Möbius transformations in several dimensions

MOBIUS T~SFO~~TIONS IN SEVERAL DIMENSIONS ~-====~=~===~=====================~~~==~====;= . Lars V. ~hlfors Copyri...

Author: Lars Ahlfors

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MOBIUS

T~SFO~~TIONS

IN SEVERAL DIMENSIONS

~-====~=~===~=====================~~~==~====;=

. Lars V.

~hlfors

Copyright 1981 by Lars V.. Ahlfors

INTRODUCTION

======~=====

These are day to day notes from a course that I gave as Visiting Ordway Professor at the University of Minnesota in the fall 1980. I had used some of the material earlier for a similar course at the

university of Michigan and for a month each at the University of Vienna and the University of Graz. The present notes are a little more detailed, but still not in a definitive ferm.

My aim was to give a rather elementary course which would acquaint the hearers with the geometric and analytic properties of Mobius transformations in

n

real dimensions which could serve as

an introduction to discrete groups of non-euclidean motions in hyper-

bolic space, especially from the viewpoint of solutions of the hyperbolically invariant Laplace equation as well as quasiconfor:mal deformations. I wish to thank the University of Minnesota and the Chairman of the Mathematics Department for having invited me, and above all my friend and former student Albert Marden for having initiated my visit. I also thank the many faculty rrembers and students who had the patience to sit in at roy lectures to the very end.

I am also indebted to Dr. Helmut Maier who helped me edit the notes and supervised the typing. The typist, unknown to me, has also

my sympathy.

Lars V. Ahlfors

'TABLE OF CONTENTS ===~=~===========

I. The classical case

p.l

II. The general case

p.13

III. Hyperbolic geometry

p.31

IV. Elements of differential geometry

p.41

v.

p.57

Hyperharrnonic functions

VI. The geodesic flow

p.72

VII. Discrete subgroups

p.79

VIII. Quasiconformal deformations

p.101

I.

The classical case.

Everybod¥ is familiar with the fra.ctional linear transformations

1.1. (1)

y(z):=

where

a, b, c, dEC

C = C U (CO)}

and

ad - be

f

az +b cz + d O.

They act on the extended complex plane

which we identify with the 1 - dimensional proj eetive space

In terms of homogeneous coordinates

w = 'V(z)

P1 ( C) •

can be expressed through

the matrix equation

(2 ) This has the advantage that the Mldbius group of all

y

can be represented

as a matrix group either by means of the ~eneral linear group by the unimodular or sRecial linear group

S~ (C).

G~(C)

or

More precisely the

Mobius group is isomorphic to

where

C*

where

r=(Ol) •

is the multiplicative group of nonzero complex numbers and to

1 0

Mobius group by

In spite of this identification we shall denote the

~ ( C)

rather than

PS~ (C)

for the simple reason that

the identification does not carryover to higher dimensions. We shall of'ten sittplify normalized by

ad - bc

= 1.

y( z )

to

yz

and we think 0'£

Y as a matrix

We should be aware, of course, that

y

and

-y

2

represent the same M"obius t=:-a.n.sformation. formula for the inverse in

S~

It is useful to memorize the

:

(3)

1.2.

There is a natural topology on

}~ (C).

G~ ( C)

G~ (C)

S~ (C)

and hence also on

is connected for the space of all complex 2

X

2

and

matrices has

eight real dimensions and the space of singular matrices has only six.

The

mapping

ad~ be ) of

G~(C)

consider

on s~(C) S~{C)

shows that

S~(C)

is likewise connected.

We can

~~(C) •

as a double covering of

The subgroups with real coefficients, are denoted. by G~(JR) , S~(:R) and

~(:R) • There is also a subgroup Gr;C1R)

+

G~ (:Ia)

and S ~ (JR )

G~ (JR )

are connected, but

pass continuously from a

ma~rix

with positive determinant. is not for one cannot

with positive determinant to one with negative

determinant. Any

YEM2(JR)

maps the real axis, the upper half-plane, and the lower This is obvious from

half-plane on themselves.

...

(4)

yz - yz =

z-z

jcz+d·1 1.3. (5)

From (1) one computes y( z) - y( z ')

:=

2

3

and in the limit for

zt-+z

y'(z) =

(6}

ad. - be

(Cz + d) z ~ yz

In particular,

2

We rewrite (5) as

is copf'Orr.lal.

1/2 1/2 y(z) - Y(ZI) = Y'(z) l' Y'(ZI) I' (z- z')

(7)

where it is understood that :=

with a f:xed choice of the square 1 .. 4.

