

definite

(Eu-

10.1.

The Skew F i e l d o f

Quaternions

(d) The i n v e r s e of q i s (qq)

1

357

q+u

=

q + u

qu

=

u q

-q.

( e ) The c e n t r e Z(IH) of IH i s (R-1 £ R. ( f ) The c e n t r a l i z e r o f a q u a t e r n i o n q i s t h e l i n e a r span of 1 and q, q is real.

unless

In t h i s c a s e t h e c e n t r a l i z e r i s IH.

In o r d e r t o p r e p a r e t h e f o r m u l a t i o n of t h e n e x t t h e o r e m ,

we o b s e r v e

t h e n a t u r a l d e c o m p o s i t i o n of q i n t o t h e r e a l p a r t a = Re(q) and t h e

that

imagi-

n a r y p a r t ν = Im(q) = bi+cj+dk l e a d s t o q2

[a 2

=

+

ν2]

+

[ 2 a v]

The s q u a r e b r a c k e t s in t h i s e q u a t i o n i n d i c a t e t h e s e p a r a t i o n i n t o r e a l imaginary

and

parts. 2

One n o t i c e s t h a t f r o m q

e R at l e a s t one of t h e e q u a t i o n s a = 0 and ν = 0

follows.

The f i r s t c a s e y i e l d s p u r e l y i m a g i n a r y q u a t e r n i o n s ,

is

negative

then

(if

and o n l y if q i s r e a l

q * 0),

while

the

alternative

itself.

occurs

if

2

The c o m p l e t e s o l u t i o n s e t of t h e s p e c i a l e q u a t i o n q s i s t s of all pure

and t h e s q u a r e

possibility

= -1 c o n s e q u e n t l y

con-

norm-l-quaternions.

It i s n o w no l o n g e r d i f f i c u l t t o d e t e r m i n e a l l s u b a l g e b r a s o f IH:

Theorem 10.1.3: The s u b a l g e b r a s U of IH a r e ( 1 ) t h e z e r o a l g e b r a 0 (dim U = 0); ( 2 ) t h e a l g e b r a of r e a l q u a t e r n i o n s IR-1 as IR (dim U = 1); (3) those

s u b a l g e b r a s w h i c h a r e s p a n n e d ( a s v e c t o r s p a c e s ) by 1 and any 2 = -1. They a r e a l l i s o m o r p h i c t o C (dim U = 2); and

element ι with ι ( 4 ) IH i t s e l f

(dim U = 4 ) .

Proof: Suppose that U is a subalgebra

of IH w h i c h i s n o t c o n t a i n e d

in t h e

above

358

list.

10. Quaternions

Then dim U e { 1 , 2 , 3 } ,

because the dimensions'0 and 4 correspond only

to 0 and IH, respectively.

I If dim U = 1, we select some q e U = U\{0>. Its square is, as an element 2 of U, linearly dependent of q and both, q and q , are unequal to 0. Like in the introductory remark to the present theorem, we split up q = a + ν with a = Re(q) and ν = Im(q). If q is not real (a * 0), the above formula implies a

(compare

the real

2

+ ν a

and imaginary

2

_

parts

2 a 1 separately).

After

simplification,

this is equivalent to

But a

2

is positive,

while ν

2

is not.

Hence,

in contrast to the assumption,

q is real, and U = IR. On the other hand, if dim U is 2 or 3, there are nonzero purely imaginary quaternions in U. The square of such an element is, as we saw b e f o r e ,

real

and negative. We are t h e r e f o r e allowed to replace the given quaternion by a 2 suitable real multiple ι with the additional constraint ι = -1. If the dimension of U equals 2, we have case (3) of theorem and are done. Thus, only the possibility dim U = 3 must be discussed. We determine an arbitrary element κ * 0 in U which is orthogonal to 1 and t. Then iK is also perpendicular to 1 and i . This brings about that LK and 2 (LK)

are

linearly

dependent,

cause of IK l 1, however,

which

is only conceivable

this leads to LK = 0,

for

contradiction.

real

LK.

Be-

10.2. Left and Right Multiplication

359

10.2. Left and Right Multiplication

T h e p r o d u c t of t w o

quaternions

q

=

a + b i + c j

+ dk

u

=

a + ß i + y j

+ Sk

and

is (see Tab.

qu

10. 1.1)

=

( a a - b ß - c y - d S )

l + ( a ß + b a + c S - d y )

i

+ ( a y - b S + c a + d ß ) j + ( a ö + b y - c ß + d a ) k

The m u l t i p l i c a t i o n with a q u a t e r n i o n u * 0 f r o m the l e f t or f r o m t h e i n d u c e s a l i n e a r t r a n s f o r m a t i o n of IH. We d e s c r i b e i t w i t h t h e h e l p of m a t r i c e s L(u) a n d R(u),

respectively.

If w e ( a s we s h a l l do t h r o u g h o u t

book) i n t e r p r e t q u a t e r n i o n s as r o w v e c t o r s ,

q

=

i. e.

set

(a,b,c,d)

we i m m e d i a t e l y g e t f r o m t h e m u l t i p l i c a t i o n

table

α

β

y

δ

α

δ

-y

-y -δ

α

β

y -β

a

-β

L

=

L(u)

=

L(a,β,y,δ)

=

-δ

and

right the this

360

10. Quaternions

α

R(u)

=

ϋ(α,β,γ,δ)

β

γ

-β

α -δ

-7

δ

δ r

=

-δ

-jr

α

-β

β

α

The associativity of ΙΗ implies several useful relations;

among others,

we

derive f r o m (uq)u' = u ( q u ' ) the commutativity of l e f t and right multiplications: L(u) R ( u ' ) and vice versa.

This identity

=

R(u') L(u)

can as well

be obtained by direct

calcula-

tion, but this much more tiresome and not obvious at all. Two more important relations can also be considered as direct consequences o f ' the associative law, now written in either of the f o r m s q ( u u ' ) = (qu)u' or (uu')q = u ( u ' q ) .

They are

R(uu')

=

R(u) R ( u ' )

and L(uu':

L(u')

L(u)

One has to be careful with the order of the f a c t o r s ; since we use row vectors, we must multiply f r o m l e f t to right, when we apply the matrix notation.

Clearly,

all

equations

could

(except

for

the

technical

difficulties

caused by the clumsiness of vertical arrays and the waste of space) equally well be written as column vectors. Then we would get the transposed m a t r i ces and equations.

For the reasons expounded in chapter 2, we stick to the

row representation. The determinants of L ( q ) and R(q) are det L ( q )

=

det R(q)

=

2

They d i f f e r f r o m zero whenever q * 0. The last statement is obvious because the invertibility of q implies that of L ( q ) and R(q). The multiplication matrices associated with a real quaternion α are identi-

10.2. L e f t and Right M u l t i p l i c a t i o n

361

cal w i t h t h e s c a l a r m a t r i x a. From t h i s and t h e p r o d u c t f o r m u l a , we d e d u c e the orthogonality

conditions L(q) L(q)

=

L(q) L(q)

=

-l

R(q) R(q)

=

R(q) R(q)

=

-l

a s well a s

In p a r t i c u l a r ,

t h i s gives L ( q ) , R ( q ) e SO^tlR), if q h a s t h e n o r m 1.

362

10. Quaternions

10.3. Rotations of the Quaternion Algebra The norm in Η is by section 10.1 Euclidean with 1, i, j , k as an o r t h o normal basis.

The makes it possible to characterize the 2 - and 3-spheres

very elegantly with the help of

quaternions.

To this end, we only need to compile some results of the f o r e g o i n g considerations:

Theorem 10.3.1: ( a ) The standard-3-sphere in IH s R4 is the set

I q e IH

= 1 J

of all quaternions with norm 1. ( b ) The subset

q € IH

Re(q) = 0, = 1

of the pure quaternions in !f

Μ

q 6 and the 4

associated covering

2 a S 0

We shall

see

later

that

4

(IR)

this result

ss

S

3

Χ

S

3

can be generalized

considerably:

all

SO (IR) with η ϊ 3 possess double covers, the spin groups. We shall have to π deal with them and their elements, the spinors, much more thoroughly in chapter 13. The special f o r m of the extension provided by theorem 10.3.3 and its de3

scription with the sphere, mensions,

however,

is limited to IR

nothing like that remains true.

peculiar nature of these two spaces.

4

and IR . In higher

di-

This emphasizes once again the

10.4. R e p r e s e n t a t i o n by Complex M a t r i c e s

367

10.4. Representation b y Complex Matrices

The n a t u r a l i s o m o r p h i s m of C w i t h t h e s u b a l g e b r a R·1 ® R·1

c m

s p a n n e d by 1 and i s u g g e s t s t o w r i t e t h e t y p i c a l q

=

a + b l + c j

quaternion

+ dk

(a, b , c , d e R) in t h e f o r m q

=

w + ζ j

with w =

a + bi

e C

z

c + di

e C

=

Since j w =

j (a+bl)

=

a j - b k

=

(a-bl) j

=

w j

however, the product formula (W+Zj) (w+zj)

=

(Ww + ZJzJ) + (WzJ + Zjw)

=

(Ww - Zz) + (Wz+ Zw) J

which i s v a l i d f o r a l l W,Z, w , z e C, c o n t a i n s t h e c o m p l e x c o n j u g a t i o n . map q i-» w + z j t h e r e f o r e is n o t a r e p r e s e n t a t i o n

of IH. It is

The

nevertheless

u s e f u l on a c c o u n t of i t s e x t r e m e l y s i m p l e a r i t h m e t i c . F o r e x a m p l e , t h e c o n j u g a t e of q is q

=

a - b l - c j - d k

=

(w+zj)

=

w - j z

=

w - z j

368

10. Quaternions

The norm of q reduces to

q q

=

(w+zj) (w+zj)

=

w w + ζ ζ

and is consequently identical with the sum of the norms of the two

(com-

p l e x ) constituents of q. A genuine representation of H as a complex matrix group is also easy to find. The Pauli matrices i and η e { 2 , 3 } . 2 4 2 Let us consider the case η = η = 2 f i r s t ! We obtain Ν = 2 η with η a 3. 1 2 3 3 The subgroup U g has index 2 in G and is, of course, cyclic. Every element of G outside l>3 is contained in a subgroup conjugate to either U^ or U^ and because of η

= n^ = 2 an involution. We choose two of them, f o r example a

and b, such that the product ab generates

Then G = is a dihedral

group. We are now l e f t with η

ι

= 2 and η = 3. The fundamental relation in this 2

case implies 12 η

»

=

W 3

This is admissible only f o r η = 3 , 4 , 5 . 3 The f i r s t alternative, (n ,n ,n ) = ( 2 , 3 , 3 ) , 12

contains precisely — = 6

12

12

yields

Ν = 12.

The

t w o f o l d and — + — - 8 t h r e e f o l d poles,

group

G

lying on

10.5. F i n i t e Groups o f

Quaternions

half as many axes (3 + 4).

375

G t h e r e f o r e possesses e x a c t l y 8 elements of

der 3, each of them f i x i n g only one axis and the two associated poles.

orThis

implies that the permutation representation of G on the 4 t h r e e f o l d axes is f a i t h f u l , and G isomorphic to a subgroup of the symmetric group S . The o r 4

der of G immediately leads to G s A^. For

(n^, η , n3> = ( 2 , 3 , 4 ) ,

the argument runs analogously.

i n j e c t i o n into the S4 acting on the 4 t h r e e f o l d axes. tion,

however,

We again f i n d an

In the present

situa-

we have Ν = 24 and G = S . 4

At last we come to more involved.

