Axiomatic characterization of physical geometry

Lecture Notes in Physics Edited by 1. Ehlers, Mtinchen, K. Hepp, Ziirich R. Kippenhahn, Mtinchen, H. A. Weidenmtiller, and J. Zittartz, K61n Managing Editor: W. BeiglbGck, Heidelberg

Heidelberg

111 H.-J. Schmidt

Axiomatic Characterization of Physical Geometry

Springer-Vet-lag Berlin Heidelberg

New York 1979

Author Heinz-Jiirgen Schmidt Fachbereich 5 Naturwissenschaften/Mathematik Universittit Osnabriick Postfach4469 D-4500 Osnabrtick

ISBN 3-540-09719-8 ISBN o-387-09719-8

Springer-Verlag Springer-Verlag

Berlin Heidelberg New York New York Heidelberg Berlin

Library of Congress Cataloging in Publication Data Schmidt, Heinz-Jtirgen, 1948. Axiomatic characterization of physical geometry. (Lecture notes in physics; 111) Bibliography: p. Includes index. 1. Geometry. 2. Axiomatic set theory. I. Title. II. Series. QC20.7.G44S35 530.1’5162 79-23944 ISBN 0-387-09719-E

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 5 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0 by Springer-Verlag Printed in Germany

Berlin

Printing

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1979 Hemsbach/Bergstr.

PREFACE This book will deal with the basis of a theory, which can be considered as the most ancient part of physics, namely Euclidean geometry.

For

about 100 years there has been a debate on the physical space problem, especially stimulated by the creation of tivity.

(non-Euclidean)

General Rela-

In spite of this, contrary to the impression generated by some

textbooks on physics,

the topic is far from being in a final form. The

problems of interpretations often neglected,

and definitions of physical concepts are

partly because methodological rigor is

replaced by physical intuition,

and partly because these problems are

inherently difficult and inextricably intertwined. situation in mathematics, pre-Bourbaki millenium.

(successfully)

In contrast to the

the foundations of physics are still in their

I think, however, G. Ludwig has made an impor-

tant step toward an adequate understanding of physics,

and this book

may be viewed as a partial realization of one point of his program. A large class of physical applications of Euclidean geometry concerns constructions with rigid bodies.

Thus geometry yields propositions

about the behaviour of these bodies and is, in this sense, an emperical theory. This standpoint was adopted by H. v. Helmholtz

[HELl and

A. Einstein, who wrote: "Feste K~rper verhalten sich bez~glich ihrer Lagerungsm~glichkeiten wie K~rper der euklidischen Geometrie von drei Dimensionen;

dann enthalten die SMtze der euklidi-

schen Geometrie Aussagen ~ber das Verhalten praktisch starter K~rper." Consequently,

([EIN] p. 121)

G. Ludwig suggested

[LUD2] going one step further and

formulating geometry explicitly as a theory of possible operations with practically rigid bodies, and "transport".

using as basic concepts

"region",

"inclusion"

IV In 1977 I started carrying out this program in detail. completed by connecting mathematical

the theory of regions and transports with the

results on the Helmholtz-Lie

a second part dealing with mobility this approach was presented [SCHI]. Following conference, completed

problem

in Osnabr~ck

to the Fachbereich

schaften der Universit~t

Osnabr~ck

and accepted

in November

1978.

In conclusion

I should

at this

"Zum physikalischen

5, Mathematik/Naturwissen-

as the author's

like to thank K. B~rwinkel,

Habilitationsschrift

J. Ehlers, A.

Hartk~mper,

A. Kamlah,

couragement

and interest have been of great value to me. Further I

express my gratitude

G. Ludwig,

D. Mayr and G. S~Bmann, whose en-

to T. and M. Louton

for revising the translation

of my manuscript.

I have also much appreciated

Frau A. Schmidt's

rapid and accurate

August

1979

Frau P. Ellrich's

typing of the manuscript.

~ "

~'~

1977

with rigid bodies, which

The German version entitled

was presented

by chains,

in November

generated by the discussions

I added a chapter on operations

Raumproblem"

[FRE]. Together with

and distance measurement

at a conference

suggestions

this work.

One part was

~'-~" ~

and

CONTENTS

I

I. I n t r o d u c t i o n

17

2. O p e r a t i o n s w i t h rigid bodies 2.1 General e x p l i c a t i o n

17

2.2 C o n s t r u c t i o n of regions

26

2.3 C o n s t r u c t i o n of transport mappings

48

3. Regions and t r a n s p o r t mappings

61

3.1 4 Axioms

61

3.2 Points

66

3.3 Regions as point sets

71

3.4 C o n g r u e n t mappings

79

3.5 Chains I

83

3.6 C o m p l e t i o n of the group

91

3.7 Chains II

I O0

4. The H e l m h o l t z - L i e p r o b l e m

108

4.1 I m p l i c a t i o n s of the theorem of Yamabe

108

4.2 M o b i l i t y and distance m e a s u r e d by chains

118

4.2.1 Proof of "(i) =>

(ii)"

4.2.2 Proof of "(ii) => 4.2.3 Proof of "(iii)

=>

122

(iii)"

126

(i)"

132

4.3 T i t s / F r e u d e n t h a l c l a s s i f i c a t i o n

149

5. C h a r a c t e r i z a t i o n of E u c l i d e a n geometry

152

5.1 D i m e n s i o n

152

5.2 C u r v a t u r e

154

5.3 E u c l i d e a n r e p r e s e n t a t i o n

155

6. R e f e r e n c e s

160

7. N o t a t i o n s

163

1.

INTRODUCTION

This book p r e s e n t s

an a x i o m a t i c

p hysic a l

This will

geometry.

exploring

1.1.

some problems

Geometry,

spacetime), theory

contains

very general the

geometrical

identification

occuring

reduced

- via E. N o e t h e r ' s concepts

theorem

of space

groups.

p hysic a l

theories,

Geometry

is the m a i n m e d i u m

Another

aspect

If We d e s c r i b e

traced back

to a g e o m e t r i c a l

relation

"pre-theory" [LUD 3]).

The

statements

physical

in terms

precise

theory, "data"

can be of the

by d i f f e r e n t

between

these

theories.

This

measurement

notion

manner,

theory

is o f t e n

using

PT I, w h i c h

of "tracing the interis a

PT2, under c o n s i d e r a t i o n in

can be

(see

PT 2 consist of t h e o r e t i c a l

PT I. These statements in turn are

of basic

construction

of theories

Moreover,

and the c o r r e s p o n d i n g

every p h y s i c a l

of a p h y s i c a l

another

experiments of physics

and p r e - t h e o r i e s

in

PTI, and so forth.

would

where

consist

geometry

of a

is located

the outset. Therefore,

an a x i o m a t i c

some

momentum,

theories

role of g e o m e t r y

measurement.

in a more

experimental

a systematic

hierarchy

: (almost)

of the p r e - t h e o r y

interpretable Thus,

r.e.

physical

theory).

such as energy,

spacetime)

(resp.

of such a connection.

as follows

theoretical

of

for the m o m e n t

the same nature

fundamental

can be r e s t a t e d

space

Every

or q u a n t u m

there must be a c o n n e c t i o n

of the

intention

- to the i d e n t i f i c a t i o n

(resp.

formulated

back"

concepts

of

space.

(ignoring

in d i f f e r e n t

s ymmet r y

the

role in physics.

concepts

of p h y s i c a l

momentum,

different

physical

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

angular

with

as the theory of physical

a constitutive

versions

to the f o u n d a t i o n s

be d e v e l o p e d

dealing with

understood

plays

approach

formulation

of g e o m e t r y

as a p h y s i c a l

at

theory

is of c o n s i d e r a b l e

especially has no

for a t h e o r y w h i c h

pre-theory

Geometry,

interest

(at least

understood

for m e t h o d o l o g i c a l

is a p r e - t h e o r y in the sense

as a p h y s i c a l

for all others

indicated

theory,

research, but

above).

presents

two p r i n c i p a l

questions: I. How can the g e o m e t r i c a l sense may g e o m e t r y 2. W h e r e

do we know

If it is p o s s i b l e second

question

validity

of a p h y s i c a l

course,

this

For

instance,

as "logical

investigated

There

in

true,

in c o n n e c t i o n

for i n s t a n c e

(see

reference

whose

cases

the results

with

each

criteria

of the

a theory

is a c c e p t e d

of experiments.

as it appears

"conflict"

Moreover,

other opinions those w h i c h

which is t h o u g h t

[BOH]). This

could

theory

on the

Of

sur-

not be u n t e r s t o o d is at m o s t should

of v a l i d i t y

occupy

of the e m p i r i c a l

w.r.

assume

be

(see the dis-

brings

geometry

being

be d i s c u s s e d

us back

the r e m a i n d e r content

to the v a l i d i t y

to be the a-prior i base

o p i n i o n will

to e x p e r i m e n t s

a n s w e r will

problem

with

speaking,

the

[LUD 3]).