Jad- be cz+d

~oot.

The cross-ratio of' four points

z, z I

,

s,

S'

in that order is

d,e:fined by

zt - 6 z' - (; I

(8)

when tbis makes sense, i.e. When at most two points coincide.

B.lf use of (7)

it follow's that

(z , z' , " ,I) :for the ·derivatives cancel against each other..

In other words, the cross-

ratio is invariant under all Mobius tr&nsi'Qrmations .. The cross-ratio is real if and. only if the four points lie on a circle

or

strai~ht

lines.

Note that our definition makes (z , 1 , 0 ~ (0) = z

4

From (6) one obtains

1.5.

(10)

y"(z)

2c

yl (z)

cz+d

and hence

~:it: ~

(11)

if

c

i

O.

=

It follows that

(12) and for

l!'"

Z

(13 )

D

t:

~: ~

= -

~

•

More explici t,ly, this is equivalent to the vanishing of the Schwarzian derivative:

(14)

III It 2 II 1 (.Y:.) . " 2 ::: 0 :L _ _3 (.Y.:.) = (Y...)' __ y' 2 "VI y' 2 yl

There is obviously a close link between the cross-ratio and the Schwarzian

(15 )

for arbitrary meramorphic

f.

For those who like computing, I recommend

proving the formula

(16)

(f(z + ta) ,f(z + tb) , f{z + tc) , fez + td)):;: (a",b,c,d)(l+

t

2

6' (a-b) (c-d)S:r(Z»

+ o(t 3 ) .

5

1.6.

The half-plane.

half-plane

S~(E)

We return to the action of

H2:;:: {z :;::

X

+ iy ; y > OJ.

on the, upper

We use the notation

G;;:

S~ (JR)

(or l\~(1\) ). Let

K denote the isotropY' group of

From

i.

ai+b = i ci +d

we get

a ;;: d ,

2

a. +b

b:;:: - c ,

2

;;: 1 •

Hence

G

sin cos

COS

(17)

g;;:

- sin

K ~ 80(2)

In other words,

different elements of

Let

(

•

e

e

More precisely,

and

8+ n

M2 (1&)

•

1 u ( 0 l) •

The group co.

give

~

A be the group of homothetic transformations

simila.rities and also the isotropy group of

( : t 1) , NA

t

>

0 ,

is the group of

It is simply transitive on

as seen from ( lu),t o1 t

0 )(')

ot

Hence

2

~

-1

:;:: u+t i

G:;:: NAK ; this is tile lwasawa decomposition.

decomposition

g:;:: nak.

(18)

g:=

Every function

(lo

Rxplicit1y, if

X )

(y1/2

f

on

~

G / K.

Each

0) (

cos e sin -sin e cos

If

can be lifted to a fUnction

Conversely, a f\L~ction on

if it is constant on the left cosets K \ G ).

has a unique

e) e

y-l/2

(many authors prefer

g EG

g(i):;:: x + iy , y > a , then

0

1

defined by !(g) :;:: f(gi) on

e) e

but the same element of

K

and N the group of translations

H2

g E K has the form

g K.

~

!

on

G

G becomes a function is identified with

6

1.7.

111e circles orthogonal to the real axis, including vertical lines, are

mapped on each other by all. g E G.

model of

pyPerb~lic

z=-, Z2Erf

They are the straight lines in Poincare t s

or non-euclidean geametry.

lie on a unique

n.e. line

Any two distinct pOints

(Fig. 1)

Fig .. 1 A distance can be d.efined by the point-pair invariant

(19) the invariance is seen from the relation

(20)

For

z2 ... zl we obtain the infinitesimal invariant ds

(21) ('ire have

dropped a factor

2 ).

;:;:

~ y

This defines the Poincare m.etric on

It leads to a new distance function d.(zl ' z2) (22 )

2

H

defined as the minimum

7

over all paths from

zl

to

z2.

The shortest curves are the

To find the geodesic we use agE G which maps

zl' z2

It is readily seen that the vertical line segment from

~eodesics.

on iy1 ' iy2 • iy1

to

iy2

is

minimal, and thus

(23 ) It follows tb.at the geodeSics are the orthogonal circles. The relation between

8

and

d

is deterrrl.ned by

from which we conclude that 5

The distance function

+ d( z2 ' Z3)

if'

z2

d

:=

d 2

tanh-

is ad.ditive on

lies between

We can use (24) to verify that

fiwe .. lines, i.e.

zl and 8

d(Zl' Z3} = d(zl' z2)

z3 .

is a distance :function, :ror if

5 = tan hS!' < 2

1.8.