(η , n^, n3> = ( 2 , 3 , 5 ) ;

Ν = 60.

We have — = 30 t w o f o l d poles,

Now things are

i.e.

slightly

15 corresponding

axes

and just as many elements of order 2 in G. They are all c o n j u g a t e to each other,

since the same is true f o r the 30 poles. The c e n t r a l i z e r of an i n v o -

lution consequently

has order

4 and cannot

be anything

else

than a

2

group (isomorphic to 2 ), as there are no elements of order 4 in G,

four

because

2

the n (

are not divisible

by 4.

Each 2

contains

three p a i r w i s e

commuting

involutions. Hence,

in t o t a l ,

there are e x a c t l y 5 f o u r groups which are mutually c o n j u -

g a t e within G. (This assertion f o l l o w s easier f r o m Sylow's theorem, c f . renstein [1968], of

commuting

involutions

In the

same manner

Go-

b e f o r e , 2 we conclude that the representation of G as a permutation group on the 2 or the t r i p l e s

Huppert [1967] etc. ).

is f a i t h f u l .

This

as

induces an

isomor-

phism of G with a subgroup of order 60 in S s> which can only be A s We have h i t h e r t o shown that the groups compiled in the theorem are the only conceivable

ones.

The

existence

of

finite

subgroups

of

SOgtR) and their uniqueness up to conjugacy is also easily For the c y c l i c groups, hedral group D

these

types

we have proved this already in theorem 10. 5. 1. A d i -

is obtained f r o m the cyclic group U

2n a x i s perpendicular to the axis of U . π The positioning of the U - p o l e s is a r b i t r a r y η f o r the o r t h o g o n a l i t y condition,

by adding a t w o f o l d

π and the second

can also be chosen at w i l l .

axis,

except

All such c o n -

f i g u r a t i o n s are equivalent to each other under the action of the e n t i r e t a t i o n group S0 3 (IR),

in

demonstrated.

SO, f o r each n,

there is essentially only one

ro-

possibi-

lity. The t e t r a h e d r a l group G s A^ has 4 t h r e e f o l d poles a, b, c, d, say, which t o g e t h e r f o r m an o r b i t of a = (0,0,1).

G.

We introduce Cartesian coordinates in IR3 with

The sum a + b + c + d then is G - i n v a r i a n t .

If

it w e r e

diffe-

r e n t f r o m 0, G would act i n j e c t i v e l y on the 2-dimensional orthogonal space.

Due to theorem 10. 5. 1, this cannot happen since G is not c y c l i c .

sub-

376

10. Quaternions

The 120"-potation about the axis through a permutes the points b, c, d. this and the o b s e r v a t i o n that we may bring b by an a p p r o p r i a t e t r a n s f o r m a t i o n to the f o r m ( 0 , ? , ? ) ,

From

orthogonal

uniqueness up to the action of

SC^tR)

is immediate. The S - and A - g r o u p s can be t r e a t e d by r e c u r r i n g to the A 4

5

contained

in

4

each of them. Since the l a t t e r can be oriented in space only in one way, except f o r a r o t a t i o n , the analogous conclusion f o r the l a r g e r groups is c l e a r l y true as w e l l .

2

While the c l a s s i f i c a t i o n of

the f i n i t e r o t a t i o n groups in the plane R does 3 not present any problems and R as the next case - as seen - is also quite easy to handle, sion. ral

the d i f f i c u l t i e s g r o w

very quickly with

increasing

dimen-

Even with r e f i n e d group t h e o r e t i c a l methods one soon reaches a natu-

limit.

This is mainly caused by the already in spaces of moderately

large

sion immense v a r i e t y of groups arising through the procedure.

For this r e a -

dimen-

son, only some s c a t t e r e d results concerning a f e w of the spaces in the s e 4

5

quence R ,R , . . .

have been published,

which we shall not describe,

however,

since w e have no applications f o r them. Clearly, arbitrary

t h e r e is no hope at all dimensions,

presentations, momorphisms

e.g. into

tensor products,

to c l a s s i f y all

finite rotation

since every f i n i t e group has f a i t h f u l orthogonal

the permutation representations as w e l l suitable

etc.

groups

SOη(R)

constructed

by

reduction,

as f u r t h e r direct

of reho-

sums,

The problem under discussion thus would as a f i r s t

step r e q u i r e an e x p l i c i t table of all f i n i t e groups. But this is completely out of the question:

even the groups of prime

power

o r d e r p f have w i d e l y varying structures already if the exponent f is of

the

o r d e r of 10. Much more i n t e r e s t i n g to us is a c o r o l l a r y of theorem 10.5.2. The q u a t e r n i onic m u l t i p l i c a t i o n is a s s o c i a t i v e ;

hence the i n v e r t i b l e elements of

H form

a group H x . I t s f i n i t e subgroups can be c l a s s i f i e d easily with our methods. The bulk of the work has indeed already been done.

Theorem 10.5.3: ( a ) Every f i n i t e subgroup G of IHX = H\{0} is isomorphic to one of the lowing:

fol-

10.5. F i n i t e Groups of

Quaternions

377

(1) a c y c l i c group of a r b i t r a r y order Ν € IN; ( 2 ) a double cover of a dihedral group D

2n

(3) the Schur extension 2AA

4

;

of the t e t r a h e d r a l

group;

( 4 ) the Schur extension 2aS^ of the octahedral group; (5) the Schur extension 2AA g of the icosahedral In all of

these cases,

the c y c l i c normal

or

group.

subgroup of

order

2 consists

of the t w o quaternions 1 and - 1. 3 ( b ) G l i e s in the standard-3-sphere if . ( c ) Isomorphic f i n i t e subgroups of Hx are c o n j u g a t e .

Remark: The G mentioned in part tetra-,

octa-,

( a ) under

(2), . . . . (5) are also called binary

and icosahedral groups,

respectively.

are known about them, f o r example representations, rings,

Many important presentations,

di-,

details

invariant

etc. ; see C o x e t e r & Moser [1965].

Proof: If

u e G is of

consequently tion

with

order n,

|u| = 1 ,

we obtain f o r its norm

that is proposition (b).

norm-l-quaternions

induces

a

|u|n = |un| = |l| = 1 and

By theorem 10.3.2,

representation

of

the

conjugastandard

sphere S = f 3 with kernel {±1} and image SO^ilR). We use the symbol " - " to indicate this.

Then G is a f i n i t e subgroup of S s SO^flR) and t h e r e f o r e c o n -

tained in the l i s t given in theorem 10.5.2.

Beyond this,

we have shown in 2

section

10.3 that ± 1 are the only solutions of

the equation

q

= 1 in H.

This implies that there is only one involution in (H, namely - 1. Every quaternion of the even f i n i t e order 2n thus obeys u° = - 1, which in turn y i e l d s un = ( - 1 )

= 1 and o r d ( u ) = n.

We discuss the single p o s s i b i l i t i e s f o r the image group one a f t e r the er.

If

G = is c y c l i c

of

even o r d e r ,

then # ( G )

is also

divisible

othby 2,

and G contains an involution which by what has been said above must be - 1. The o r d e r of a preimage u of u is then t w i c e as l a r g e as that of u i t s e l f . This immediately leads to G = . If G is c y c l i c of odd o r d e r ,

we must distinguish between # ( G ) even or odd.

In the f o r m e r case the g e n e r a t o r u has t w o inverse images u and - u, one of which has o r d e r n. The other,

in c o n t r a s t ,

say,

has t w i c e that o r d e r and

g e n e r a t e s G. If the order of G is odd, the kernel of the c o n j u g a t i o n r e p r e -

378

10. Q u a t e r n i o n s

s e n t a t i o n G —> G i s t r i v i a l ,

i. e. G ss G.

Anyway, t h e c y c l i c i t y of G i m p l i e s t h e same p r o p e r t y of G. There

remain

the groups

listed

under

(2), (3), (4), (5) in t h e

compilation,

n a m e l y3 D , A , S , and A , a s c a n d i d a t e s f o r G. 2n 4 ^ 4 S In t h e s e c a s e s G and c o n s e q u e n t l y also G have even o r d e r s .

The k e r n e l of

t h e h o m o m o r p h i s m G —» G now is {1, -1} = 2, and G a d o u b l e cover of G. To finish the proof,

we m e r e l y have t o v e r i f y t h a t t h e e x t e n s i o n i s of

Schur

t y p e f o r G e {A4> S 4> Ag}. All t h r e e of t h e s e g r o u p s c o n t a i n a s u b g r o u p U i s o m o r p h i c t o t h e Klein f o u r 2

g r o u p 2 . T h a n k s t o t h e o r e m 1 0 . 5 . 2 we can, by c o n j u g a t i n g w i t h a s u i t a b l e invertible quaternion,

choose t h e l a t t e r such t h a t i t c o n t a i n s ,

besides

the

identity, the three 180°-rotations around the coordinate axes. The i n v e r s e image of Ü t h e n is U = {±1, ±i, ± j , ±k}, [ 1, J ]

=

I"1 j " 1 i j

=

(-i) (-j) i j

=

and (Ij)2

=

k2

l i e s in t h e c o m m u t a t o r g r o u p U' of U, and hence a l s o in G ' .

=

- 1

Since - 1 is

t h e only i n v o l u t i o n in G, it m u s t e v i d e n t l y be a l s o c o n t a i n e d in t h e

centre

Z(G). T h i s s h o w s t h a t G is a S c h u r e x t e n s i o n of ( - 1 ) - 2 w i t h G, a s s t a t e d in (a). The only c l a i m n o t y e t e s t a b l i s h e d ,

(c),

follows directly from the

analo-

g o u s p r o p o s i t i o n in t h e o r e m 10. 3. 2 by going back t o t h e p r e i m a g e u n d e r o u r representation.

Several

a p p l i c a t i o n s of t h i s r e s u l t

s h a l l play an i m p o r t a n t

r o l e when

we

i n v e s t i g a t e e f f i c i e n t n u m e r i c a l i n t e g r a t i o n m e t h o d s on t h e 2 - s p h e r e in p a r t F of t h i s book.

11.1. Doubling Method of Cayley and Dickson

379

11. Octaves 11.1. Doubling Method of Cayley and Dickson The geometric relevance of the complex numbers and the quaternions, 2

cially with regard to the parameterizations of S

espe-

3

and S , is caused

basi-

cally by the existence of a multiplicative norm (or modulus) and the associated Euclidean scalar product. For this reason, we want to investigate both of them more thoroughly. particular

interest

is the question whether similar

Of

structures can be de-

fined also in other algebras. To begin with, we formulate the problem more precisely,

at the same introducing some notation:

Definition 11.1.1: Let U * 0 be an algebra over the ground f i e l d K. ( a ) A norm on 11 is a positive definite quadratic f o r m N: 11 —> K. The induced scalar product is, as always, defined via

Clearly,

=

this is possible

not essential,

| j^N(x+y) - N ( x )

only if

-

N(y)j

char(OC) * 2,

but this restriction

is

since we shall later consider only the f i e l d s DC e {R, C}.

In particular,

N(x)

=

( b ) The conjugation associated with Ν is the K-linear map κ: 11 —» 11, which leaves 1 invariant and in the orthogonal complement

l1

=

{ χ € 11

< x , l > = 0 ]·

of 1 corresponds to the multiplication by - 1. The image of χ under κ is

380

11. Octaves

χ

=

κ(χ)

=

2 ·1 - χ

( c ) The n o r m is c a l l e d multiplicative (and V a normed algebra) if it is p o s i t i v e d e f i n i t e , and f o r a l l a, b e U t h e e q u a t i o n

=



holds true.