"protophysics", physics

Roughly

theory,

question

and the role of a p p r o x i m a t i o n

are n e v e r t h e l e s s

geometry,

The

in most

in what

(if at all)?

as a p h y s i c a l

is not as trivial

contradiction".

approximately

is "true"

to the general

theory.

or,

"real things"?

geometry

not c o n f l i c t

statement

be i n t e r p r e t e d

to

that g e o m e t r y

is r e d u c i b l e

if it does

cussion

be a p p l i e d

to f o r m u l a t e

as true

face.

concepts

a part of of e m p i r i c a l

briefly

to the initial

of this book.

of geometry,

of

below.

question,

To a p p r o a c h

3 scales

of d i m e n s i o n

n e e d to be d i s t i n g u i s h e d : The m i c r o s c o p i c

(~),

the m a c r o s c o p i c

the

(or "!aboratory") (L) and the

astronomic

(A) dimension.

restricted

to the l a b o r a t o r y

perception

arises

Moreover,

we will

operating

with

physical

for example

(e.g.,

our g e o m e t r i c a l

to that part of L - g e o m e t r y rulers

and compasses,

since we feel,

of g e o m e t r i c a l

aspects

or

that this g e o m e t r y

optics

is doubtful

relativity.

(e.g.,

whether

independently

as well

is

as of other

utilizing

Clearly

there

is a close

The e x p e r i m e n t s

which

a small

take place

by L - g e o m e t r y

- e.g.

permit

of a u n i v e r s a l

occur,

pretheory

On the other hand, of these more

scale

structure, interpret could

imply

that

formulate

or general

the L - g e o m e t r y

the nature

should

theories,

as a theory

of

encounter

theories has

geometries. space

processes,

are d e s c r i b e d

Hence L - g e o m e t r y

and only L - g e o m e t r y as m e n t i o n e d

the various

Such p r o c e s s e s

of such

must

in w h o s e

the

be context

characteristics

above.

be p o s s i b l y

viewed

as a limit

not only due to its m a t h e m a t i c a l

but also due to its rules L-geometry

theory

ultimately

"L-theories".

L-geometry

extensive

It

of q u a n t u m

us to deduce

as one of the p r e - t h e o r i e s

~- and A - g e o m e t r y

theories.

can be f o r m u l a t e d

between

in the L-dimension.

and other

and

mechanics.

connection

or large

of certain

geometries

to this we shall

classical

by means

or telescopes)

the c o r r e s p o n d i n g

In c o n t r a s t

and the a s t r o n o m i c

indirectly

using m i c r o -

of such theories

without

on e i t h e r

of the m i c r o s c o p i c

can only be e x p l o r e d

L-experiments

this

is

geometries.

dimensions

viewed

approach

it works well.

ourselves

and joists),

The g e o m e t r i c a l

which

confine

axiomatic

dimension, from w h i c h

and in w h i c h

rigid bodies

building-stones a pre-theory

The present

of interpretation.

of the a s s e m b l a g e

If we

of r i g i d bodies,

general r e l a t i v i t y t o g e t h e r w i t h equations of matter, respectively,

q u a n t u m theory of solid state,

p o s s i b i l i t y of a s s e m b l i n g

c e r t a i n bodies,

or,

could p r o v i d e the

thus revealing

e u c l i d e a n s t r u c t u r e of space in l a b o r a t o r y dimensions. of such a p r o b l e m of c o n s i s t e n c y

pre-theory

=

\

the

The solution

s y m b o l i z e d by the d i a g r a m

r e s t r i c t e d theory

/

m o r e e x t e n s i v e theory

w o u l d l e g i t i m i z e and e x p l i c a t e the a f o r e m e n t i o n e d

i d e n t i f i c a t i o n of

various c o n c e p t s of space in d i f f e r e n t theories.

E v e n in the case of l a b o r a t o r y g e o m e t r y the r e l e v a n t concepts - "point",

"line",

(from now on just geometry) "plane",

"angle" - have no d i r e c t p h y s i c a l meaning. r e p r e s e n t e d by "small" e x p l a i n in p h y s i c a l

spots or m a r k i n g s

"distance"

and

W h e r e a s points may be

it is m o r e d i f f i c u l t to

terms w h a t a line or a d i s t a n c e b e t w e e n two

points is. Of course,

it is p o s s i b l e to c o n s i d e r c e r t a i n p r o c e d u r e s

p r o d u c i n g s t r a i g h t edges or for c o m p a r i n g d i s t a n c e s

and to "define"

the c o r r e s p o n d i n g c o n c e p t s o p e r a t i v e l y by these procedures.

Basically, proceeds

the p r o t o - p h y s i c a l

in this way.

a p p r o a c h of the E r l a n g e n - K o n s t a n z

They f o r m u l a t e standards

group

for m e a s u r i n g devices

and s o - c a l l e d "principles of h o m o g e n e i t y " ,

from w h i c h they seek to

derive a Euclidean geometry

83 ff.).

(see [BOE]

p.

This a p p r o a c h seems to d e p r e c i a t e the e m p i r i c a l basis of g e o m e t r y favour of a n o r m a t i v e basis.

However,

one can argue,

e m p i r i c a l c o n t e n t of g e o m e t r y is then m a n i f e s t e d

that the

in the tacit

in

a s s u m p t i o n of p r a c t i c a b i l i t y of the standards or w o r k a b i l i t y of the procedures. When one tries to s t r i n g e n t l y analyze the conditions of g e o m e t r i c a l p r o c e d u r e s one must translate the p r i m i t i v e g e o m e t r i c a l operations into a m a t h e m a t i c a l

language and formulate the conditions of

w o r k a b i l i t y as m a t h e m a t i c a l axioms:

If,

for example,

rods,

the goal of the present volume.

one compares distances by t r a n s p o r t i n g m e a s u r i n g

these rods must not be deformed during transport.

s a t i s f a c t o r y to claim:

It is not

"experience shows that they are not deformed",

b e c a u s e d e f o r m a t i o n would need to be m e a s u r e d by other n o n - d e f o r m e d m e a s u r i n g rods. A universal d e f o r m a t i o n is not detectable. and does not exist,

Hence it is m e a n i n g l e s s

says the operationalist,

m e a s u r e d by t r a n s p o r t i n g m e a s u r i n g rods.

d i s t a n c e is what is

In p r i n c i p l e we agree,

w o u l d still try to improve on this argument at two points. conditions of the w o r k a b i l i t y of the p r o p o s e d operations e x p l i c i t e l y formulated. distance

but

First,

the

should be

One apparent condition in the c o m p a r i s o n of

is, that two m e a s u r i n g rods made from different m a t e r i a l s

have the same length before some transport, the same length after the transport as m a t h e m a t i c a l

axioms,

if and only if they have

(see ( 2 3 1 8 ) ) .

these conditions make it p o s s i b l e to define

the concepts under c o n s i d e r a t i o n as m a t h e m a t i c a l f o r m a l i z e d physical

When f o r m u l a t e d

theory.

terms w i t h i n the

This has the additional advantage that

we are now no longer r e s t r i c t e d to one specific method of m e a s u r i n g a quantity. theory

Each appropriate theorem of the m a t h e m a t i c a l part of the

(e.g., on the e q u i v a l e n c e of two definitions)

another possible corresponds

"operational definition",

to the same physical concept.

now yields

which n e c e s s a r i l y

This is the second i m p r o v e m e n t of an o p e r a t i o n a l i s m w h i c h does not take into a c c o u n t the de facto p l u r a l i s m of m e a s u r i n g methods. (Admittedly it w o u l d be n e c e s s a r y to study other p h y s i c a l theories such as g e o m e t r i c a l optics and their c o n n e c t i o n to rigid body g e o m e t r y in order to c o n s i d e r the full p l u r a l i s m of g e o m e t r i c a l measurements.)

In short,

the above is the s t a n d p o i n t of G. Ludwig

w h i c h we adopt.

Given a m a t h e m a t i c a l

(see [LUD 1,2,3]),

f o r m u l a t i o n of a p h y s i c a l theory,

c e r t a i n sets and r e l a t i o n s of the theory play the role of p h y s i c a l l y i n t e r p r e t a b l e terms;

the i n t e r p r e t a t i o n being either direct or

J d e r i v e d from s p e c i f i c pre-theories.

H o w e v e r the d e v e l o p m e n t and

s u b s t a n t i a t i o n of the m e a s u r i n g p r o c e d u r e s

for the n o n - i n t e r p r e t e d

terms is a c h i e v e d by a p p r o p r i a t e m a t h e m a t i c a l

constructions within

the t h e o r y . If it is p o s s i b l e to derive all terms and t h e o r e m s of the theory from the i n t e r p r e t e d terms by means of some axioms expressible

in these terms,

one has reached the a x i o m a t i c basis of

the theory.

T u r n i n g back to geometry, relations)

we have to decide w h i c h terms

of E u c l i d e a n g e o m e t r y of the 3 - d i m e n s i o n a l

s u i t e d as i n t e r p r e t a b l e terms,

(sets,

space E 3 are

i. e. w h i c h terms are as close as

can be p o s s i b l e to the p h y s i c a l a p p l i c a t i o n s of geometry.

Following

the a p p r o a c h of G. L u d w i g

([LUD 2] II and IV), w h i c h in some aspects

is due to H. v. H e l m h o l t z

(see [HEL]), we choose a class of subsets

of E 3, called

(spatial)

by c o n f i g u r a t i o n s

regions, w h i c h are e x p e r i m e n t a l l y r e a l i z a b l e

of fixed bodies.

The inclusio__nn of regions w o u l d

c o r r e s p o n d to the r e a l i z a t i o n of "sub-bodies" w i t h i n these configurations.

F i n a l l y the group of c o n @ r u e n t m a p p i n g s

lations and proper rotations

(and their products)

formed by transcould be inter-

p r e t e d as d e s c r i b i n g the t r a n s p o r t of rigid bodies.