There is a dif:'erent way of computing

A.ciditivity at once.

d(zl' z2)

We refer to Fie· l and compute

which shows the

(z2' zl' ~l' S2) -

If the circle is mapped on the imaginary axis the cross-ratio becomes

8

This proves the relAtion

In this form "We may even regard

negative when the order of'

d(z~,

zl' z2

is

z2)

as a signed. distance which becomes

reversed~

The additivity on a geodesic

::ollows at once from

1.9.

The length of an arc is

Jc My and the area of' a measurable set E is

J E

dx~Y Y

It is a useful exercise to compute the area of a n.e. polygon

P

(Fig. 2)

"I

/

I

,',

I

, 1

I

(

I

r I

f

.....

,

.....

,

"

" \'

\ \

\

\

, 1

I I

t

\

I Fig. 2

9

By Stokes I formula

u

Each side is an arc change change~

0=

dx Ady 2

r

A =

p

Y

z - a = re

o~

i8

~

and,

the a.ngle of the tangent.

y

;::: -

A =-1:: f3

a

v

2n.

If the outer angles are

one obtains

v - 2:1( = (n - 2)1C - E a v

' .. 10.

Tt is easy to :;>ass from the half-plane

('E C ;

I 'I

< I}.

measures the

For a simply connected polygon the total

including the jumps at the angles, is

and the inner angles

J de

ae

tf

t,o the uni.t disk

We want to choose a canonical Meoius mapping

A good choice is to let

z = 0 ,. i ,CJ)

correspond to

,=

B2;;;

.; ~B2 •

-i , 0 " i .

This

gives

z-i

(26)

,=i-;+i' We introduce the notation

Z

,*;:::

, with respect to the unit circle. over into

=

r+ 1.. • .;, -1.

C-=i'

1 =

,

...L-

1,1 2

for the symmetric point of

B.1 the reflection principle

"C* and by the invariance of the cross-ratio

(27)

This shows that the point-pair invariant in

B2

'2 (1- tl '2 I 1'1-

1

is

-

z ,z

go

10

a..l1d the correaponding Poincare metric is

(28)

(r'emember that we had suppressed a factor

2).

It is again trivial that diameters are geodesics and hence the same is true of all circles orthogonal to the unit circle. !:

The di stance from

0

> 0 is r

d ;;; d(O,r) =

J

o

Note that

l+r 2dt ;;; log l-r' 2

I-t

r = tanh ~

'

8(o,r); r .

Another observation is that

1+r (r , 0 , -1 , 1) ;;; - 1 -r If the geodesic through

'1" 2 meets the unit circle

in

(1)1' ~

we have

thus

From now on we prei'er to use the variable

z

in the unit disk, and

with the notation of Fig. 3 we have thus

Fig. 3

to

11

(29)

l.li.

The self-rnap'pings of

(30)

l! Z

B2 which take = e

ie

a

into

0

are of the form

z-a 1-

az

We want t:l derive this formula in a geometric way.

Fig. 4 For this center

a*

carries

a

pur~ose

we construct

a*

as shown in the picture. into

0

and the orthogonal circle with

The reflection in this orthogonal circle

and an arbitrar.y point

z

into

2 * a cr z = a* + (a* I 1 -1) (z - a*) = - '::" a a

To obtain a sense-preserving mapping we let in the line through the origin perpendicular to

reflection is expressed by

(31)

zl-?Tz =

a

as. a..

z-a

1- aZ

be followed by reflection If'

a.rg a =

a

this

and we end up with the mapping

z- a 1 -' 8.z

The most general mapping is and it is hence of the farm (30).

T

a

followed by a rotation about the origin,

We make a note of the formulas

2

T' (z)

a

l-ITa z

, 1- laJ (1- az)2

l2

1 1 ..

all well known to conformal mappers.

of the Poincare metric. T' (0) > 0 a.

fz{

,

2

The last formula expr'esses the irtvariance

The norwAlization

o~

Ta

•

Note that

TO:; -a

a

and hence

T

-~

= T-

a

1

is such that

T'(a) > 0 and a

13 II. 2.l.

TPe general cas,e.