We l i s t a f e w of t h e m o s t i m p o r t a n t p r o p e r t i e s of t h e c o n j u g a t i o n :

Theorem 11.1.1: Let κ be a c o n j u g a t i o n mapping on some a l g e b r a o v e r K. 2 = (1) κ is i n v o l u t o r y : κ = id, and hence χ = x; (2) The f i x e d s p a c e of κ c o n s i s t s of t h e m u l t i p l e s of 1: x=x

β

xeK-1

(3) Each a l g e b r a e l e m e n t c o m m u t e s w i t h i t s c o n j u g a t e : — χ χ =

— χ x

=

2 χ - χ

2

Proof: Check t h e d e f i n i t i o n s .

It is o b v i o u s t h a t t h e c o n j u g a t i o n s and n o r m s in C and H f u l f i l l t h e s e m a n d s and,

beyond t h i s ,

are multiplicative.

The l a t t e r p r o p e r t y is

de-

excee-

dingly r a r e and a l m o s t c h a r a c t e r i z e s t h e IR-algebras known t o us, n a m e l y IR, C, and IH; e x c e p t f o r t h e s e t h r e e ,

t h e r e is only one f u r t h e r

finite-dimen-

s i o n a l n o r m e d a l g e b r a w i t h u n i t e l e m e n t over IR. Our n e x t g o a l w i l l be t o c o n s t r u c t t h i s o c t a v e a l g e b r a 0 = 0(IR).

Its

is d e r i v e d f r o m t h e f a c t t h a t dim (0) = 8. We u t i l i z e a p r o c e d u r e IK by Cayley and Dickson, t h e s o - c a l l e d duplication method:

proposed

name

381

11.1. Doubling Method of Cayley and Dickson

D e f i n i t i o n 11.1.2:

(a) To any algebra II of dimension η over Κ with conjugation κ, norm N, and s c a l a r product we construct by the adjunction of some Ω the "dou2 U with underlying space tl ® U-Ω. The additional element

bled" algebra

Ω is orthogonal to all of 11 and subject to the normalization 2 = 1. The multiplication in

II is determined by the rules

(a Ω) b

=

(a b) Ω

a (b Π)

=

(b a) Ω

(a Ω) (b Ω)

=

- b a

valid f o r all a, b € B. When written in full, elements in 2I is (a +a Ω) (b +b Ω) 1 2 1 2

=

condition

a 1 b1 -

L

b~a 2 2

J

+

the product of two typical

L

b2 a1

+ a 2 b~ Ω 1 J

(with a r b i t r a r y3 a , a , b , b e U). 1 2 1 2 2 (b) We extend the conjugation to Ϊ by the formula a +a fi = 1 2

The set

a

- a Ω 1 2

2 U = U © β·Π obviously is an algebra of twice the dimension of V.

It is also easy to see by a simple calculation that the duplication of R by adjunction of Ω = i leads to C, from which in turn we get IH by repeating the procedure with j .

If we go one step further,

we arrive at

D e f i n i t i o n 11.1.3:

2 The octave algebra 0 = IH is obtained from the quaternion algebra IH by duplication.

We now follow common practise, call the additional element Ε r a t h e r than Π

382

11.

Octaves

a n d i n t r o d u c e t h e s y m b o l s I, J, Κ f o r t h e p r o d u c t s iE, j E , kE. T h e n w e h a v e

Theorem 11.1.2: The m u l t i p l i c a t i o n

r u l e s of

the basic octaves

1, i, j , k. Ε, I, J, K,

a n o r t h o n o r m a l s y s t e m w i t h r e s p e c t t o , a r e g i v e n in t a b l e

Tab.

1

J

k

Ε

I

J

Κ

1

-1

k

"j

I

-E

-K

J

J

-k

-1

i

J

Κ

-E

-I

k

J

-1

-1

Κ

-J

I

-E

Ε

-I

-J

-K

-1

i

j

k

I

Ε

-K

J

-1

-1

-k

j

J

Κ

Ε

-I

-j

k

-1

-1

-k

"j

i

-1

-J

In t h e n e x t f e w s e c t i o n s ,

I

Ε

we s h a l l c o n s i d e r s o m e of t h e m o s t

l a w s f u l f i l l e d by t h e o c t a v e s in g r e a t e r d e t a i l ,

11.1.1.

fundamental

b u t b e f o r e w e do s o ,

u s e f u l t o g e t r i d of t h e i n c o n v e n i e n c e of t h e C a y l e y - D i c k s o n b a s i s , p r o d u c t s m a y n o t be e a s y t o

form

11.1.1

1

Κ

which

it

is

whose

remember.

T h e s o l e o b v i o u s s y m m e t r y of t h e r e p r e s e n t a t i o n u s e d up t o n o w i s t h e triality a u t o m o r p h i s m τ w h i c h i n d u c e s s i m u l t a n e o u s c y c l i c p e r m u t a t i o n s of triples

(i,j,k)

and

(I,J,K)

in t h a t

order,

the

w h i l e f i x i n g Ε (and n a t u r a l l y

1

also). We g e t a much n i c e r b a s i s if w e r e n a m e a c c o r d i n g t o t a b l e 11.1. 2. T h e s u b -

11. 1. Doubling Method of Cayley and Dickson

383

s c r i p t s of the new b a s i s vectors are to be read mod 7 or, even better,

in-

terpreted as elements in F . The symbol e^ f o r the unit element is j u s t a name without deeper meaning; we shall not use the f a c t that the index set { 0 0 : 0 , 1 , 2 , 3 , 4 , 5 , 6 } is the proj e c t i v e l i n e P 1 (F 7 ) = F7u{oo}.

Tab. 11.1.2 1 e to

1

j

-e

-e

1

k -e

4

2

Ε

I

-e

-e

0

J 3

-e

Κ 5

-e

6

The arithmetic in 0 in the new nomenclature is governed by the following description:

Theorem 1 1 . 1 . 3 : (a) The eight elements e 00 = 1 and e L (L € F7 ) form an orthonormal b a s i s of 0 = 0(R). (b) The conjugation is the linear mapping from 0 onto itself which is determined by e00 = e CO and e L = - e L f o r L € F 7. The norm obeys '

=

xx

=

xx

e

IR-1

and the s c a l a r product is calculated via

=

^ x y

+ y x j

=

| | χ y + y χ j

(c) The products of any two basic elements are (L e F ): e e

2

2 = L

=

, = 1

e

- e 00

=

- 1

384

11. Octaves

g

e

6

L+l L+2

C L+2

— — g Q L+2 L+l

— Q L+4

g

— —

— g L+l

L+4

= —

G

q

e

L+4 L+l

q

L+4

L+l

L+2

6

=

L+4

g

L+2

Note t h a t 1 = I 2 , 2 = 3 2 , 4 = 2 2 a r e t h e s q u a r e s in F* = J^MO). All

formulas

above

are

invariant

under

multiplication

of

the

subscripts

w i t h one of t h e t h r e e s q u a r e s 1 , 2 , 4 a n d / o r a d d i t i o n of F ^ - e l e m e n t s . ther,

Toge-

t h e s e t r a n s f o r m a t i o n s g e n e r a t e a p e r m u t a t i o n g r o u p of o r d e r 21 (and

s t r u c t u r e 7: 3) which c o n t a i n s only a u t o m o r p h i s m s of 0. With t h e s e s y m m e t r i e s , identification

e

=1

00

t h e e n t i r e a l g e b r a can be r e c o n s t r u c t e d

and

a

single

nontrivial

pair

e -e = - e · e = e . 1 2 2 1 4 A s i m p l e r · d e s c r i p t i o n could s c a r c e l y be imagined.

of

products

from

the

like,

for

e x a m pr l e ,

The c o m p l e t e

multiplica-

t i o n r u l e s r e l a t i v e t o t h e e - b a s i s a r e a t t a c h e d as t a b l e 11.1.3 b e l o w .

Tab.

e e e e e e e e

00 0 1 2 3 4 s 6

e -e -e -e e -e e e

0 00

3 β 1 s 4 2

e e -e -e -e e -e e

1 3 00

4 0 2 6 Β

e e e -e -e -e e

-e

2 6 4

00

s 1 3 0

11.1.3

e -e e e -e -e -e e

3 1 0 s 00 6 2 4

e e -e e e -e -e -e

4 5 2 1 6 CO 0 3

e -e e -e e e -e

-e

s 4 β 3 2 0 00

1

e -e -e e -e e e

-e

6 2 s 0 4 3 1 00

11.1. Doubling Method of Cayley and Dickson

385

We c o l l e c t t h e t w o m o s t f u n d a m e n t a l a l g e b r a i c p r o p e r t i e s of t h e Cayley o c t a v e s in

Theorem 11.1.4:

The o c t a v e a l g e b r a 0 = O(IR) is n o r m e d and s t r i c t l y

alternative.

Proof:

Here,

"strictly

alternative"

means t h a t

t h e alternative law h o l d s , 2

c o n s i s t s of t w o p a r t s ,

which 2

namely t h e i d e n t i t i e s a ( a b ) = a b and ( a b ) b = a b ,

w h i l e t h e m o r e r e s t r i c t i v e associative law, i. e. a(bc) = ( a b ) c ,

is n o t

true

f o r a l l a, b, c e 0. If we d e v e l o p a and b w i t h r e s p e c t t o t h e b a s i s

{»e ,0e ,1e ,2 e ,3 e 4, e , e S, e 6} , we see i m m e d i a t e l y t h a t t h e a l t e r n a t i v i t y of 0 is t a n t a m o u n t w i t h t h e a s sertions (e e )e L Μ Ν

(e e )e = Μ L Η

e (e e ) + e (e e ) L Μ Ν Μ L Ν

and (e e )e + (e e )e LMM LNM

=

e (e e ) + e (e e ) LMN L NM

w h e r e t h e s u b s c r i p t s L, Μ, Ν r u n t h r o u g h p'tlF^). Whenever one of t h e m e q u a l s oo, i . e . ,

if 1 i s among t h e f a c t o r s ,

t h i s is t r i v i a l ;

c o n s i d e r t h e t r i p l e s L, Μ, Ν e IF . 7 Then e L > e M a n < l e „ a n t i c o m m u t e p a i r w i s e ,

we t h e r e f o r e need only

and by c o n j u g a t i n g and a f t e r w a r d s

i n t e r c h a n g i n g L «—» Ν b o t h r e l a t i o n s t r a n s f o r m i n t o each o t h e r . it

is s u f f i c i e n t to e s t a b l i s h

first.

the validity

of

only one of

them,

Therefore e. g.

the

Because of t h e c y c l i c a u t o m o r p h i s m e g —> e s + j and t h e i n v a r i a n c e of

a l l e q u a t i o n s in 0 u n d e r t h e s u b s t i t u t i o n e s —> e

, we may s u p p o s e in a d -

d i t i o n t h a t L = 0 and Μ is one of t h e n u m b e r s 0 , 1 , - 1 . t h e r e m a i n i n g 21 c a s e s w i t h t h e h e l p of

The v e r i f i c a t i o n of

t a b l e 11. 1.3 is l e f t t o t h e

dili-

gent r e a d e r . Thus,

t h e o c t a v e a l g e b r a 0 is a l t e r n a t i v e ;

duced f r o m

t h e n o n - a s s o c i a t i v i t y can be d e -

386

11. Octaves

k Κ =

(I J) Κ

I 1

I ( J K) and Ο is s t r i c t l y

Ε - Ε

alternative.