The

formulation

considering

of an a x i o m a t i c

an a b s t r a c t

given by a r e l a t i o n relations

T on R

is subject physical These

over

points

class

structure,

basis

We will

regions

of b o u n d e d

namely

tell

open

subsets

of as

(R, fig.

minimal Other

length

chains w i t h

restriction the

of these

chains w o u l d

"globular"

of m o b i l i t y

form of the links,

means level.

Plastically

A Riemannian

minimal

each

is k n o w n

chains.

of section

3-space

on

from

if one p o s t u l a t e s

in s e c t i o n

of a m e t r i c

on the space

3 by

(R, r A m}.

C(l,k)

= {m 6 C(1)Im ~ k or m A k}. def

A substitution

is d e f i n e d

to be a 6 - t u p l e

( 1 1 , k 1 , ~ 2 , a , 1 2 , k 2) satisfying: a) i i 6 Bk.

for i = 1,2,

1

b)

n2 = P ° S ( 1 2 ' k 2 ) '

c) a: C(ll,k I) + C ( 1 2 , k 2) is a s u r j e c t i v e

mapping,

d) e k I = k2, e) Vm 6 C(ll,kl),

m A k I =>

(am = m and pos(ll,m)

= pos(12,m),

f) ~m ~ an m C n. By f),

~ is i n j e c t i v e .

In order

(2215)

to i l l u s t r a t e

the above

Example:

/JJ

//I// •

o

d e f i n i t i o n , we

/ /

provide

the f o l l o w i n g

JJ

/ I /

.

.!..,;: :

Q::': : :'.,. 11

~2 C(ll,kl)

= {n, m, kl,

klm,

klmn,

mn}

C(12,k2)

= {n, m, k 2, k2m,

k2mn,

mn}

12

3~

(2216) Lemma:

L e t be 11 6 Bml,

substitution.

Then

= ~IC(ll,ml) def

k I ~ m I and

C(ll,ml)

and ~ 2

( 1 1 , k 1 , ~ 2 , ~ , 1 2 , k 2) a

c C(ll,kl)

holds.

= P°S(12'm2)' def

If m 2

= ~ml, def

then

~

( 1 1 , m 1 , ~ 2 , ~ , 1 2 , m 2) is a s u b s t i t u t i o n .

Proof:

L e t be m 6 C(ll,ml). mAr

mI C

m implies

kI E

m and m A ml,

6 R i m p l i e s m A k I, r. H e n c e m 6 C(ll,kl).

since for all m 6 C(11):

~ is s u r j e c t i v e ,

am ~ m 2 = ~m I => m ~ m I = > m

6 C ( l l , m I)

and am A m 2 = ~m I => m A m I => m 6 C 11,mi). The o t h e r d e f i n i n g

properties

of a s u b s t i t u t i o n

are i m m e d i a t e l y

clear.

(2217)

Definition: T = (i)

Let S =

(11,k1,~2,e,12,k2)

( 1 2 , k 2 , ~ 3 , B , 1 3 , k 3) be s u b s t i t u t i o n s T 0 S

= def

(ll,k1,~3,Se,13k3),

and ~, c o n s i d e r e d (ii)

S -I

= def

(12,k2,n1,~

of the r e l a t i o n

-I

groupoid

and n e u t r a l

,11,ki),

where

are g i v e n by

We o m i t the proof,

which

(2219)

L e t ~i 6 Pos(ki)

Definition: (i)

is s t r a i g h t

~I ~ ~2 V11 def such

of

a

-I

is the i n v e r s e

, w h e r e i is the i d e n t i t y

of s u b s t i t u t i o n s

to the m u l t i p l i c a t i o n

elements

B~ is the p r o d u c t

~.

T h e class

w.r.

= P°S(11'kl)" def

as r e l a t i o n s .

=(ll,k1,n1,1,11,kl) def on C(ll,kl).

Proposition:

and ~I

where

(iii) E(ll, k l )

(2218)

and

has the s t r u c t u r e

defined (2217)

(ii) and

forward.

for i = 1,2.

6 ~I Hl 2 6 ~2 Ha

that

{Vn 6 B, (nil I and nAkl)

above.

=> n ~ k 2} =>

The

of a

inverse

(iii).

32

(ll,k1,~2,~,12,k2) We will called (ii)

say: the

~I N ~2 ~1 a n d

n2 can be



(n1'ki)

(2220)

Lemma:

equivalent

the

result

of

6 Pos

for

Proof:

n of

L e t m 6 B be a minimal

It is r e a s o n a b l e only

fixate

and v i c e

complication Therefore (i.e.

(2221)

has

we want

bodies

sucht

with

the

Let

exists

mentioned omitted,

without

interested

the

considered

when

loss

in t h e s e

s.c.

for m i n i m a l

same

there

properties.

substitutability that

no p a r t s

s.c.

in

in some

(n'k')

exists Hence

for

of k 2 are u s e d

(2219) (i).

of the

equivalent

This

following

sub-bodies

with

to

trivial proofs. "copies"

and p o s t u l a t e :

E Pos

such

that

subset. "~ ~ w'

Then

and

=> n { L).

serve

These

the

6 Pos

p ~ m 2. mI = ~ kI E

~ k 2 = m 2.

p is m i n i m a l ,

p C ~ and

p A kI

p ~ m I.

Proof: 1.1.1.

Consider

the

case

p A k 2. N o w

p ~ m 2 by

the

above

s.c.,

hence

P ~ m I• 1.1.2.

1.2.

1.3.

In t h e

ease

p E k 2 we

hence

p ~ mI.

Since

K I = Pl

there

exists

We

assert:

Proof:

and

the

infer

s.c.

a substitution

from

p A k I that

is p r o v e d

p = ~p A ~ k I = m I,

for minimal

(~,k1,~1,~,~,mT).

8k 2 = ek2 (= m 2 ) .

We write

M(k2)

= M ( k I)

~ R and

conclude

p

(see

(2220)),

34

M(~k 2) = M(m I) ~ R = M(Sk2) Further, i,i'

pos(~,m)

1.2.

= pos(llm)

£ ~1" The latter

by definition By virtue

of

since ~m 6 R, am = Bm = m.

implies

(2210) (2216)

and pos(~,ml)

(iv). By

V n 6 M(ml) , pos(l,n) (2212),

there exists

(T,k2,~2,B,l,m2) , which 2.

~2 ~ C(122,n 2)

I--

,6 "~ D ~

C(122,k2) ~

f_

!

~T

IT

%

C (121, k2) C--~. C (121 ,n I )

(The a r r o w ~ uniquely

- denotes

the i n c l u s i o n

f

determined

embedding.)

by this p r o p e r t y

and thus

a,

are

6, Y are ~-isomor-

phisms.

(iii)

Let

~

= ~ 0 ~ 0 ~, def

pos(111,n1).

Then

ak I = k I and k I & n 2. follows

from the i m p l i c a t i o n s :

k I = ak I 6 C(112,n2) k I & n 2 or k I ~ n 2 ~k I = k I ~ n 2 = ~n I => k I ~ nl,

in c o n t r a d i c t i o n

to

kI A nI. 1.1.

1.1.1.

We assume

m 6 C(111,ki)

and will

Clearly,

am 6 C(112).

Consider

the case m ~ k I. H e n c e

showam

6 C(112,ki).

am ~ ~k I = k I and

86

a m 1.1.2.

In

6 C(112,ki). case

m

A k I assume:

3 h r-am,

a k I.

It

h F" k I, k I A n 2

follows:

(see

I.

h A n2 h 6 C(112,n2) i r-m

such

~i F" ~mt i ~ m, Hence 1.2.

that

ak I

k I in c o n t r a d i c t i o n am

A mk I = k I and

We~assume

First

s E C ( 1 1 2 , n 2)

that

s = at.

It remains

r 6 C(111)

1.2.1.

If

s " l k I , ~r

1.2.2.

If s A k I, a s s u m e h 6 C(111)

(iii)

~ak

and

s = mr.

s 6 C ( 1 1 2 , k I)

3h such

(see

r 6 C(111,ki).

6[D I] = D 2 a n d

We

have

a)

a n d b)

Let

to

to

show are

m 6 C(111)

pos(111,m)

I. A s s u m e hence

m

points

s "lk I or

to

f)

c),

are

in t h e

d),

f)

s A k I.

r 6 C(111,ki).

infer to

proved

analogously.

definition

follow

s A k I.

from

(2214)(ii). (ii) ; it

e). and

m

A k I. W e

either

m

A n I . It f o l l o w s ,

m = 6m.

a)

such

k I. N o w

and we

¥ [ D 2] = C ( 1 2 1 , k 2)

= pos(121,m)

n I is m i n i m a l ,

implies

F ak I = k I i n c o n t r a d i c t i o n

immediate;

show

h P- r,

(i))

Thus

~ r 6 C(111,ni)

r ~ k I and

that

ah

3.

thus

r 6 C(111,ki).

I = k I, h e n c e

= C ( 1 1 1 , n I)

r A k I and

and

to s h o w :

~h E mr = s a n d

remains

Since

that

show:

such

note

A k I.

mm 6 C(112,ki).

s 6 D I and will

Clearly,

and

to m

3 r 6 C ( 1 1 1 , k I)

that

2.

h = al

Moreover,

have

to d e r i v e

6m = m a n d

. A n I or m that

m

pos(111,m)

"I n I h o l d s .