Before passing to the general case we shall pay special attention to

the group

M(If)

of' !'-1'obius transformations acting on the upper half- space

,X ) ,x > 0) ~ Already ?oincare was well aware that the 3 3 action of M(C) can be lifted to H3 or if' one pref'ers to all of JR3 • *)

H3 : (x ::; (Xl' x

In fa.ct, any

2

y E M( C)

is a product of reflections in circles (or straig..ht

lines), the circle determines an orthogonal hemisphere (or half-plane) and the reflection extend.s to a ref'lection in the hemisphere.

One has to show,

of' course, that the end result does not depend on the particular factorization of'

y.

The three-dimensional upper half-space is very special because of' the .fact that one can make use of quaternions in a very elegant manner. The quaternions can be identified with matrices (~ ~) -v u

(1) where

k= (~~)

and

i=(10)

In fact, if' we write

u, vEe

u=,\ +i~ , v=v1 +1v2

O-i

'

we obtain

u uy) = u + ' . k ( -v ~~ + v IJ + v 2 l

(2 )

= u + vj

The conjugate of

Izl

is given by

have used the rule

Izl2 aj

z::; u + vj = zz = =

ja

luJ2

is +

-z = u- -

vj

and the absolute va,lue

Iv/ 2 • When computing tr_€ product we

for arbitrary complex a.

We have replaced the notation

M2(C)

by

M(C) •

14

Points i!l. (y

>0

if

1R3

~ow be denoted by z:;: x + y j

will

z E~).

Suppose

yES~(C)

a b Y = (c d)'

x E C , y E lR

with

is given by

ad - be = 1

•

We define the action by yz == (az+b)(ez+d)

We need to show that these quaternion

-1

= (zc+d) -1 (za+b)

expressions axe equaJ., and that

yz.

is a special

or the same form as z.

In the first place, the two expressions (3) are equal if and only i f

(zc+d)(az+b) - (za+b)(cz+d) = 0

•

This works out to zcb + daz - zad - bcz = 0

which is true because

ad - be

is real.

The next step is to actually compute yz =

yz..

To begin with

~az+b)(cz+d)

Icz +d1 2 Here cz

+ a. = ex + d + cyj

cZ"+d.

= ex + a - cyj

az +b ;::; ax+b + ayj

- 2 + [-(ax+b) cy+ay(cx+d)] (az + b)(cZ+d) = (ax+b)(cx+d) + acy : (ax + b)(Ci: + d) + a'i:.y2 + yj

j

15

and thus

yz.::;

(4 )

- 2 ( ax + b )( -ex + d) + acy + -yj Icz + dJ 2

which is of the desired form.

2.2.

Also,

fez

+d1 2

==

ICX+dI 2 + Jcl 2 y2

We shall also compute a. formula for the difference

"fZ - yz t .

0

We

write it in the form.

y.z - yz. T = (az + b)( cz + d) -1 - ( z 'c + d.) -1 ( Z f a + b) (5)

= (z'c+df1[(ztc+d)(az+b)- (z'a+b)(cz+d)](cz+d(l ::; ( Z I C

+ d ) -1 (z _ Z f ) (cz + d) -1 •

This is at least similar to the corresponding formula (1.4) in the ccmplex co,se. On

passing to the absolute values we obtain

Iyz

(6)

- yz t

j --

Iz-z'l

Icz + d I Iez' + d [

and infinitesimally fdy(z) I

(7)

Comparison with

(8) is invariant.

(4)

:;;

laz\ Icz +dl

shows that ds::;

~ y

2

16

The cross-ratio should be defined by

By

use of (5) one obtains

Thus the matrices correspond.ing to the quaternions (Zl ' z2 ' Z3 ,z4)

are similar.

(YZI' YZ2 ' "(Z3 ' YZ4)

and

Because the absolute value is the square root

of the determinant and the real part is half of the trace of the matrix (1) it follows that the absolute value and

~he

real part of the cross-ratio are

invariant. The stabilizer o:r

The full group M(H3 )

j

is the group of unitary matrices

is ~ransi tive, for

yj::; u

+ vj,

v > 0 , by taking

-1/2)

uv

v -l/2 (,

2.3.

The quater nion technique works only for

elegant it is not particularly useful. on the entire space

JRn •

:M(~ )

and even if it looks

We shall now again deal with actions

We shall use the notation x:;::o (Xl' •.•

and when operating with ma-;rices we treat

of similarities consists of all mappings x~mx+b

x

a.s a column vector.