The u n i t e l e m e n t by d e f i n i t i o n 11. 1.2 is i n v a r i a n t u n d e r c o n j u g a t i o n , a l l l i n e a r c o m b i n a t i o n s of i, j , k, Ε, I, J, Κ change t h e i r s i g n s .

while

Consequently,

χ —» χ is a l i n e a r i n v o l u t i o n in 0. The p r o d u c t of t w o t y p i c a l o c t a v e s a = a +a Ε and b = b +b Ε y i e l d s ,

when

we a p p l y t h e c o n j u g a t i o n ,

which t u r n s o u t t o be i d e n t i c a l w i t h t h e p r o d u c t of t h e c o n j u g a t e s in r e verse order,

This i m p l i e s t h a t x x is r e a l .

A c o m b i n a t i o n of b o t h r e s u l t s w i t h t h e

thus

verified formula ab

b a

s h o w s t h a t χ —> χ indeed is t h e c o n j u g a t i o n which b e l o n g s t o t h e n o r m N(a) in t h e s e n s e of

d e f i n i t i o n 11. 1.1.

a a

The s c a l a r p r o d u c t

f o r e . We t h e n have t h e i d e n t i t i e s

and

a a

N(a +a E) 1 2

=

( a +a E) (a - a E) 1 2 1 2

N(b +b E) 1 2

=

(b +b E) (b - b E) 1 2 1 2

is e x p l a i n e d

as

be-

1 1 . 1 . D o u b l i n g Method of C a y l e y and Dickson

N(ab)

=

Tab - b a b a I 1 2 2 j [ l l

=

+ a 1 1 +

=

a

a b b a 1 2 2 1

I I +

is i d e n t i c a l

b b 1 1

- a b + a b + b a b a + a b [ 2Zj [ 1 2 1 2_] 2 1 2lJ

-

a b a b 1 1 2 2

-

b a b a 2 2 1 1

+

b

a a b 2 2 2 2

+

a b a b 1 2 2 1

+

b a b a 1 2 2 1

+

b

a a b 1 2 2 1

+ 1 2

a b a b 1 2 2 1

+

387

+ 2 1

b a b a 1 2 2 1

+ 2 2 -

1

a

1

b

2

a

b

2

-

b

2

a

2

b

1

a

1

with

N(a) N(b)

=

| < a , a > + [ 1 1 2 2 J

=

1 1

+ 1 2

K b , b > + . ι ι 2 2 J

+ 2 1

+ 2 2

s i n c e f o r a l l q u a t e r n i o n s a, b, c , d t h e e q u a t i o n

a b c d

holds.

To p r o v e t h i s ,

+ d c b a - a d c b - b c d a

=

0

we r e m a r k t h a t t h e l e f t - h a n d s i d e e q u a l s 2 X,

X

=

-

e

where

OM

For homogeneity reasons,

we may r e s t r i c t w i t h o u t loss to t h e s p e c i a l

vec-

tors

Of t h e

ele-

a, b, c, d e { l , i , j , k } .

4

= 256 w a y s t o c o m b i n e t h e b a s i s

m e n t s , h o w e v e r , t h a n k s t o t h e n u m e r o u s s y m m e t r i e s of X, o n l y v e r y f e w h a v e t o be e v a l u a t e d . As a l l s c a l a r p r o d u c t s l i e in R - l , abed

is

real.

This

simple

to essentially three cases,

b o t h t e r m s in X v a n i s h i n d i v i d u a l l y

observation

reduces

the

remaining

lest

possibilities

namely

(1) all f o u r v a r i a b l e s a r e d i f f e r e n t ; (2) a, b, c, d s p l i t u p i n t o t w o p a i r s of e q u a l e l e m e n t s ;

and

(3) a = b = c = d. If (1) o c c u r s ,

w e m a y n o r m a l i z e t o a = 1, b e c a u s e a s i m u l t a n e o u s

multipli-

c a t i o n of a , b, c, d w i t h a common f a c t o r i, j , k d o e s n o t c h a n g e t h e v a l u e of X. If w e t h e r e a f t e r a p p l y a s u i t a b l e p o w e r of t h e c y c l i c a u t o m o r p h i s m τ of H (triality),

i t i s even a l l o w e d t o s e t b = i in a d d i t i o n .

So w e n e e d

only

388

11. Octaves

consider

( a , b , c, d) = (1, i , j , k )

or

(l,i,k, j).

Both

quadruples

result

in

X = 0. In c a s e ( 2 ) ,

we may by the same argument assume a = 1 and,

t h a t the second pair equals i. ed,

as

there

are

( a , b , c , d ) = (1,1, i, i)

them again yield X = 0 . F i n a l l y ,

furthermore,

This gives t h r e e p o s s i b i l i t i e s to be or

(l,i,l,i)

or

a l t e r n a t i v e (3) is t r i v i a l .

(l,i,i,l).

checkAll

of

11.2. A l t e r n a t i v e D i v i s i o n Algebras

389

11.2. Alternative Division Algebras

The o u t s t a n d i n g p r o p e r t i e s of t h e Cayley-Dickson o c t a v e a l g e b r a c o n s t r u c t e d in

section

11. 1 a r e

the

nonexistence

of zero divisors

and

the

validity

of

the a l t e r n a t i v e law. We w a n t t o show n e x t t h a t , e x c e p t f o r t h e u n i n t e r e s t i n g O - d i m e n s i o n a l a l g e b r a , t h e e x a m p l e s known t o us, t h e a l g e b r a s IR,C,IH, 0 of t h e r e a l and c o m p l e x n u m b e r s , t h e q u a t e r n i o n s and t h e o c t a v e s , r e s p e c t i v e l y , ed by t h e a b o v e - m e n t i o n e d

can be c h a r a c t e r i z -

peculiarities.

To a c h i e v e t h i s aim, we u t i l i z e some s t a n d a r d m e t h o d s and t e r m s of m o d e r n a l g e b r a which we r e c a p i t u l a t e b r i e f l y . An identity in an a l g e b r a II is an e q u a t i o n which depends on c e r t a i n terminates a , b , c , . . . ,

say,

and y i e l d s a l w a y s t r u e s t a t e m e n t s if we

indeinsert

a r b i t r a r y e l e m e n t s of U f o r t h e s e v a r i a b l e s . S i m i l a r t o t h e conmutator [a, b]

=

a b - b a

of t w o e l e m e n t s a, b € U, d e f i n e d e a r l i e r ,

we i n t r o d u c e a r e l a t e d

expres-

sion, which in a c e r t a i n s e n s e m e a s u r e s "how l a r g e " t h e d e v i a t i o n s f r o m t h e associative law a r e .

Definition 11.2.1:

(a) The associator of t h r e e e l e m e n t s a, b, c of an a l g e b r a U is t h e m a n i f e s t ly t r i l i n e a r

expression [a, b, c ]

( a b) c - a (b c)

(b) A s u b s e t Μ £ 11 is c a l l e d associative if f o r a l l m , m , m € Μ t h e a s s o 1 2 3 c i a t o r [m , m , m ] v a n i s h e s . V 1 2 3

One of t h e m o s t f e r t i l e m e t h o d s t o deduce new a l g e b r a i c

identities

known ones is by polarization. It is a p p l i c a b l e w h e n e v e r t h e given

from

identity

390

11. Octaves

contains

a variable,

e. g.

a,

to

an order

higher

than the f i r s t .

In

this

case we have to replace the variable a by a linear combination of the general shape a + Xb, where b is a new indeterminate and λ an element of

the

underlying f i e l d DC. Then we develop this relation into powers of λ. If Κ contains " s u f f i c i e n t l y many" elements, such that the λ-power occurring in the derived

identity

(as functions on K) are linearly

independent,

this

implies that the c o e f f i c i e n t s of all λ 1 are zero. (The f i e l d s R and C, which shall be the only ones considered in the sequel, are large enough since f i n i t e l y many d i f f e r e n t power functions are always linearly independent over them). An example may serve to illustrate this. If in the R-algebra U the identity [a,a,a]

=

0

holds, we f i r s t get by the above prescription [a+Ab, a+Xb, a+Xb]

=

0

(λ e ER) and f r o m this the expansion 0

=

j

[a,a,a] + λ ^ [b,a,a] + [a,b,a] + [a,a,b]

+ λ 2 ^ [a, b, b] + [b, a, b] + [ b , b , a ] j + λ 3

[b,b,b]

Aside of the original equation [a, a, a] = 0 as the term independent of λ, we find the new identity [b, a, a ] + [ a , b , a ] + [ a , a , b ]

=

0

( c o e f f i c i e n t of λ ) as well as (belonging to λ ) the same with a β b and the identity [ b , b , b ] = 0, which is equivalent to the one we started with. In the

derived

relation,

a still

occurs to

second order.

It

is

therefore

possible to repeat the procedure. This leads to the identity [a, b , c ] + [a, c, b] + [ b , a , c ] + [ b , c , a ] + [ c , a , b ] + [ c , b , a ] We now come to the central topic of our investigations.

=

0

Many of the f o l -

lowing arguments are adapted f r o m Braun & Koecher [1966], where even consi-

11.2. A l t e r n a t i v e Division Algebras

391

d e r a b l y m o r e g e n e r a l r e s u l t s can be f o u n d a l s o .

Definition 11.2.2:

An a l g e b r a V i s alternative if t h e

[ a , a , b]

i s t r u e in II. More e x p l i c i t l y ,

identity

=

[ a , b, b ]

=

the two relations 2 a ( a b)

=

( a b) b

=

0

are

a b

and

ab

2

The a l t e r n a t i n g law h a s numerous consequences, f e w very i m p o r t a n t

of w h i c h w e o n l y c o m p i l e a

examples:

Theorem 11.2.1: L e t U be a n a l t e r n a t i v e a l g e b r a o v e r IR o r €. (a) The a s s o c i a t o r

of t h r e e e l e m e n t s in V d o e s n o t c h a n g e i t s v a l u e

cyclic permutations we a p p l y a n odd ( b ) In U, t h e

of t h e a r g u m e n t s ,

permutation.

identity

[a,b,a]

or,

while it is m u l t i p l i e d

=

0

equivalently,

( a b) a

holds.

=

a (b a)

An a l g e b r a f o r w h i c h t h i s is t r u e ,

i s c a l l e d flexible.

( c ) F o r a l l a , b, c e II, w e h a v e

[a,b,ca]

=

a

[a,b,c]

under

by - 1 if

11. Octaves

392

[ca,b,c]

=

[a,b,c] c

and [a,b2,c] ( d ) Furthermore,

=

[a,b,bc] + [ab,b,c]

the Moufang i d e n t i t i e s (a b a) c

=

a (b (a c ) )

c (a b a )

=

( ( c a) b) a

as well as (a b) (c a) are valid.

Here,

=

a (b c ) a

we used the f l e x i b i l i t y to save the brackets in

ex-

pressions like (ab)a = a(ba). ( e ) The subalgebra generated by any associative set Μ £ U is itself

associ-

ative. ( f ) The same is true f o r all subalgebras of U which are generated by any two elements.

Proof: Polarization of the defining relations [a, a, b] = 0 and [a, b, b] = 0 provides us with [a,c,b] + [c,a,b]

=

0

[a,b,c] + [a,c,b]

=

0

The associator thus goes over into its negative if we interchange the f i r s t two or the last two arguments. This implies (a). Statement (b) is then obtained by an application of ( a ) to one of the initial

identities.