A n 2 and m = am = = pos(112,m)

Bm = 7m,

= pos(i22,m)

=

37

pos(121,m). 2. N o w

consider

is d i s j o i n t it can be

from

ml, .... mk,

~m ~ which From

hence

$-2m ~

(2211).

substitutions

Lemma:

of

according

in 1.1.

we

conclude event

=

6m ~ nl, of m

# ~m, we

sequence

to

which

(228).

we

infer

Hence by

~m = m.

(2213):

[]

expresses sub-bodies,

a sort

of c o m p o s a b i l i t y

can be e x t e n d e d

for

to the case

sub-bodies.

L e t kl,

ni

k, w h e r e

n i A k I for i = 1 , . . . , k . are

A k I. As

....

of d i s j o i n t number

m

pos(111,nl)

In the

infinite

= pos(112,n1)

lemma,

because

Any m i

6m. = m. and l l

(ii)

= pos(112,m).

preceding

(2225)

6n I : n I and

is i m p o s s i b l e

of a f i n i t e

latter

= {nl,ml,...,mt}.

By a s s u m p t i o n ,

~m ~ m by

pos(111,nl)

let M(m)

V i : 1,...,k,

a countable

~-Im ~

pos(111,m)

The

Using

obtain m G

that

= pos(112,mi).

pos(112,n1).

and

n I and k I, the

shown,

pos(111,mi)

would

the case m ~ nl

Further,

the n i are m i n i m a l the

following

and

substitutions

assumed:

©©

I©©I

@ @ fig.

(2226)

38 SIo

=

(110,n1,~,~1,111,n~) !

!

$11 = (!11,n2,~2,e2,112,n2)

$I ,i-I = (ll,i-1'ni'~!'~i'll li'n!i) !

S1,i

l

= (ll,i,ni+1,vi+1,~i+1,11,i+1,ni+1)

!

$I , L-I = (11,L_1,nL,v L',~L,II

L,nL )

SI,L

= (11,L,k1,K1,6,12,L,k 2)

S2,L

= (12,L,n~,vh,Yt,12,L_1,nL)

$2,i+I

= (12,i+1,nL+1,gi+1,Yi+1,12,i,ni+1)

S2,i

= (12i,n~,vi,Yi,12,i+l,n i)

$21

= (121,n~,v1,Y1,120,n I)

If V i=I,..

,L, pos(12,i_1,ni)

6

= YI' .... YL 6 ~L,...,~I def

i, w h i c h

4.2.

n. C l

imply by c o n s t r u c t i o n

4.4.

n i A nk,

k 3. T h i s w o u l d

under

n¶3' k3

violate

which are

since this w o u l d

s.c.

nI 6 NI. for ~2 ~

n3"

In this c a s e n i is i n v a r i a n t

83 . H e n c e n i A k 2 is in c o n t r a d i c t i o n

k2•

$IO

= pos(12,i_1,ni)

... SIL , S2L,...

( 1 1 , k 1 , ~ 3 , 6 , 1 3 , k 3) and t h e r e b y

w e can a s s u m e

need only apply (2228)

by

since n k is m i n i m a l .

however

the a b o v e

for all k > i,j.

5. We h a v e p o s ( l l , i _ 1 , n i ) sequence

impossible,

n I• { N 3. m NI,

all Yk' k > i a n d

to n i C

are s a t i s f i e d ,

is i m p o s s i b l e

n[ for s o m e j ~ i is also 3

ni C

is p o s s i b l e

13,i:

4.1.

4.3.

s.c.

This

to ni ~ 13, i. S i n c e the n i are minimal, t h e r e

4 possibilities

In sequel,

a substitution

substitutions

$3, i : 1 3 , i , n i , ~ i , Y i , 1 3 , i _ 1 , n i) for i=k...].

only

the

(i'2 , k , k 2 ,~3,~3 '13, k ,k3).

4. Set Sl, L = S~, k 0 S~, L a n d c o n s t r u c t

simply

implies

s.c.

Vn, (nCl~, k and nAk2) $3, k :

t h a t the s.c.

R(~)

spatial regions. written

to the

~1 ~ ~3"

is a l w a y s

satisfied

u s e d in the p r e v i o u s

since we

proof.

= PoS(~P

/////////

l I "

Y3

r-

12

11

fig.

(238)

13

50

(239)

Definition:

The

f r a m e ~ 6 F w i l l be c a l l e d

iff the f o l l o w i n g that k ~ m,

property

holds:

F o r all k, m,

In this c a s e w e w i l l w r i t e s u b s e t of i n e r t i a l

pos(12,k) (k,m, ll,n)

= pos(n,k). ÷ 12 .

jwl///////

,ni!

m

m

11

12 fig.

Axiom

J1:

T h u s we p o s t u l a t e the

further

arbitrary The

(2310)

J % ~. that

inertial

developement

inertial

following

a configura-

f r a m e s w i l l be d e n o t e d by J c F.

@iI (2311)

frame,

11, n £ B such

11 6 Bm N ~ and n £ B k D ~p,there e x i s t s

t i o n 12 6 Sm n ~ s a t i s f y i n g

The

an i n e r t i a l

frames

can be f o u n d and w i l l

of p r e - g e o m e t r y

within

some

perform

f i x e d but

f r a m e ~ 6 J.

lemma

is an i m m e d i a t e

consequence

of the d e f i n i t i o n

(239). (2312)

Lemma:

(i)

k ~ n E m and

(ii)

1 6 B m and

(iii)

(k2'm'h1'q)

(k,m,h,p)

(k,m,h,p) ÷ q1'

÷ q implies

q £ Bm a n d k I ~

p o s ( q , k I) = p o s ( q l , k l ) . (Use a x i o m B2.)

+ q implies

(k,n,h,p)

(k,m,l,p) m implies

÷ q

÷ q.

51

(iv)

(k,m,h,p)

÷ q and pos(p,k)

(k,m,h,p)

÷ q.

(v)

h [-k

and

(2313)

Definition:

(i)

A

(k,m,l,p)

(positional) =

(2210) (vi)).

(ii)

~ is c a l l e d

(iii)

Two chains

11

(~2,k2)

cyclic

danger

12 ...

iff

of p o s i t i o n s

where

11

(~2,k2)

12

T ~ (Pl,kl)

11

(P2,k2)

12 ...

=

...

two p o s i t i o n s

I

for

is e i t h e r f " o r

of c o n f u s i o n

--I

we may write:

(~N,kN).

congruent,

{ (~i,kI)

, such that

(nN,kN).

(nl,kl)

d, ~ are c a l l e d

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

÷ q.

(~ ,k ) 6 Pos

I v (~v+1,k +i), Without

implies

(h,m,l,p)

is a s e q u e n c e

(~1,kl;~2,k2;...~N,kN),

~ (nl,kl)

(iv)

~ q implies

N-chain

~=I...N-I I (~v,kv) (see

= pos(p,k)

~ [] ~, iff

(~N,kN)

and

(PN,kN). (~i,ki),

(~2,k2)

are c o n g r u e n t

iff

kI = k2. (v)

An N - c h a i n

~ =

transportable an N - c h a i n Clearly, (vi)

Any

A cyclic N - c h a i n

2-chain

(2214) Proof: (2315)

Proposition:

and each

is:

V

~ (~1,k])

T

6 Pos

there

and s a t i s f i e s

satisfying

transportable.

exists

o ~ T.

iff it is w e a k l y

(v) will

frames

(239)

This can easily

is w e a k l y

be cyclic. is j u s t : be g e n e r a l i z e d :

transportable.

[] Any

(~],k]), 11

V (p,k)

transportable

of inertial

By induction.

That

is c a l l e d w e a k l y

(using a x i o m B3).

Any N-chain

Proposition:

(~N,kN)

(p,k)

a is c a l l e d

property

is w e a k l y

...

contains

T is unique

the d e f i n i n g

11

iff V ~ 6 {I...N}

Y which

transportable Thus

(~I,ki)

two p o s i t i o n s (~5,k5)

(~2,k2)

...

can be joined by a 5-chain.

6 Pos (~5,k5).

there

exists

a 5-chain

52

Proof: Aldef=

By

(2234) (i) there exists

reg(~i'ki)


p

pi t

?..,> p/~

commutes

(~ d e n o t i n g

the c a n o n i c a l

If g : P ÷ P is a n o t h e r

map,

surjection).

we have

f o g = ~ 0 g and

map

60

= ip/~.

We will

so is ~ a n d ~

I = ~-I

write

~(~I,~2,k) mappings

(2330)

If f is b i j e c t i v e ,

= T(~1,~2,k) obtained

in this

Theorem:

Proof:

By

: R ÷ R

and

that A < B implies immediately

Finally

we

are

to

T is a s u b g r o u p

from

lift

set of t r a n s p o r t

of A u t ( R , < ) .

TA ~ ~B for all

follows

by T the

way.

T is a s u b g r o u p

(2329),

denote

of Bij(R).

If r e m a i n s

T 6 T and A,

B 6 R. B u t

(2324) (vi).

transport

to show, this

[]

mappings

to the

lattice

of r e g i o n s

(R,O}.

o n the a r r a n g e m e n t

REGIONS

3.3

by

intervals

altering

a I < a l + I b y al ~ al+ I, t h e r e prefilters,

]a,b[.

T b 6 x

with

of a,

at l e a s t o n e p o i n t " .

F the o r e f i l t e r

U D F and x = lim

generated

U. It f o l l o w s :

by

72

~ b A b % O T b < a a6x

x6a.

(333)

Proposition: uniformity

Proof:

(BI )

We in

[BGT]

Nd c b d

have

to

P

6 x.

Set

prove

a ^ T b such

N U ~ y

L NvY ,•

~-

NV

~

y,

where y £ P and ~ 6 T is arbitrarily In order to show property -I (y) of points ~U,V transitively

chosen.

(iii) we notice

y 6 P/N V are homeomorphic

on P/N V and ~U,V is equivariant.

= NvX~-C---~eJN V in order to calculate

:

that all inverse

~

~

because

T operates

Therefore

the inverse

images

we may take

image:

Ivc e JNv

= nU[~vI[eJNv ]] = ~u[JNv] = JNv/JN u.

Now consider notation

[]

an arbitrarily

by writing

chosen U £ U. We shall simplify

our

N U = N, Nx = x, T/N = T, P/N = P, JxN/N = J.

115

Let T be the Lie algebra of T and d the subalgebra subgroup

J.

Consider

any 3 6 J. The C~-mapping

onto e, its derivation

belongin~

to the

t ~ J t 5 -I , t 6 T, maps e 6

at the point e thus being a linear

endomor-

phism of TaT ~ T, which will be denoted by J

:

T-+

(419)

T

Lemma:

Let t 6 T and I 6 ~ , then

exp(It)

Proof:

~-I : exp(l~(t)).

The left hand side defines

thus is of the form exp(It'), the possible

definitions

transformation

j ~

is a restriction

9.6).

It is faithful,

(j~x=~jx=~x=j=e). definite

represented invariant

complement, Consider

inner product,

[]

of the group J, of T

no central~elements T may be equipped

(see [F&V] except with a

denoted by , such that J is

endomorphisms

to all 9, hence this holds

(see [F&V]

35.1).

d is

also for its -orthogonal

which will be denoted by K.

the following

K C-~T exp~ ~

(4110)

representation

Since J is compact,

by one of

5.6).

a linear representation

since J contains

of ~,

~, namely by the induced

(see [ABR] Def.

of the adjoint

by -orthogonal

w.r.

= t' follows

of the derivation

forms

which

positively

t' 6 T. ~(t)

of smooth curves

The assignement

a l-parameter-subgroup

~ ~T/J

C=-map

f:

~ P, that is:

Definition:

Its derivation

composed

at O, f'

f(X)

= def

(expX)

x for all X 6 K.

= Tof, decomposes def

in the following

way:

116

K~-~T

To,~

~ r& '~

~ r3.; (,~/~) ~ r::: ~'

Td

T = J • K

where

z denotes

Therefore,

the c a n o n i c a l

j

= K

quotient

up to i s o m o r p h i s m s ,

local C~-diffeomorphism

J~K

~

map

f' is e q u a l

a n d for a s u i t a b l e

O • K + O • K/J. to Id K. H e n c e

f is a

x-neighbourhood

U the

restriction f-11U

: U ÷ K ~ ~P

is a C ' - c h a r t canonical

of P a r o u n d

chart. (see

[K&N]

(4111)

Proposition operates

coincides f-1

(compare

[FRE ]

on P by l o c a l l y

= ~(k)

exp and ~ are l o c a l l y

2.7.): orthogonal

transformations

via

chart.

~IK is o r t h o g o n a l

w i t h 3~ w h e n

3 f(k)

since

to as the

I, I 4.2.).

the canonical

Since

x. It w i l l be r e f e r r e d

It is e v e n a n a l y t i c a l ,

analytic

Proof:

the p o i n t

computed

for all

it s u f f i c e s

to show that

in the c a n o n i c a l

chart.

3 locallY

T h i s means:

3 6 J and k 6 K.

It follows: f(k)

=~ 5 ( ~ o e x p ) ( k ) = 3 (exp k) : 3 (exp k) 5 -I

J.

f-1 ~ f(k) = l o g ( j ( e x p =

(4112)

3 (k),

Lemma:

k)) ~-1 by

There

(41 9 ) .

exists

[]

a local C ~ - i n j e c t i o n

T : U ÷ T such

117

that

~(y) (x) = y for all y in some n e i g h b o u r h o o d

of x.

oo

Being

Proof:

a submersion,

v o T = Id U

(see

[B&C]

Identifying

T/J = P, we may

v has a local

Prop.

6.1.4.). express

v T(y)

= ~ J where

a 6 T such that ox = y.

Hence

T(y)

6 J

(4113)

Theorem:

and T(y)(x)

such that T o p e r a t e s

Proof

(see

[F&V]

inner p r o d u c t

= aj(x)

P may be e q u i p p e d

6.4.3.1,

= y in the form

= a(x)

with

isometrically

[K&N]

T,that means:

Let y 6 U.

T(y)J

= aj,j

C -section

2, X 3.):

= y.

a Riemannian

metric

g,

on P.

The p o s i t i v e

on K ~ T ^ P has to be t r a n s f e r r e d

definite

to the o t h e r

X

tangent

spaces

T~ P, z 6 P. This m a y be done by the linear

I^ f : T^ P ÷ T ^ P, w h e r e x x z The c o n s t r u c t i o n satisfying

by

on the choice

of f. Let g 6 T also

f-] gx = x, j

Y~ g = T~ f 0 Ti j, w h e r e T ^ P invariant

f 6 T, fx = z.

does not d e p e n d

gx = z, then

isomorphism

= flg £ ~, def leaves the inner p r o d u c t

Ti j

in

(4110).

X

In this way an inner p r o d u c t transporting

maps

differentiable Consider with

Hence

-I

= T^z g

o T~ g

-I

-I

to d e p e n d

(4112),

l o c a l l y.

hence

The

on z^ in a

C ~ -

z ~ g~ is C ~ too.

fixed

and T~(hog) , the p r o d u c t

= T~ h is an isometry. isometrically

g is u n i q u e l y

the claim only

way by use of

T operates

Whereas

can be chosen

globally.

any z 6 P and h 6 T. Let g 6 T such that gx = z. T o g e t h e r

(T~g)

T~(h0g)

g~ on T~ P is d e f i n e d

on

determined

that T C o n s i s t s up to p o s i t i v e

(P,g).

[]

by the inner p r o d u c t

of isometrics, definite



on T P and

the inner p r o d u c t

linear maps

commuting

is

with

the

118

r e p r e s e n t a t i o n of J on K. At any case, remains

4.2.

a p o s i t i v e scaling factor

facultative.

M O B I L I T Y AND D I S T A N C E M E A S U R E D BY CHAINS

We now turn to the last and d e c i s i v e a x i o m of Freudenthal. developed

It is

from earlier p o s t u l a t e s c l a i m i n g for i n s t a n c e that t w o

sets of points, w h o s e internal d i s t a n c e s are p a i r w i s e equal, may be m a p p e d onto each other by a

(global)

using the notion of a metric, Freudenthal

(421)

isometry

"mobility"

(and s i m i l a r l y by Tits)

(Birkhoff). W i t h o u t

is now f o r m u l a t e d by

in the following way:

There exist two points x, y E P such that the orbit Jx y dissects

the space P,

w h e r e "dissection" means:

(422)

Definition:

Let B be a t o p o l o g i c a l

space, A c B. A is

said to d i s s e c t B, iff B~A is not connected.

The p o s t u l a t e

(421)

m e a s u r i n g distances. (3514))

is i n t i m a t e l y c o n n e c t e d w i t h the p o s s i b i l i t y of The c o n v e r g e n c e of the chain q u o t i e n t

(see

cannot be p r o v e d on the basis of the axioms RI to R5, as is

Shown by the f o l l o w i n g

(423)

C o u n t e r example:

R is the lattice of b o u n d e d open subsets

of 2 2 , T is induced by translations.

Since the chain links do not freely rotate, l(x,y,a) h l(u,v,a)/a £ R

the limit of

o

depends on the shape of the links as is shown in the following

119

figure:

] I j i

]

I I

I....

I

fig.

Clearly,

t

(.424)

the counter example does not satisfy the r e q u i r e m e n t

(421).

This is of course not accidental as c o n v e r g e n c e of the chain q u o t i e n t and m o b i l i t y are e s s e n t i a l l y equivalent,

(425)

Theorem:

as we will show.

In a system in which axioms RI to R5 hold,

the

following 3 assertions are equivaleDt: (i)

There exist two points x, y 6 P such that the orbit Jx y dissects the space P

(ii)

(Freudenthal).

There exists a point x 6 P and a n e i g h b o u r h o o d V of x such that each orbit Jx y' x # y 6 V, dissects the space P

(Tits). Further,

(iii) The chain q u o t i e n t w.r. on regions

T is a Lie group. to

(R, 0

a I < a ° and

a 2 < a ° V y 6 b:

c

conclude

of the

is f u l f i l l e d ,

~,

the

space

following

the

chain

quotient

A(x,y,u,v).

assumptions the

of

space

in

follows:

V f 6 R

completeness

and we m a y

will

image

b ~l f = ~

< I)

the

thus

that

l(x,y,a I )

Q(x,-,-)

filter

H a ° 6 R ° V a I, a 2 6 R ° s u c h

(429)

each

is C a u c h y - c o n v e r g e n t .

of the m a p p i n g

formulation

x 6 f

(428)

x 6 P and

to c o n v e r g e

on r e g i o n s ) .

The

By

is said

f 6 S ° containing

directed

Fc(P~f,~)

quotient

some

(y~ l(x,y,a)/l(u,v,a))

and

The

iff

: [R o] ÷ F c ( P ~ f , ~ )

[R o] = R o / T

gence

(R, I.