The remaining relations under (c) are found through a short

calculation,

11.2. Alternative Division Algebras

393

d u r i n g which we make r e p e a t e d use of the a l r e a d y deduced s y m m e t r i e s of t h e associator. [a,b,ca]

The f i r s t equation f o l l o w s f r o m

= =

[ca,a,b] = ((ca)a)b - (ca)(ab) = (caZ)b - (ca)(ab) 2 2 2 ( c a )b - c ( a b) + c ( a ( a b ) ) - ( c a ) ( a b ) = [ c , a , b ] - [ c , a , a b ]

=

[a2,b,c] - [a,ab,c]

=

(a2b)c - a2(bc) - (a(ab))c + a((ab)c)

=

a((ab)c) - a(a(bc))

=

a

[a,b,c]

w h i l e the second is proved a n a l o g o u s l y [ca,b,c]

= = = =

[b,c,ca]

=

via

(bc)(ca) - b(c(ca)) 2

=

(bc)(ca) - b(c a)

2

2

( b c H c a ) - ( ( b c ) c ) a + (be )a - b ( c a ) = [ b , c , a ] - [ b c , c , a ] 2 2 2 [a,b,c ] - [a,bc,c] = (able - a(bc ) - (a(bc))c + a ( ( b c ) c ) ((ab)c)c - (a(bc))c = [a,b,c] c

The f l e x i b i l i t y implies beyond t h i s [a,bz,c]

=

(ab2)c - a(bzc)

=

( ( a b ) b ) c - (ab) (be) + ( a b M b c ) - a ( b ( b c ) )

=

((ab)b)c - a(b(bc))

=

[ab,b,c] + [a,b,bc]

w h i c h is the t h i r d of the r e l a t i o n s we need. The p r o o f of t h e Moufang i d e n t i t i e s is now e a s y .

In the f i r s t ,

we f o r m the

d i f f e r e n c e of both s i d e s and s i m p l i f y w i t h the help of the a l r e a d y

verified

s y m m e t r y p r o p e r t i e s of the a s s o c i a t o r : ((ab)a)c - a(b(ac))

Here,

=

((ab)a)c - (ab)(ac) + (ab)(ac) - a(b(ac)

=

[ab,a,c] + [a,b,ac]

=

- [b,c,a] a - [c,b,a] a

(c) has been employed.

c(a(ba)) - ((ca)b)a

and a t l a s t

=

- [ab,c,a] =

[ac,b,a]

0

We g e t the second r e l a t i o n

similarly:

=

c(a(ba)) - (ca)(ba) + (ca)(ba) -

=

- [c,a,ba] - [ca,b,a]

=

[a,c,ba] + [a,b,ca]

((ca)b)a

=

a [a,c,b] + a [a,b,c]

=

0

394

11. Octaves

(ab)(ca) - a((bc)a)

=

(ab)(ca) - a(b(ca)) + a(b(ca)) - a((bc)a)

=

[a,b,ca] - a [b,c,a]

= Ο

again utilizing (c). This is part (d) of the theorem. To establish (e), we start out from the four identity [ab, c, d] - [a, be, d] + [ a , b , c d ]

=

[a,b,c] d + a [b,c,d]

which is true in every algebra and can be corroborated without d i f f i c u l t y by direct calculation. Polarization of the relations in (c) shows that an associator which depends linearly on four elements changes its sign if we permute any two of the a r guments. This implies, among others, - [a,bc,d]

=

[bc,a,d]

=

- [cb,a,d]

=

[ab,c,d]

and analogously [a,b,cd]

=

[cd,a,b]

=

- [ad,c,b]

=

[ab,c,d]

The three terms on the left-hand side of the four identity are therefore all equal. We obtain the new identity 3 [ab,c,d]

=

[a,b,c] d + a[b,c,d]

Restricting now to a, b, c, d e M, the associators [ a , b , c ] and [ b , c , d ] vanish, since Μ by definition is associative. There only remains [ab,c,d]

=

0

and the cyclically permuted equations. Together, they say that Μ υ (ab) is associative as well. In conclusion, we can add to an associative set the products of two arbitrary

elements without

losing associativity.

This implies

(e)

and -

since

all two-element sets are obviously associative - also ( f ) , and the proof the theorem is complete.

of

11.2. A l t e r n a t i v e D i v i s i o n Algebras

By t h e way,

395

t h e r e v e r s a l of ( f ) is a l s o c o r r e c t :

An R - a l g e b r a is

alterna-

t i v e if and only if a l l s u b a l g e b r a s g e n e r a t e d by t w o e l e m e n t s a, b e U a r e associative.

This is a t r i v i a l c o n s e q u e n c e of t h e f a c t t h a t t h e

alternative

l a w s depend on no m o r e t h a n t w o a r g u m e n t s . We f u r t h e r m o r e need

Definition 11.2.3:

(a) If

the

elements

a, b e 11\{0)

a b = 0, w e c a l l t h e m

of

an

algebra

II f u l f i l l

(left a n d right) zero divisors.

the

Clearly,

condition 11 h a s

z e r o d i v i s o r s if and only if ab = 0 implies a t l e a s t one of t h e

no

equa-

t i o n s a = 0 o r b = 0. (b) II is a division algebra if f o r a l l a, b e 11 w i t h a * 0, t h e e q u a t i o n s χ a

=

b

a y

=

b

and

have u n i q u e s o l u t i o n s χ and y,

respectively.

F o r t h e i n t e n d e d c l a s s i f i c a t i o n of t h e a l g e b r a s IR, C,H, 0 t h r o u g h t h e i r

es-

sential properties,

re-

n o n e x i s t e n c e of z e r o d i v i s o r s and a l t e r n a t i v i t y ,

quire another elementary information,

we

namely

T h e o r e m 11.2.2:

A d i v i s i o n a l g e b r a does n o t c o n t a i n z e r o d i v i s o r s .

If t h e dimension is f i -

n i t e , t h e r e v e r s e s t a t e m e n t is a l s o t r u e .

Proof:

The n o n e x i s t e n c e of z e r o d i v i s o r s in division a l g e b r a s is t r i v i a l , f o r a * 0, t h e e q u a t i o n ab = 0 h a s m e r e l y one s o l u t i o n ,

because

which is c l e a r l y 0. £

Vice v e r s a ,

a s s u m e tl h a s no z e r o d i v i s o r s and choose a e U = U\.

Left

and r i g h t m u l t i p l i c a t i o n s w i t h a a r e t h e n 1 i n e a r maps of V i n t o i t s e l f

with

396

11.

trivial kernels.

Octaves

From the homomorphism theorem we deduce t h a t the

images

a r e i s o m o r p h i c t o a n d t h u s ( f o r d i m e n s i o n a l r e a s o n s ) i d e n t i c a l w i t h II.

Be a w a r e t h a t t h e l a s t s t a t e m e n t i s i n d e p e n d e n t of t h e u n d e r l y i n g f i e l d ! We n o w h a v e t h e m e a n s a t o u r d i s p o s a l t o f o r m u l a t e t h e c e n t r a l r e s u l t this

of

section:

Theorem 11.2.3: Let U be an a l t e r n a t i v e

d i v i s i o n a l g e b r a of

IR. T h e n V i s i s o m o r p h i c t o

either

( 1 ) t h e f i e l d R of t h e r e a l

numbers,

( 2 ) t h e f i e l d C of t h e c o m p l e x

positive f i n i t e dimension

over

numbers,

( 3 ) t h e n o n c o m m u t a t i v e s k e w f i e l d IH of t h e q u a t e r n i o n s ( 4 ) t h e a l t e r n a t i v e r i n g 0 of t h e o c t a v e s ,

or

neither commutative nor

associa-

tive.

Proof: We s e t η = dim II a n d s e l e c t a n a r b i t r a r y n o n z e r o e l e m e n t a e U. T h e t i o n e of t h e

solu-

equation

e a

a

i s u n i q u e l y f i x e d . By t h e a l t e r n a t i v e l a w ,

we g e t

e (e a )

e a

2 and e is i d e m p o t e n t :

e

= e.

Moreover,

o b v i o u s l y e * 0. F o r a l l b 6 11 w e

also have

and

2 be'

(b e ) e

=

e(eb)

2 = e b

b e

similarly

=

eb

11.2. A l t e r n a t i v e D i v i s i o n Algebras

After cancellation,

397

the two last relations reduce to b e

=

e b

=

b

Hence U c o n t a i n s ( e x a c t l y ) one u n i t e l e m e n t , namely e, which we f r o m now on call

1.

It c o m m u t e s w i t h a l l e l e m e n t s of 11. The same h o l d s f o r a l l

real

m u l t i p l e s of 1, which f o r m a o n e - d i m e n s i o n a l s u b a l g e b r a IR-1 i s o m o r p h i c t o R. As u s u a l , we i d e n t i f y it w i t h t h e f i e l d IR. If η = 1, we have U = (R and a r e done. We may t h e r e f o r e a s s u m e w i t h o u t r e s t r i c t i o n t h a t η £ 2. Every n o n r e a l χ e 1I\R g e n e r a t e s by t h e o r e m 11.2.1, p a r t (e) or ( f ) , an a s s o c i a t i v e s u b a l g e b r a of U: t h e p o w e r s χ

do n o t depend on t h e o r d e r of m u l -

t i p l i c a t i o n (k e IN). In o t h e r w o r d s : ΐ is power-associative. The (n+1) v e c t o r s

Ι,χ,χ2

χ" are

linearly

dependent.

Thus,

there

is

a

n o t i d e n t i c a l l y v a n i s h i n g p o l y n o m i a l Ρ of d e g r e e s η w i t h P(x) = 0. Ρ has real

orem 6 . 5 . 5 ;

c o e f f i c i e n t s a n d d u e t o t h e fundamental theorem of a l g e b r a

see Gauß [1799]),

(the-

it s p l i t s into i r r e d u c i b l e p o l y n o m i a l s of

de-

g r e e 1 o r 2. Since Μ h a s no z e r o d i v i s o r s , χ must be a z e r o of one of t h e s e f a c t o r s . The d e g r e e of t h i s p o l y n o m i a l is 2, s i n c e we a s s u m e d χ t IR, and χ does n o t obey a l i n e a r e q u a t i o n w i t h c o e f f i c i e n t s in IR. We t h u s have a r e l a t i o n of t h e f o r m χ* + Μ χ + Ν =

0

w i t h s u i t a b l e x - d e p e n d e n t r e a l n u m b e r s Μ = M(x) and Ν = N(x). The irreducibility of

t h i s minimal

polynomial

of

χ

is e q u i v a l e n t

to

its

discriminant being n e g a t i v e : M2 - 4 Ν
,

where Μ+ 2 χ 1

In p a r t i c u l a r ,

U = IR -1 © [R-x = R-l © IR - i = 11^,

and

after

having

inserted

t h e d e f i n i t i o n i n t o t h e minimal e q u a t i o n of x, we o b t a i n

and U = U 3 (C. χ If η = 2, we even have U ξ €, and t h e s t a t e m e n t of t h e t h e o r e m is t r u e . We t h e r e f o r e a s s u m e η > 2 and f i x t h e e l e m e n t i. Then tt is a p r o p e r

subalge-

b r a of U i s o m o r p h i c t o t h e a l g e b r a C of complex n u m b e r s : IR ® IR · i c II. We c o n s i d e r an a r b i t r a r y y 6 II \ 1 . Lying o u t side U , it c a n n o t be r e a l , and by t h e p r o c e d u r e we a l r e a d y a p p l i e d b e f o r e t o c o n s t r u c t i f r o m x,

we

g e t a new e l e m e n t t e U = IR-1 © R-y obeying

The s u b a l g e b r a g e n e r a t e d by χ and y (or, w h a t a m o u n t s t o t h e same t h i n g , by i and t ) ,

w i l l be c a l l e d 11. Due t o t h e o r e m 11.2.1, U is a s s o c i a t i v e .

The minimal p o l y n o m i a l s of i ± t a r e q u a d r a t i c : ( 1 + ι ) 2 + Μ+· t i + t ) + Ν

0

(i-t)2 + Μ ·(i-t) + Ν

0

w h e r e Μ+ = M(i+t), Μ- = M ( i - i ) , Ν+ = N ( i + i ) , Ν With t h e a b b r e v i a t i o n

= N(i-i) are real numbers.

11.2. A l t e r n a t i v e D i v i s i o n Algebras

λ

399

=

i ι +

i

L

we d e r i v e ( - 2 + λ + Ν+ ) + M + i + M + ι

=

0

( - 2 - λ + Ν ) +M

=

0

i - M

ι

Summing up y i e l d s ( - 4 + N+ + N ) + (M+ + M ) i

+ (M+ - Μ ) ι

= 0

The t r i p l e 1, i, ι is l i n e a r l y i n d e p e n d e n t over R; hence M+ and Μ zero.

must

be

The d i f f e r e n c e of t h e two minimal e q u a t i o n s of i ± ι s h o w s t h a t 2 λ

=

Ν - Ν

+

and A € R. We d e f i n e a new e l e m e n t j t h r o u g h

j

Ai+2t — —

=

getting J2

=

- 1

and i j + j i

=

0

If we now d e f i n e in a d d i t i o n k = i j and t a k e t h e a s s o c i a t i v i t y account,

of U i n t o

t h e n a l l a r i t h m e t i c r u l e s of t h e q u a t e r n i o n a l g e b r a a r e s a t i s f i e d ,

as the examples k2

= j k

(ij)(lj) =

j(ij)

= =

i(ji)j

=

- i(ij)j

(Ji)j

=

- (ij)j

= =

- iZ jZ - i j2

= =

- 1 i

400

11. Octaves

indicate. U is thus a homomorphic image of IH. The conjugation χ —> χ corresponds to that in IH. This implies the relation

x+y which clearly holds as well f o r

=

χ + y

any pair

( x , y ) of

elements in U (the

as-

sumption about χ and y, namely that they are not both contained in a subalgebra

isomorphic

to C,

is irrelevant,

since the conclusion

evidently

re-

mains true f o r IR c IH and C c IH). For every λ e R, we moreover find

λχ

λ χ

such that the conjugation has been shown to be IR-linear. Except f o r this,

it is involutory:

χ = χ and has the (precise) f i x e d space

IR. The norm thus becomes a positive definite quadratic f o r m on U whose associated scalar product we, as usual, denote by

=

i

N(x+y) - N(x) - N ( y )

A direct v e r i f i c a t i o n proves that the elements l , i , j , k all have norm 1, are mutually -orthogonal,

and linearly independent.

T h e r e f o r e , ft is 4 - d i -

mensional and isomorphic to IH. In case η = 4 again the theorem is valid, since then II = Μ. So, f r o m now on, we suppose η > 4 and choose a norm-l-element Ε which is perpendicular to 1, i , j , k . names I , J , K .

Then,

For the products iE, jE, kE we introduce the alias

by our prior results,

(k, E , K ) obey the same rules as ( i , j , k ) ,

the triples ( i , E , I);

(j,E,J);

and

in the correct order.

The remaining products of two basis elements are easily found by the Moufang identities,

e.g.

f o l l o w i n g the prescription below:

If both f a c t o r s are in { I , J , K } , J Κ

=

(jE)(kE)

The products,

=

- (Ej)(kE)

we calculate ä la =

- E(jk)E

=

- E i E

in which one f a c t o r is equal to E,

=

- E I

=

- i

while the other is not,

have been treated already. A typical example of the f e w formulas which are still missing is

11.2. A l t e r n a t i v e D i v i s i o n Algebras

J Κ

=

(ΚΙ)Κ

=

401

- (ΙΚ)Κ

=

- I Κ2

=

I

The only case not yet considered is EZ

=

(Ii Η Ii)

Putting things together,

=

- (IiHil)

=

- I i2I

=

I2

=

- 1

we have seen that {1, i, j, k, Ε, I, J, K} is an

ortho-

normal basis of a subalgebra in 11 isomorphic with 0, and particularly that η a 8. If η = 8, we get U = 0, and the theorem holds. The only possibility

left

is η > 8.

From what we said b e f o r e it

that Β contains an octave algebra.

follows

We p r e f e r to express this not in terms

of 1, i, j , k, I, J, K, but relative to the cyclic standard basis

According to section 11.1, the subscripts are taken f r o m the set P*(F )

=

7

F

7

U {oo>

Like in the last step, we seek an element which is orthogonal to all e^ and has the norm 1. Let us call it f . The products f

=

L

e -f L

00

(L 6 F ) complete a system

=

{eL,fL

j Lep'(F7l

}

of 16 pairwise orthogonal vectors of norm 1, whose squares are all equal to - 1, with the sole exception of e

2

00

= e

00

=1.

The 15 remaining e's and f ' s

anticommute with each other. Aside f r o m the alternative law, since

theorem

11.2.1

applies.

we have to s a t i s f y the Moufang identities, This

uniquely

determines

all

arithmetical

rules in the subalgebra Β £ U generated by 6. The reason is that f o r each pair L, Μ Ε

the product e^ e^ is contained in 0 and of the f o r m ε e^ with

suitable e = ± 1 and Ν s P 1 (F ). This yields

11. Octaves

402

f f

L M

=

(e f

=

- ε f

L

Me f ) CO Μ 03 00

e

f

Ν

=

- (f e He f ) Co L Μ C O

=

- f

= ε ί ( ί β ) = ε ί CO 00 Ν

CD

2

(e e )f L Μ C O

00

β CO

Ν

=

- ε β

Ν

and a l l o w s us to evaluate

f

Μ

e

Ν

=

- ε f

=

- ε f Ν

L

and

f

without ambiguity.

Finally,

e L H

the anticommutativity

implies a l l equations

of

the f o r m

e

The missing rules

follow

L

f

Μ

=

- ε f

Ν

f r o m the observation

by the v e r y construction of

the f

that

the t r i p l e s

(e , f L

are imaginary units in quaternion

bras which s a t i s f y the same conditions as ( i , j , k ) do (in this But the 16-dimensional a l g e b r a 8 is not a l t e r n a t i v e .

,f

00

L

)

alge-

order!).

To v e r i f y this,

we on-

ly need to compare the a s s o c i a t o r s [ e , e , f ] 0

1

2

=

( e e ) f - e ( e f ) = e f + e f 0 1 2

0 1 2

3 2

0 4

= f - f S

= 0

S

and

[e , f ,e ] 0

2 1

=

(e f )e 0 2 1

- e (f e ) 0 2 1

=

- f

6

e

1

- e

0

f

4

= f

S

+f

= 2 f

S

The sum of both does not vanish as it should do in an a l t e r n a t i v e where the identity

[ a , b, c ] + [a, c, b ]

is true.

=

[a,b+c,b+c] - [ a , b , b ] - [ a , c , c ]

=

0

5

algebra,

11.2. A l t e r n a t i v e D i v i s i o n Algebras

The p a r t

of

theorem

1 1 . 2 . 3 which is c o n c e r n e d w i t h a s s o c i a t i v e

403

algebras

d a t e s back t o F r o b e n i u s [1878], A s i m i l a r c h a r a c t e r i z a t i o n of a l g e b r a s w i t h a m u l t i p l i c a t i v e n o r m h a s been given by H u r w i t z . We s h a l l d i s c u s s it in t h e next section.

404

11.

Octaves

11.3. The Theorem of Hurwltz

The g e n e r a l i z e d t h e o r e m of F r o b e n i u s i n d i c a t e s t h e s p e c i a l p o s i t i o n of

the

a l g e b r a s R, C, H, 0. However,

s i n c e i t i s b a s e d on an a l g e b r a i c a l p r o p e r t y ,

tive law,

w h i l e o u r m a i n c o n c e r n is c o n n e c t e d w i t h t h e g e o m e t r i c a l

tions,

we shall discuss also a d i f f e r e n t ,

r i z a t i o n f o u n d by H u r w i t z in 1898,

namely the

though closely r e l a t e d

alternaapplicacharacte-

namely

Theorem 11.3.1: The only

finite-dimensional

normed

algebras

with

unit

elements

over

the

f i e l d of r e a l n u m b e r s a r e IR, C, IH, a n d 0.

Proof: L e t U b e an n - d i m e n s i o n a l

algebra over R which s a t i s f i e s the

w h e r e η e IN. We c o n t i n u e t o u s e o u r s t a n d a r d

nomenclature.

In II, t h e r e i s s o m e e l e m e n t a d i f f e r e n t f r o m 0 , norm

a n d t h e e x i s t e n c e of

implies

=

< a · 1, a · 1>

=



f r o m w h i c h a n d t h e d e f i n i t e n e s s of w e m a y d e d u c e t h a t

Polarizing the

=

1

identity

=

< a , a >

we o b t a i n

+

=

assumption,

< a , c > + < c , a >

the

11.3. The Theorem of Hurwitz

405

By the symmetry of the scalar product, this simplifies to

=



A second polarization provides

+

=

2

Another identity in U is (a b) b To prove it,

=

a

we decompose b = ß-1 + bQ with

< l.t> Q >

= 0 and assume f i r s t

that the corresponding claim with bQ in place of b is correct. (a b) b

=

(ab +ßa) TF+βϊΓ 0 Ο

=

Then

(ab )ΪΓ + ßaET + ßab + ß2a 0 0 0 0

while a

coinciding with (ab)b,

=

a + a β 2 o o

=

(ab ) b~ + a β 2 o o

because b

= -b . o o Thus, we have to establish the formula (ab)b = a only under the additional hypothesis = 0. We introduce the abbreviation = λ·1 with λ e R and calculate the norm of χ

=

( a b) b - λ a

It turns out to be

=

- 2 λ + λ2

The f i r s t term can be shortened considerably:

=



=



=

λ2

since II is normed by assumption. We transform the second summand with the help of the above identity.

Substituting

406

11. Octaves

(a,b,c,d)

ι—»

(ab,b,a, 1)

we obtain the formula

=

2 - < ( a b ) l , a b >

=

=

λ

The norm of χ now reduces to

=

λ 2 - 2 λ 2 + λ 2

=

0

and we have χ = 0, as desired. If

we

insert

the

special

value

a = 1

in

the

just

verified

equation

(ab)b = a, we are led to · 1

=

b b

and t h e r e f o r e also (a b) b

=

a

=

a (b b)

Since b is a linear combination of 1 and b, we get the f i r s t variant of the alternative law, (a b) b

=

ab2

b (b a )

=

b2 a

The second,

is proved in exactly the same manner; we merely have to w r i t e down all p r o ducts in reverse order. Hence U is an alternative algebra. of zero,

Moreover,

it does not contain

f o r ab = 0 implies

and consequently a = 0 or b = 0.