We shall i d e n t i f y K and ~ P

(see

(4110))

such that the inner p r o d u c t

on K c o i n c i d e s w i t h the usual inner product on ~ P . W e recall that w.r. linear,

to the c a n o n i c a l chart f-1

orthogonal,

: U + ~ P , f,1 ~ f consists of

locally d e f i n e d m a p p i n g s

and dim Jy ~ p - I a c c o r d i n g to

~P

÷ ~P.

(4213). We define r =

Let be y E U lif-lyll and

S(O,r)

= {~6~PILI~II=r} . We may choose y 6 P such that f-1[U] def c o n t a i n s S(O,r) and hence the image of the orbit y

= f-1 ~ c S(O,r). Y is a c o m p a c t s u b m a n i f o l d of S(O,r) of def d i m e n s i o n p - I, hence open in S(O,r). By v i r t u e of p > I, S(O,r) c o n n e c t e d and we infer Y = S(O,r). £ ~P

~

linearly.

l~,

proves:

The c o n t r a c t i o n s

O < I < I map J - o r b i t s onto J-orbit,

Hence in some n e i g h b o u r h o o d

a sphere w.r.

is

since J operates

f-1[V] c f-1[U]

each orbit is

to the c a n o n i c a l chart and thus d i s s e c t s P. This

125

(_4216)

Proposition"

In a certain n e i g h b o u r h o o d V of x each

o r b i t Jy, ~ # 9 6 V, dissects _D, if dim P > I.

Now we are ready if T can be shown to be a Lie group, = T and

(425)(ii)

follow

either by

(4215)

since then

for p = I or by

(4216)

for p > I.

(4217)

Lemma:

For all U, V 6 U the following holds:

dim P/N V = O or dim P/N v = dim P/N U.

Proof:

Let us assume U c V and p = dim P / N u > dim P/N V = q > O.

The s u b m e r s i o n VU, V : P / N u ÷ P/N V is e q u i v a r i a n t w.r. o p e r a t i o n of J J-orbits.

By

(see

(417)

(418) (ii))

to the

and hence it maps J - o r b i t s onto

the same holds for its d e r i v a t i o n

TNuX ~U,V : TNux(P~Nu)

÷ TNvx(P/Nv)'

which can be identified w i t h a

n o n - t r i v i a l linear p r o j e c t i o n ~ : ~ P

÷ ~q.

J - o r b i t s are the spheres in ~ P

~q.

resp.

As m e n t i o n e d before,

But a sphere S p-1

the

is not

p r o j e c t a b l e onto a sphere S q-l, 0 < q < p, because for instance -I (O) D S p-1 tion.

# ~ implies O 6 ~[S p-I] = sq-1 , w h i c h is a c o n t r a d i c []

(4218)

Theorem:

T is a Lie group.

Proof:

Let us assume the c o n t r a r y and consider a c o u n t a b l e infinite

d i r e c t e d s u b - f a m i l y of U according to ding i n v a r i a n t subgroups Ni,

(411)

i 6 ~ . By

and call the correspon-

(4217)

the sub-family may

be chosen in such a way that the spaces P/N i have the same dimension. Hence the submersions images

vi+1,i

(see [B&C] Prop.

: P/Ni+I

6.2.1.).

÷ P/Ni have discrete inverse

126

By

(418) (ii),

open

(see

compact spaces

[BGT]

space and

subgroups only

for

all k ~ k By

J N i / J N i + I is d i s c r e t e .

(413)

III§

2.5 Prop.

is the

topological

consequently of the

contains

f o r m JNj.

finitely

14)

many

Further, subgroup

only

i E ~ . Thus

such we

two that

that

we may

we have

P/N

different z ~ W.

~ T/JN,

hence

points

y,

subfamily

we obtain

PROOF

(4215)

that

OF

P/N k = P / N k + I ,

we may

~ : P x P+

We

consider

confine

~

of

JN k = J N k + I for

let a £ R ° w i t h

means

N k % {Id},

a neighbourhood

(see

U in such

(313))

a way,

there W of y

containing

that

because

that

N k c T(V)

z.

If

for

N k z is c o n t a i n e d 0

(iii)" ourselves

connected The

to the

case

Riemannian

dim P >

manifold

corresponding

metric

I. We

recall

on w h i c h

shall

be d e n o t e d

.

two p o i n t s

be a s s u m e d

consider

~

isometrically.

by

later

of d i f f e r e n t

which

Because

of W

a contradiction,

"(ii)

P is a c o m p l e t e ,

operates

of

J N i + I ~ JN i is v a l i d

in W and y E N k z is i m p o s s i b l e .

By

number

assume

z 6 N k y and

L e t V be a k e r n e l

the

k Z ko,

4.2.2.

number

.

O

construct

some

a finite

a finite

We conclude

and

of JN i. But JN i as a

s u m of o n l y

for all k Z K ° and y £ P, N k y = N k + I y. exist

J N i + I is a c o m p a c t

to be

x, y 6 P and

a region

sufficiently

small

a < ao,

(R,c,T)-chains

be a r b i t r a r i l y instead

of

a O £ R O which

(depending

chosen.

By

(R, 0 so small,

is an i s o m e t r i c m a p

because

there 3.6)

a l w ays

f[S(O,r)] map,

t h a t each

("minimizing"

exists

in

"convex"

a n d P is a h o m o g e n e o u s

space.

r I < r 2 ~ 6 (r I) < 6 (r 2) .

r I < r2,

such that ~(0)

f[S(O,rl)]

at a p o i n t

z 2 6 f[S(O,r2)]

and ~ : [O,~(r2)]

= x, ~(~(r2) ) = z 2. It m e e t s

÷ P being

the o r b i t

z I # z 2. H e n c e

< $(X,Z 2) = 8(r2).

The c o n v e r s e

holds

Thus

the f o l l o w i n g

locally

S(O,r),

to x. r ~ ~(r)

I, IV Th.

a geodesic

= ~(X,Zl)

I, IV Th.

of x, in w h i c h

h e n c e all p o i n t s

1 S ~(r o)

is p o s s i b l e

(4222)

~(rl)

are s p h e r e s

This w i l l

a

f-1.

f is c o n t i n u o u s .

neighbourhoods

[K&N]

y : [O,d] ÷ P such that ¥(0)

of ~ - i s o m e t r i e s ,

This

we h a v e to c o n s t r u c t

minimal.

j o i n i n g x and y. By

h a v e the same d i s t a n c e because

is "almost"

L e t V be the n e i g h b o u r h o o d

the c a n o n i c a l J

for X(x,y,a)

for any s t r i c t l y m o n o t o n e

notions

real

[]

function.

are s y n o n y m o u s :

O r b i t Jx y' I m a g e w.r. Equidistant

to f of the s p h e r e sphere

S(O,r)

{ z 6 P I 6 ( x , z ) = 6 ( r ) }.

N o w a s s u m e a ° 6 R ° is s u f f i c i e n t l y Further These

let Xo, Yo 6 a be p o i n t s

exist because

c ~P ,

a is compact.

small that

A(a)

I

S A(a o) < ~ ~(ro).

of m a x i m a l

distance

6 ( X o , Y O) = A(a).

Consider

a congruent

mapping

128

6 T such implies

that

Tx O = x.

If Yl

= ~Yo' 6(x'Yl) def 0 < r < r o.

J x Yl = f [ S ( O , r ) ] ,

Define

K(O,r)

Either

the

f[S(O,r)]

=

= ~(Xo'Yo)

< ~(r°)

{ ~ E ~ P l II~II< r}.

geodesic

y[O,d]

at a p o i n t

fs c o n t a i n e d

x I, s i n c e

in f [ K ( O , r o ) ]

f[S(O,r)]

= Jx Yl

or

dissects

it m e e t s P.

Yl

....

....../ y

x2 xI

Yo o

fig.

In the

first

In the

second

case we put m = O and proceed

c a s e we h a v e

there

exists

chain

to be c o n s t r u c t e d .

Otherwise

we

some

j 6 Jx

repeat

the

yl[A(a),d]

for y etc.

Evidently,

after

a such

that

(4223)

m steps

6 (Xm,Y)

x I = y(t),

such

that

x I = j YI"

construction

< A(a)

and

indicated

0 < t S d.

If y = x I, w e

we obtain

as

are

Because

We put

x I 6 Jx Yl

T I = jT for the

finished.

substituting

a chain

below.

between

x I for x,

x and x m of o r d e r

129

(4224)

The

m = [ A(a)

latter

geodesic ~(x,y)

~ "

follows,

and

because

are p o i n t s

~ ( x , x I) = ~ ( X l , X 2) = ... g ( X m _ 1 , x m)

on an

= g(a),

(isometric) hence

= ~ ( x , x I) + g ( X l , X 2) + ... 6 ( X m _ 1 , x m) + ~(Xm,y).

N o w we h a v e to c o n s t r u c t chain,

X,Xl,...Xm,Y

the last two c o n g r u e n t

mappings

of the

Tm+ I, Tm+ 2. L e t be ~, ~ 6

Xm+ I

xm

Y

fig. s u c h that ~ x ° = x m,

(4225)

~ x O = y and put x' = ~ Yo' y' = ~ Yo" We n e e d

the f o l l o w i n g

(4226)

Lemma:

Proof: defined

J

x

x' D J m

The geodesic

y

y'

# ~.

yI[d(X,Xm),d]

can be e x t e n d e d

for all t 6 ~ , s i n c e P is c o m p l e t e

Consider obtaining

extensions a geodesic

([K&N]

to a g e o d e s i c I, IV Th.

at b o t h ends Xm, y by a l e n g t h of A(a),

4.1.). thus

130

y'

: [~(X,Xm)-A(a),d+A(a)]

2A(a)

+ 6(Xm,Y)

< 3A(a)

Hence

the p o i n t s

÷ ~

with

< ~(ro).