=

=

0

divisors

11.3. The Theorem of Hurwitz

407

The generalized Frobenius theorem 11.2.3 then immediately yields the r e quired result.

It is natural to ask whether the conditions of Hurwitz' theorem can be r e laxed. In f a c t , there are also normed algebras without 1. They are all constructible f r o m R, C,IH, 0 by the following simple procedure:

Definition 11.3.1: We choose two orthogonal transformations A and Β of the normed algebra II. The (A, Β)-modification of 11 is the algebra with the same elements, but the new product x*y

=

x A yB

=

A(x) B(y)

The essential properties of the modification are

Theorem 11.3.2: ( a ) All modifications of a normed algebra 11 are also normed and have the same dimension as 11. (b) Every normed algebra is a modification of IR,C,IH or 0 and in particular of dimension 1,2,4 or 8.

Proof: That the (A, B)-modification of 11 is normed can be seen f r o m |x*yI *

=

|A(x)B(y)|

=

|A(x)| |B(y)|

=

|x| |y| *

Now let U

be any normed algebra. We write the multiplication in II in the » form ( x*, y ) i-» x*y and select an arbitrary a e 11 with a * 0. Since II has no zero divisors, we derive the existence and uniqueness of some e e 11 which satisfies a*e = a. Then |e| = 1, and l e f t and right multiplication by e are orthogonal transformations of tl . Call them A and B, respectively:

11. O c t a v e s

408

A(x)

=

BCx)

=

χ » e e * χ *

We n e x t i n t r o d u c e a new p r o d u c t on II χ y

by

A _ 1 (x) * B 1 ( y ) *

=

which d e f i n e s an a l g e b r a U. E v i d e n t l y , To f i n i s h t h e p r o o f ,

U

is t h e (A, B ) - m o d i f i c a t i o n of U.

we t h e r e f o r e have t o d e m o n s t r a t e t h a t U c o n t a i n s

a

u n i t e l e m e n t , s i n c e in t h a t c a s e , by t h e t h e o r e m 11.3.1 of H u r w i t z , we m u s t have 11 e induced by the quadratic form μ without comment. An immediate consequence of the definition is λ(1)

=

μ(1)

=

=

1

and, more generally, f o r all χ € 11, λ(χ) Moreover,

=

it is again possible to introduce a conjugation mapping in U. It

is given with the help of λ by χ

=

2 λ(χ) · 1 - χ

=

2 < 1, x>· 1 - χ

The conjugation obeys the rule λ(χ) and is thus an involution:

=

2 λ(χ) λ(1) - λ(χ)

=

λ(χ)

416

11.

χ

=

We n o w c o m e t o t h e a l r e a d y

2 λ(χ)·1 - χ

=

Octaves

χ

announced

Theorem 11.4.3: L e t II b e a q u a d r a t i c a l g e b r a o v e r t h e f i e l d K. T h e n t h e d o u b l e d a l g e b r a is q u a d r a t i c as well.

2

If w e d e n o t e t h e a s s o c i a t e d f o r m s on U a s λ and μ, t h o s e on

11 a s A a n d M,

we d e d u c e f r o m t h e d e f i n i n g e q u a t i o n s ( a £2) b

=

( a b) Ω

a (b £J)

=

(b a ) Ω

( a Ω) (b Ω)

f o r a l l a , b £ II t h e

=

- b a

relations:

Λ(a+bΩ)

M(a+bil)

=

=

ACa)

μ(Ά) + μ ^ )

while the conjugation is given by

a + b Ω

=

ä - b Ω

Proof: We s t a r t

bb

with

=

2 X(b)-b - b2

U

=

2 X(b)-b - 2 *(b)-b + μ ^ ) · 1

and c a l c u l a t e t h e s q u a r e of a+bΩ;

=

μ(^·1

11.4. Quadratic

(a+bfi)2

417

=

a Z + a ( b f i ) + (b£2)a + (bfi)(bi2)

=

a2 - M(b)-1 + 2 M a ) b Ω

A comparison

2 *(a)

Algebras

=

j a 2 - b b j + |b a + b a j Ω

with

(a+bΩ) -

j/i(a)

+

·1

=

2 A(a)-a + 2 *(a) b Ω -

^(a)

+

M(b)J-l

leads to the f o r m u l a s f o r Λ and M. The r e s t is obvious.

The doubling method can thus be i t e r a t e d . H o w e v e r , ly l o s e e s s e n t i a l a l g e b r a i c

in doing so, w e g r a d u a l -

properties:

T h e o r e m 11.4.4: Let I (a)

be a q u a d r a t i c a l g e b r a o v e r the f i e l d K. Then we have

21I

a l t e r n a t i v e «=» II a s s o c i a t i v e . 2 ( b ) U a s s o c i a t i v e ] · 1

1 1 . 4 . Quadrat i c A l g e b r a s

419

We s u b s t i t u t e t h e f o r m e r e x p r e s s i o n and o b t a i n äb - b ä or,

=

2 jx(ab) - X ( a ° b ) j - 1

shorter, äb - b ä

=

(ab) - A ( b a ) j - 1

On t h e o t h e r h a n d , one e a s i l y deduces f r o m t h e g e n e r a l l y v a l i d e x p a n s i o n a(ba) - (ab)a

=

a»(ba) - a»(ab) + a(a»b) -

(a°b)a

( v e r i f i e d by e x p l i c i t l y m u l t i p l y i n g o u t ) t h e i d e n t i t y [a,b,a]

=

(ab)a - a(ba)

=

Since 11 by t h e o r e m 11.2.1 is f l e x i b l e , tion vanishes.

A(ab-ba)-a - -l t h e l e f t hand s i d e of t h e l a s t

equa-

Hence, f o r a t Κ·1 we f i n d by c o m p a r i s o n of c o e f f i c i e n t s A(ab-ba)

=

0

which n a t u r a l l y r e m a i n s t r u e if a is a m u l t i p l e of 1. C o l l e c t i n g t h e mediate results,

inter-

we a r r i v e a t X(ab)

=

A(ba)

and ab - b a

=

0

t h a t means ab

=

b a

T h e s e r e l a t i o n s a l l o w us t o s i m p l i f y t h e above a s s o c i a t o r s t o a c o n s i d e r a ble e x t e n t :

420

11. Octaves

[Α,Α,Β]

j\aj8)a - ß(aa)

=

+ ^ ( a a ) b + (aä)b [B, A, A]

=

- ß(aa)

+ a(ßa)j

- ( a b ) a - ,

the e q u i v a l e n c e

of

( b ) and

(c)

is

ob-

vious. We again i n t r o d u c e the Jordan product a°b = - ( a b + b a ) . Then in e v e r y a l g e b r a

422

11. Octaves

the relation

(a b) a - a (b a )

=

a « ( a b) - a » ( b a) + (a»b) a - a (a»b)

h o l d s , which can be p r o v e n by d i r e c t e v a l u a t i o n . [a,b,a]

=

( a b) a - a (b a )

=

An a l t e r n a t i v e w r i t i n g i s

[a»b,a] - a°[b,a]

The e q u a t i o n a«b which, bra,

=

b»a

=

X(b) a + A(a) b - < a , b > - l

a s we have v e r i f i e d b e f o r e ( t h e o r e m 1 1 . 4 . 4 ) ,

is valid in any

alge-

then yields [a, b , a ]

=

X(a) [ b . a ] - \ ( [ b , a ] ) a - X(a) [ b . a ] + < a , [ b , a ] > - l [a, b, a ]

=

- λ( [b, a ] ) a + < a , [ b , a ] > - l

The l a s t f o r m u l a is s t i l l t o t a l l y g e n e r a l .

Under t h e c o n d i t i o n s (b),

it

re-

duces to [a,b,a] In c o n t r a s t t o t h i s ,

=

-l

we have

=

2 X(a) A(b) - X(a«b)

By a s i m p l e t r a n s f o r m a t i o n we o b t a i n [ a , b, a ]

=

^2 X(a) A ( [ b , a ] ) - A(a» [b, a ] ) j • 1

=

- X(ao[b,a])-l

and w i t h - 2 X(a»[b,a])

=

X(a(ab)-a(ba))

=

λ((ab)a-a(ba))

=

X([a,b,a])

=

0

we s e e t h a t [ a , b , a] = 0. This p r o v e s t h e i m p l i c a t i o n (b) => (a). We now a s s u m e t h a t U is f l e x i b l e . The above i d e n t i t y t h e n a t t a i n s t h e s i m -

11.4. Quadratic Algebras

423

pie f o r m

- λ ( [b, a ] ) a + • 1

=

[a,b,a]

f r o m which A ( [ b , a ] ) = Ο immediately f o l l o w s ( i f

=

0

a is linearly dependent on

1, by comparing c o e f f i c i e n t s ; otherwise the claim is t r i v i a l ) . The linear f o r m λ thus vanishes on all

A(ab)

=

commutators:

A(ba)

This has

A(ab)

=

as a d i r e c t consequence.

( a ° b ) » c - ao(boc)

=

A(a°b)

Moreover,

=

2 A ( a ) A ( b ) -

we get f o r the Jordan product

A ( a ° b ) c + A ( c ) a»b -

-l

- A ( a ) b°c - A ( b « c ) a + < a , b » c > - l =

^ - A ( b ) A ( c ) j a + ^X(a) A ( b ) - j c + ^A(c) - A ( a )

We apply λ to this equation.

The result is

A(a«(b°c))

=

Using the already obtained identities,

or,

j-l

A((a»b)oc)

this can also be w r i t t e n as

A(a(b°c))

=

λ((a°b)c)

A(a(bc)) + A(a(cb))

=

A((ab)c) + A((ba)c)

=

A((ab)c) + A((cb)a)

expanded,

A comparison with the r e l a t i o n

A(a(bc)) + A(c(ba))

424

11. Octaves

which is derived f r o m the f l e x i b i l i t y law by p o l a r i z a t i o n ,

and

substitution

into λ y i e l d s

A(c(ba))

and f i n a l l y ( a ) => (b).

=

A(a(cb))

=

A((cb)a)

Hence the f i r s t three statements are equivalent.

It remains to be shown that ( b ) implies the corresponding 2 linear f o r m Λ belonging to

p r o p e r t y of

the

11.

For the g e n e r i c commutator, the calculation runs as f o l l o w s . We again w r i t e A = a + αΏ, Β = b + βΩ, C = c + y£2 and take the d e f i n i t i o n of A as w e l l as the mutual o r t h o g o n a l i t y of ΙΙ·β and 1 into account and get Λ( [Α, Β ] )

=

λ(ab-ßa-ba+öß)

=

A(ab-ba) + λίΰβ-βα)

=

0

because λ is z e r o when applied to the commutator [a, bl = ab - ba, and βα is c o n j u g a t e to αβ. The argument f o r the a s s o c i a t o r of A, B, C is s i m i l a r :

A([A, B,C])

=

λ((ab)c-a(bc)) + A(a(yß)-y(ßa)) + X ( ( b F ) a - y ( o c b ) ) + λ( ( c ß ) a - ( ß a ) c )

The f i r s t contribution vanishes by assumption, terchange the f a c t o r s in the others;

A([A,B,C])

=

while we are a l l o w e d to i n -

this leads to

λ((yß)a-y(ßa)) + A(a(by)-(ab)r)

+ Μ(cß)a-c(ßa))

=

0

We do not want to c a r r y on with these considerations and only remark as a c o r o l l a r y that all algebras in the sequence

R — > C — > H - * 0 — >

are f l e x i b l e .