= y' (~ (X,Xm)-A (a)) def

and

= y' (~(X,Xm)+A(a)) def

satisfy

~(Xm,X)

= ~(Xm,~)

= A(a)

length

By the choice

and t h e r e f o r e

of ro,

y' is i s o m e t r i ~

lie on the orbit

Jx

x'. m

Further: 6(x,y)

= ~(x,x m) + 6(Xm,Y)

> A(a)

(~,y)

= 6 (~,x m) - ~ (Xm,Y)

< A(a)

Hence

x lies

A(a),

which is

and Jx

x' m assertion.

(4226)

in the i n t e r i o r

is

J

Y connected

y',

o

of the sphere

x lies

in the

and contains

x,

around

exterior,

y with J

Y ~. T h i s p r o v e s

y'

radius dissects

the

D

proves

Xm+ 1 6 J x

just

and

the e x i s t e n c e

of a

point

x' n Jy y'. we put Xm+ I = ~ x' = ~ y',

and Tm+ I = j ~,

Tm+ 2 = ~ ~. This

yields

the chain

~ 6 Jxm'

~ E Jy,

[x,y,a,T I .... T~]

where m

=

if x

= y,

m

m + 2 if x m # y. A minimal hence

chain

k(x,y,a)

(4227)

between S m, and,

l(x,y,a)

x and y m u s t by

_< [ A(a)

Let d £ R ° be such that ll(x,y,a)-l(x,y,a')

] + 2.

a' = Nd(a)

I -< I (see

~(x y,a)

< [~(x-t-~!)] + 3 _< ~!x,y) [

A(a)

J

satisfies

(3711)).

> l(x,y,a'),

-

(not strictly),

(4224):

k(x,y,a)

'

be s h o r t e r

hence

A(a)

+ 3 "

a c Nd(a)

implies

P

131 Together

(4228)

with

(4221)

follows:

6(x,y) A(a)

< l(x y,a) '

< 6(x,y) - A(a)

6(U,V)


62

62

61 11 - 3 ~ ~--

From

pair

x % y and u % v in sequel.

61

61 / 1 ' ~ A : ~22 { ~ /

+ 3,

A -

A

62

1 1+__3_3___ X2-3

follows

B,

def

62

I A ~ 11_3

, hence:

361 61 + XI-3 B

61 /I +

3


r2

137

SSnce a has a diameter

less than ¢/3, there exists a point w on the

a-chain between x and y such that r 2 + ~2 ~ < fi(.x,w)

minimal

~ r 2 + E.

Now

fi(~,~)

s ~(.~,w,a)

(&,~,a)

~ ^

^

+ E/3

+ ~/3

^

~(u,v,a) 2 2 Sr2÷~c, which is a contradiction. NOW consider the canonical

f-l:u÷

D chart around x £ P (see section 4.1)

K

and the dilatations Dt : K ÷ K Z ~ t-Iz,

t C ]0,1].

Each congruent mapping

~ £ T induces a local congruent m a p p i n g

Dt d -1 • f Dt 1, w h i c h o p e r a t e s Intuitively canonical

speaking,

in the O-neighbourhood

f-l[u]

one looks at the congruent mappings

chart with a m a g n i f y i n g

glass. One then expects

o f K.

in the the space

to look almost Euclidean. Indeed,

by

(4111)

f-1 ~ f consists of local,

commuting with dilations.

It remains

orthogonal

to examine transformations

occuring

in exp K. We set for t E ]O,1] and X £ K

(4238)

~(t,X)

= Dt f-1 exp(tX) f Dt I def and notice that the local transformations generally be non-linear.

However,

a p p r o x i m a t e d by translations (4239) Lemma:

are defined,

~(t,X) will

fort t.T_Q1__thgy may be

of K, as we will show.

Let X,Y,K 6 K and e,B £ ~

expression

transformations

be such that the following

and put ~(t,X,Y)

= T(t,X)Y. def

138 ^

(i)

K = T(t,X,Y)

(ii)

T(t,X,O)

(iii)

T(t,~X,BX) ~(t,X)

(iv)

~ exp(tK)x

= exp(tX)

exp(tY)x,

= X,

-I

=

(~+B)X,

= ~(t,-X).

Proof:

(i)

By

(4110),

f(X)

K = ~(t,X,Y) hence (ii),(iii) locally

(iv)

are

chosen

(4239) (i) y i e l d s

and,

f(tY)

exp(K1)x

a method

f-1[U]

of

T(t,X),

= exp(tX)

exp(tY)x,

following

fact:

exp(tY)x.

consequences

from

by d e f i n i t i o n

= exp(tX)

= exp(tX)

diffeomorphic,

K I and K 2 are

(expX)x

~ f(tK)

exp(tK)x

and

=

of the

= exp(K2)x c

implies

Since

f is

K I = K 2, p r o v i d e d

K.

[]

of c o m p u t i n g

K = T(t,X,Y).

Find

a Z 6 T

satisfying

(42310)

exp

Z = exp(tX)exp(tY). !

This

can be a c c o m p l i s h e d

(see

[VAR]

written

(42311)

2.15.).

Z(X,Y)

as an a b s o l u t e l y

Z =

[ haO

where for c

o

by

the

Baker-Campbell-Hausdorff

is a n a l y t i c convergent

w.r.

to

(X,Y)

t n Cn(X:Y) ,

the c n are p o l y n o m i a l

mappings

Y × I + I of d e g r e e

instance: = O

c 2 = l[x,y]

Further

= 7!~[x, Ix,Y] ] - 7~[Y, Ix,Y] ]

we o b t a i n

a n d c a n be

series

cI = X + Y

c3

formula

a locally

unique

decomposition

n,

139

(42312)

exp Z = exp K' exp J', K' Z ~ K', chart

(42313)

K'

=

Z ~ J' are l o c a l l y

and exp:

tK,

J'

a n a l y tic,

T ÷T are analytic.

since

the a s s i g n e m e n t s the c a n o n i c a l

If we set

tJ

=

we h a v e K = ~ (t,X,Y)

(42314)

6 K, J' 6 J w h e r e

and

K =

[ t n Kn(X,Y), J = [ t n J n ( X , Y ) , from w h i c h K n can be n~O n~O c o m p u t e d for any p o w e r of t in the f o l l o w i n g m a n n e r . Inserting exp(tX)

into

(42310)

exp(tY)

and u s i n g

= exp(tK)

exp(tJ)

(42311)

we o b t a i n

and

tn Cn(X:y ) = [ tn Cn(K:J ) n~O

n~O

tn Cn(

= n~O

~ tVKv(X,Y) : ~ t ~ J ~ (X,Y)) vaO u~O

The p o w e r

series

coincide.

Since c

are i d e n t i c a l o

iff the t m - c o e f f i c i e n t s

= O, no t ° - t e r m s

occur.

For the t l - t e r m s

we o b t a i n

t 1 Cl(X:Y)

= t 1 Cl(K:J)

Thus o n l y the t ° - p o r t i o n

modulo tm-terms, of c1(K:J)

m > 1.

counts,

which

is K o + Jo"

Hence: t(X+Y) and,

(42315)

= t(Ko+Jo) ,

s i n c e X + Y 6 K,

K O = X + Y, Jo = O.

T h e n e x t terms I KI = ~ [ X ' Y ] K '

are I X Y] J1 = 2[ ' J _

K 2 = I[X-Y,[X,y]]K

I

~ [ X + Y , [ X , Y ] O]

J2 = I ~ [ X - Y ' [ X ' Y ] ] J ' where

the s u b s c r i p t

corresponding

subspace.

c o u l d be e x t e n d e d homogeneous

K

(resp.

Further,

to o b t a i n

spaces.

j) d e n o t e s

the p r o j e c t i o n

[K,d] c K

a B.C.H.

formula

is used. for

onto the

The c a l c u l a t i o n s

(reductive)

140 Now we consider K-K ____90 = t

the norm of

[ tn-1 K = L(t,X,Y), n n>1 def

Since L is analytic

in

(t,X,Y),

t 6 ]O,1]. it has a b o u n d e d derivative

at

(0,0,0),

hence IIK(t,X,Y)-Ko(X,Y) II = tlIL(t,X,Y) II S t(llL(O) ii+itlliXiiliY1iliL I It), where ilL(O) II = llK1il = II½[X,Y]KII ~ CliXliliYli and t,X,Y are sufficiently We recall K(t,X,Y) (42316)

Lemma:

small.