20

—

This is true in p a r t i c u l a r f o r the doubled octave algebra

2 0.

11.5.

P a r a l l e l i z a b i l i t y and R e g u l a r V i e l b e i n e

425

11.5. Parallelizability and Regular Vielbeine

In

this

section,

division algebras

are

always

understood

to

be

finite-

d i m e n s i o n a l o v e r t h e f i e l d IR. Our a l g e b r a i c

i n v e s t i g a t i o n s s u g g e s t t h e q u e s t i o n f o r w h i c h η e IN d i v i s i o n

a l g e b r a s in R n e x i s t .

All e x a m p l e s w e k n o w h a v e d i m e n s i o n s η € { 1 , 2 , 4 , 8 } ;

b u t w e c o u l d p r o v e t h e c o m p l e t e n e s s of o u r l i s t o n l y u n d e r c e r t a i n nal a s s u m p t i o n s like,

additio-

e. g. , normedness o r alternativity.

In f a c t , t h e s t a t e m e n t t h a t e v e r y d i v i s i o n a l g e b r a h a s one of t h e f o u r m e n tioned

dimensions

d a y no p r o o f argument Instead,

is

true

without

has been found,

any

restriction.

Nevertheless,

w h i c h i s e n t i r e l y o r a t l e a s t in i t s

to

this

essential

algebraic. o n e h a s t o r e s o r t t o e x t r e m e l y d e e p and d i f f i c u l t t o p o l o g i c a l

a s , w h o s e d i s c u s s i o n by f a r w o u l d e x c e e d t h e s c o p e of t h i s b o o k .

we shall only c i t e the r e l e v a n t topological t h e o r e m s and deduce f r o m t h e g e o m e t r i c and a l g e b r a i c consequences we a r e i n t e r e s t e d

ide-

Therefore, them

in.

More i n f o r m a t i o n c a n be f o u n d in t h e o r i g i n a l p a p e r s r e f e r r e d t o b e l o w a n d t h e a r t i c l e by H i r z e b r u c h [1983] w h i c h i s v e r y w o r t h We r e c a l l

definition

"parallelizability",

4.5.1,

where

we

introduced

reading.

the

terms

"n-Bein"

as well as section 4 . 7 ("cohomology groups"),

and

and sup-

p l e m e n t t h e s e c o n c e p t s by

Definition 11.5.1: ( a ) T h e Euler characteristic ( o r E u l e r number) of a c o m p a c t r e a l manifold Μ is the a l t e r n a t i n g

sum

π

of t h e d i m e n s i o n s of t h e c o h o m o l o g y g r o u p s of M.

Β (Μ) k

=

dim Η*(Μ)

n-dimensional

426

11. Octaves

is also called the k t h Bettl number of M.

The Euler numbers of the spheres are found directly f r o m the Poincar^ lemma. They are

zcs 2 "- 1 )

Z(S 2 n )

=

=

ο

2

Hopf [1935] succeeded in showing that it is possible to construct a nowhere vanishing vector f i e l d on a manifold Μ if and only if χ(Μ) = 0.

Thus the

even-dimensional spheres (except f o r the t r i v i a l case of S ) are not p a r a l lelizable. For spheres of odd dimension k, the existence of a regular k-bein is a very r a r e occurrence.

Complete information is provided by the f o l l o w i n g result,

which has been discovered almost simultaneously and independently by K e r vaire [1958] and Milnor [1958]:

Theorem 11.5.1: The

(n-l)-sphere

Ω s Sn 1 is only η (theorem of Kervalre and Milnor).

parallelizable

if

η e {1,2,4,8}

holds

The original p r o o f s , as well as all other which have meanwhile been found, are based on the periodicity theorem of Bott, a deep and very important r e sult of K-theory. An immediate corollary is

Theorem 11.5.2: Division algebras on IRn exist only f o r η = 1,2,4,8.

Proof: Let U be an n-dimensional division algebra over the f i e l d IR. We choose a basis b

b of II whose f i r s t vector is the unit element b = 1. I n ι For χ e Ω a S , the vectors ν = b ·χ are a basis of IR , because U conn 1 1

11.5. P a r a l l e l i z a b i l i t y and Regular V i e l b e i n e

tains no z e r o divisors.

The f i r s t of these elements,

consequently perpendicular to the sphere.

w ι

=

427

ν

1

- < v , v > v 1 1 1

ν , is equal to χ and

The (n-1) v e c t o r s

=

ν

-x ι

ι

with 2 s i s η are t h e r e f o r e tangential to Ω and, considered as f u n c t i o n s η of x, f o r m a r e g u l a r n-bein. Hence Ω is p a r a l l e l i z a b l e , and theorem 11.5.1 η implies the assertion.

Hopf [1941] had

shown

earlier

(also

with

methods

of

algebraic

topology)

that η must be a power of 2. We shall

encounter

the next chapter, citly,

another

application

of

the K e r v a i r e - M i l n o r

but b e f o r e we want to l i s t

the regular

theorem

vielbeine

in

expli-

which are derived f r o m the division a l g e b r a s C, H, and 0, since they

have many useful g e o m e t r i c

consequences.

We begin with the 1-sphere. The nowhere vanishing v e c t o r f i e l d w^ we have to f i n d is obtained by the described procedure with respect to the standard basis (b , b ) = ( l . i ) of C. At the place (x , x ) it is

Almost as simple is the situation f o r the 3-sphere. usual

basic

in Η y i e l d

quaternions

(b , b , b , b ) = ( l , i , j , k ) . 0 1 2 3

We again r e f e r t o

The

without e f f o r t the required p a r a l l e l i z a t i o n

the

multiplication

rules

by the r e g u l a r

vec-

tor fields w 1

=

w

=

2

II

c-v

+x , -X , + 0 3

V

( -* 2 ·

+x , +X , 3 0

X i

'"'S"

- x , +X , 2 1

)

L e t us now come to the 7-sphere and the Cayley-Dickson octaves! By choosing the

standard

basis

( b , b , b , b , b , b , b , b ) = (e , e , e , e , e , e , e , e ) , O 1 2 3 4 S 6 7 00 0 1 2 3 4 S 6 we obtain with table 11.3.1 at the point ( x , x , x , x , x , x , x , x ) € S the r 00 ο 1 2 3 4 s β tangent v e c t o r s

428

11. Octaves

w 1

-

(

- V +xCO,

w 2

=

(

-V

W 3

=

+X , +x , +x ,1 -X ., "Χ , +X 00 5 6 4 3 1

W 4

=

~x 1. + x o · +x 5 ,

W s

=

(

w β

=

(

w 7

=

(

, +X x ) 3 1 ~Xβ .1 +X1 :. "X5I 4 > + 2

+x , +X ,1 - X 1. ~X 1,+X ,, -X 3 00 4 2 6 0

"V

-X

2 >+

x4)

+X , +X , +X , - x 00 0 •" X 3 ) V ~x 2> 1 β -X , +Χ . - x , +x , +X . +x 4 00• - ν -V 6 3 2 0 -V

+

+X , - χ . +x , +x .,+x ) GO - V -X 2 . -X 5 . 0 4 3 1

Since t h e a l g e b r a s €, H, and Ο a r e n o r m e d , orthonormal

s y s t e m s of t a n g e n t

t h e ν in t h e s e t h r e e c a s e s a r e 1 3 7 v e c t o r s t o S , S , and S . They c o i n c i d e

w i t h t h e w . We t h u s have even c o n s t r u c t e d e v e r y w h e r e o r t h o n o r m a l ( r e g u l a r ) p o l y n o m i a l v i e l b e i n f i e l d s on t h e s p h e r e s in q u e s t i o n .

429

12. 1. Homotopy Groups of Spheres

12. Hopf Mappings 12.1. Homotopy Groups of Spheres

T h i s and t h e s u b s e q u e n t s e c t i o n s e r v e a s a m o t i v a t i o n of w h a t f o l l o w s and t o i n f o r m t h e r e a d e r a b o u t s e v e r a l deep r e s u l t s of a l g e b r a i c t o p o l o g y ;

we

s h a l l need t h e m l a t e r only o c c a s i o n a l l y .

the

For this reason,

we r e s t r i c t

p r e s e n t d i s c u s s i o n t o a m e r e d e s c r i p t i o n of t h e m o s t r e l e v a n t f a c t s w i t h o u t explaining the (sometimes extremely d i f f i c u l t ) proofs. We have a l r e a d y r e c o g n i z e d in s e c t i o n 2. 1, t h a t t h e h o m o t o p y of c u r v e s in t o p o l o g i c a l s p a c e s is a very n a t u r a l c o n c e p t . We t h e r e f o r e e x t e n d i t s r e a l m of a p p l i c a b i l i t y in t h e obvious way:

D e f i n i t i o n 12.1.1:

Two c o n t i n u o u s m a p p i n g s f : X —» Y and f : X —> Y b e t w e e n a r b i t r a r y t o p o l o g i c a l s p a c e s X and Y a r e homotopic t o each o t h e r if we can f i n d a c o n t i n u ous m a p p i n g F:

[0,1] χ X

Y

w i t h F ( 0 , x ) = f Q ( x ) and F ( l , x ) = f ^ x ) f o r all x. by t h e s y m b o l i c n o t a t i o n f

briefly

~ f^.

Homotopy is an e q u i v a l e n c e r e l a t i o n . f:

We e x p r e s s t h i s

The s e t of a l l m a p s h o m o t o p i c t o some

X —) Y i s t h e homotopy class

[f]

of f .

If X and Y a r e ( s m o o t h ) m a n i f o l d s , every h o m o t o p y c l a s s c o n t a i n s a t one C ^ - m a p .

Hence,

" s m o o t h homotopy".

t h e r e is no need t o d i s t i n g u i s h b e t w e e n h o m o t o p y

least and

430

12. Hopf Mappings

The set of all homotopy classes of mappings f r o m X to Y in general has no IT

f u r t h e r structure.

If,

however,

X is a sphere, e . g .

X = S , we can i n t r o -

duce an invariant product on it.

Definition 12.1.2: ( a ) A pointed (topological) space is a pair ( X , x o ) , gical space X and a base point χ

consisting of a t o p o l o -

contained in it.

( b ) The set of all homotopy classes of continuous maps f r o m S point Xq into a connected pointed topological

space ( Y . y Q )

with base is

denoted

by 7tk(Y). ( c ) We describe the k-sphere in the standard f o r m as the set Π of all unit vectors

ζ - (ξ , ξ ξ , ξ ) in IR1+n and choose as its base ^ k-i k the "north pole" x q = (1,0, . . . , 0 ) . The "northern hemisphere"

Ω+

=

I ξ

6

point r

Ω I ξο > 0 }

together with the "equator"

Ä

=

{ ξ e Ω I ξο = 0 }

is mapped by

V = with the scaling f a c t o r ρ obtained f r o m

continuously onto Ω. The image of Ä is the base point Xq, while the r e striction onto Ω+ is b i j e c t i v e l y transformed onto Ω \ { x Q } · ly define φ ( ξ ,ξ

ξ )

=

(1+2ξ ,ρξ

ρξ )

We s i m i l a r -

1 2 . 1 . Homotopy Groups o f Spheres

431

on the "southern hemisphere"

=

{ ξ € Ω

€

0

< θ | k

plus e q u a t o r A. To the continuous mapping f and g f r o m Ω s S a s s o c i a t e a new one, by h(£) = g»(p

h,

which is given in Ω+ by h(£) = f°