= T(t,X,Y)

Theregxists

and Ko(X,Y)

= X + Y and obtain:

a connected O - n e i g h b o u r h o o d

W c f-1[U] c K such that ¥ ~ > 0 B to > O V t 6 ]O,to[

¥ X, Y £ ~,

II~(t,X,Y)-(X+Y) II < ~IIXII I[YII. Let us summarize: local mappings (t)

on K by means of the class of

T (t)

= Dt f-1 def

T (t) = ~(t,X)

The group T operates

= Dt f-1 ~ f Dtl " Each local t r a n s f o r m a t i o n def T f D -I t £ T(t) can be w r i t t e n as a product

j such that X = T(t)o and j £ f-1 ~ f

£ T (t) has a domain of d e f i n i t i o n

containing

= J'. Each def -I D(~) = ~ [W] n W. def

If we Consider the class of regions R' a6R,acU

and O£a'}

= {a'la'=f-1[a], where def in K, it is appropriate to speak of (R',c,T(t)) -

chains between points the chain quotient? O and Y £ W, say minimal

in W. What can be said about the convergence

Notice

that a minimal

[ O , Y , a ' , ~ t) .... T~t)],

chain between

is just the image of a

(R,c,T) -chain

[x=foD -1 (O) ,foDtl (Y) .foDt1[a'],~ . . . I. . provided

(R' ,c,T(t))-

of

that a' ~ D(~i(t)))

But this can be achieved by necessarily

independent

~11

for i=I . . . 1. (4237),

if a' is sufficiently

of t). Let us make the following

small

(not

141

(42317)

Definition:

S&(X,~)

{YEKI iJX-YiJ~e} , a n a l o g o u s l y :

= def

S< and S=. T h e n w e can state (42318)

the

Proposition:

B r > O such that Ss(O,r)

V V,Y 6 S=(O,r) lim O6R' Proof:

V t 6 ]0,1[

l(t) (O'Y'a) l(t) (O,V,a)

Let

c,r 2 in

exists

(4237)

r > 0 satisfying

f[Ss(O,r)]

This

by

is p o s s i b l e

c W and

and will

be d e n o t e d

be such that Ks(x,r2+E)

by d(t) (O,Y).

c W and choose

c Ks(x,r2).

(4236) (iii).

= d(x,f0DtIY) < r 2. def Thus each p o i n t w on a m i n i m a l

Now apply

(4237)

for

rI

= f 0 D~ I Y lies this m i n i m a l This

proves

(42319)

a-chain,

in K s ( x , r 2 + E ) ,

hence

c h a i n onto a m i n i m a l

provided

a c b, b e t w e e n

in W. T h e r e f o r e

(R',c,T(t))-chain

the assertion.

x and

D t o f-1 maps

between

O and Y.

D

Proposition:

Let V,Y 6 K satisfy

V c £ ]O,1[H

t o 6 ]O,1[V t 6 ]O,to[

liVii =

iIYiJ = r.

the f o l l o w i n g

2 assertions

hold: (i)

V a £ R' V T 6 T (t)

(ii)

If an def = S ~ n n(1+~)

Combining

the inequalities

l(t) (O,Y,an) n+_____!__1~ (1+e)n l(t) (O,V,an) Now the assertion

dissects

if n ÷ ~.

to show that for some V 6 S=(O,r)

the O - n e i g h b o u r h o o d

portrayed (42320)

S n(1+~) n+1

follows

We are now p r e p a r e d

we obtain

in the introduction

Theorem:

the orbit J'V

W. The idea of the proof has been (see fig.

(131)).

There exists a V 6 K such that W~J'V is not connected.

JLVII = r and

(We recall

that W was

assumed to be connected.) Proof: I. Let V 6 K and

llVLi = r. J'V is a closed subset of S=(O,r),

latter dissecting assume

W. In order to derive a c o n t r a d i c t i o n

the

let us

143

(I) J ' V ~ S = ( O , r ) . Let V X,Y

denote

the

6 S=(O,r),

<X,Y>

(2)

H Y 6 S=(O,r)~J'V

2.

Let We

inner

= r

2

chains

K. S i n c e

V X 6 J'V,

> O be

n

on

~ X = Y, w e m a y

H 6 6 ]0,1[

n 6 ~ , t 6 ]O,1[,s consider

product

J'V

and

conclude:

<X,Y>

f o r the m o m e n t

[O,V, a n , T o . . . T n _ 1 ] ,

is c l o s e d

where

~ 6r 2. arbitrarily

chosen.

a n = b n U c n,

(3) b n c S < ( O , e n ) I V,e n) , (4) c n c S< (~ and

~i = T ( t , ~ V )

The

chain

minimal

for

property

chain

i=O...n-1.

is e a s i l y

could

only

be s h o r t e r ,

(5)

I (t) (O,V,a n)

S n.

3.

NOW

consider

6 > 0 according

(6)

e
O s a t i s f y i n g

r 8r ~ ~n < ~

Let

t

to

analogously

(> 0 s i n c e

3n > I + s),

hence:

"

is c h o s e n

so t h a t

I

< n

some

3r 2(I+E)

Further, (9)

(42319).

proved

4n2 (i+c) r

(IIO)

£n
O s i n c e

(1+2a) (I+E)

Consider

an = b n U cn according

(42319) (ii),

-

]

e n < 2-n

to

I E < ~).

(3) a n d

(4).

By

A

144

l(t) (O,Y,a n) lim n÷~

= d(t) (O,Y)

l(t) (O,Y,an)

(12)

B n

E ~ V n ~ no o

< l(t) (O,V,an) (I+2e)

and by

(5):

I (t) (O,Y,a n) < n(1+2e). From

(2) it f o l l o w s

that,

I V) , (13) ¥ X E J' (--

4.

< I + e, h e n c e

It(o,V,an)

Now consider We w i l l

This

a minimal

always

(14) n 2 ~ 2+e 16r

< ~r2/n.-

assume

chain

[O,Y,an,T1...~l].

n Z no,

further

"

implies:

8__~r > 1 2+~ - 2n 2

> En (9)

I 4r ~ ~n + 2 e e n 4r(I-~

(15)

en/8r)

4r > I-E where

~n/~

prove

¥ i=I...I (16) < Y , W >

is s t r i c t l y

the f o l l o w i n g

iIWll -< 3r.

4.2.

"i=I" = def

positive

auxiliary

by v i r t u e

assertion

of

(8).

by i n d u c t i o n

V W E ~i[an],

< ir[r(~+e)/n+4Cn(1+~)]

(17)

If U I

'

the d e n o m i n a t o r

4. I. We w i l l

(18)

Z en, hence:

and

T 1 (O) we h a v e

~I = T(t'UI)

0 J1'

Jl £ J'

We m a y w e l l

confine

W E ~l[Cn]"

Let W = T I W o, W O E c n, and W I

ourselves

to the case O E T 1[b n] and = Jl Wo" def

Hence

on i:

145

W = T (t,U 1)w I and r llW III = IIWoll < ~ + E n-

(19)

From

(13) we conclude

(20) + < r[r(6+e)/n+4

which 4.2.3.

(24)

is

(16)

Combining

IITI(O) II =

I IIYI} IIW-j I (~ V) II -< 6r2/n + 4r ~n en(1+e) ]

for i = I.

(22)

and

(15) we infer

IIUIII < 4r

and may a p p l y


HWoll

Ui+1 •

holds: r < n + ~n'

= +

< ~r2/n + r en' (13) (26) < Y , W I >

6 J',

W o, W o 6 c n,

WI = 3i+I W o , W2 = W The

Ji+1

% ~'

< r(r6/n+En).

hence

IIYII IIWo-l_n Vtl

147 ^

4.3.1.

Let W i = ~i+I Vi' Vi 6 bn, and ~ = a

--I

and ~i+I

= d o i, w h e r e

d = T ( t , W i)

^

0 Ti+ I. F r o m

~ V i = 0 we infer 0 6 ~[b n] and thus ^

we may apply particular,

the results

of 4.2.

concerning

II~(0) II < 4r by v i r t u e

of

Iio(0) II = llWill < 3r by the i n d u c t i o n

~I to T. In

(24),

and

hypothesis

(17). Thus we

^

may a p p l y

(42319)(i)

A(ri+1[bn])

= A(oo~[bn]) 2

-< (I+¢) llUi+111 (27)

for both


2,

"spherical

line".

S 6 + v(G2) , "spherical

imaginary

line".

geometry".

5.

CHARACTERIZATION

The be

classification singled

OF EUCLIDEAN

of section

out by two

Vanishing

curvature

(ii)

dimension

3.

It w o u l d (R,

rotation some

e =

of V and D

group

mapping

D

v 6 V,

is c a l l e d

: E + E, d e f i n e d

is c a l l e d

generated

a

: V ÷ V satisfying

for all x 6 V and det D > 0 is c a l l e d

f i x e d A 6 E,

A. T h e

form A ~ v 0 A,

a proper

by t r a n s l a t i o n s

by D(v0A) rotation

=

(Dv)

a proper 0 A, f o r

of E w i t h

and p r o p e r

center

rotations

of

156

E will The

be d e n o t e d

linear

mapping

reflection; fixed

S

A 6 E,

: E ÷ E,

the

L 1 : V ÷ V, L l v

T(~)

of E, R(~)

Now

let

[TF

(S;~,G)

S : V + V,

will

as the

Iv;

that

(S,T)

LI,

is c a l l e d =

(Sv)

of E w i t h

I > O,

be d e f i n e d subclass

as the

of open,

be the

0 A

center

= def

class

Recall

(at l e a s t

that

structure A glance (533)

the

N t°p

slightly

set of p o i n t s

P

(327)

with

the

at F r e u d e n t h a l ' s

list

shows

Theorem:

Assume

the

Then

exists

a structure

there

such

Consequently,

situation

that

axioms

(P,Nt°P,T)

(R,c,T)

is e n d o w e d

RI

N

with

a of the k i n d

represented

(p,Nt°P,T)

(~,=,¥)

(E(~) ,T (~) ,T(~) )

fig.

[3

is [ T F - i s o m o r p h i c

(R,