,X

n

) E lRn

The g.coup

17

where

bEEn

and

ill

is a conformal matrix, i. e.

m:;;;).,k

with

>

0

x*

,

A

and .k E O(n) Reflection in the unit sphere is defined by

(10) 2

where of Course

Ix 1

:;:

x 17 x * :::: Jx :;:

(JO

:2 + ••• +xn •

We use mostly the notation

2

~

=

m ,

J~

:;;; 0)

sometimes one needs a letter symbol for the mapping, and then we use Obviously, matrix

I

;l

= I , the identity mapping.

I

but

J.

will also stand for the unit

n

We adopt the following definition: Definition.

The full MObius group

similarities together with

J.

M(~n)

is the group genera.ted by all

M(T)

The M"obius group

whose elements contain an. even number of' factors

similarities. In other 'Words, I

do not use

n M(lR )

n M(E )

J

is the subgroup

and sense-preserving

is the sense-preserving Mooius group.

Note that

as being too J)edantic.

The derivative of a. dif'ferentiable map:ping

f

from one open set in

to another is the Jacobian matrix f' (x)

or

Df(x)

with the elements

(11)

f' ' (x). . lJ

=

of.

~

':!.v-

O-CO-j

The derivative of a similarity matrix

m.

The matriX

Jf (x)

;::;

D. f. (x) J

1

yx. = mx + b

is the constant conformal

bas the components

JRn

18 2x. x.

for

J

~

(12)

JXJ2 x

'I

0 .

It is almost indispensable to adopt a special notation far the ubiquitous lLatrix

with the entries

Q(x)

x .. x. = ...1:......J. Q() x .. 2

(l3)

Ixl

lJ

This enables us to write (12) in the for.m 1

~

Jt(x) =

(14 )

IxJ

(I - 2Q(x)}

This is perha:gs the most importa.nt formu.la in the whole theory of Mooius

transformations. 2

From Q = Q we obtain (I - 2Q)2

which means that

x"l

I - 2Q E O(n).

Thus

=I

0 •

It now fellows by the chain rule that any

is a conformal matrix for each

J' (x)

A(JRD) YEM

•

For

is a con£orm.a1 matrix for

~~'b ius transform.a t ions are In other words, al1 lYlO

With a suitable interpretation at Definition.

y. (x)

any y E M(mn )

~ E O(n) such that1Y'(x)1 cf scale at x, the same in

co

ar..d.

-1

Y

we denote by

In other words, all

directions.

C ant ormal

co.

I yt (x) I the positive number

iy'(x;]

is the linear change

19

Another application of the chain rule proves that

[yx-yyj

(l5)

=

In fact, this is triv-ial when

Iv'(x)1 I/:2 'V

IVf(Y)1

1;:2

!x-yl

is a simila.rity, and for

IJt(x)IIJI(y)llx-yl

If

'V 1

and

'Y 2

satisfy (15) then it is also true for

I'VI Y2X lyi(Y2X )I

2

'Vl 'V2yl =

J

'tore obtain

by (14) •

VI '\(2 :

IVi ('V2 X ) 11./2 I'Vi (Y2Y) [1./2 IY2 X -

{' 1V2(x)1 1/2 1Yi(Y2Y) 11/2 I'

1/2

(Y2(Y)

11(:2

*)

Y2yl

Ix-yl:= I (YlY2)'(X)(

1/2 f'

:=

(Y1 'V2)'(X)(

Ix-YI As in the complex case we wish to use (15)

"jO

prove the invariance of

the cross-ratio, but this time there is only an absolute cross-ratio

(16)

Ia,b,c,d I

--~ la::df

~J : Tb-dT

-which is obviously invariant in the sense that

(17)

Iya , 'Vb ,

yc , yd

I = Ia , b , c , d I

We can use the invariance to prove that circles are mapped on circles. Indeed, it is a classical theorem in

JR2

and

JR3 , extendab1.e to

lRn ,

Ij:2

20

that

a, b , c ) d

la-blle-d1

+

lie on a. circle in cyclic order i f and only if

Ib -clld-a!

fa -cllb . . dl • This condition can J a , d , b , c I + Ie, d , b ,al = 1 •

in the invariant form.

=

be -written

As a second application of (17) we prove the :following important lemma. Lemma 1.

Proof'.

Y leaves

If

If

ym ;::

00

fixed, then

0,

with

~

a

T

are both fini te ~ then the most general

b

Vb;:::

CD

is of the form

yx ;::: m[ (x - .0) * - (a - b) *]

(18)

m is a consta..r;t conformal matrix.

where

In fact, it is clear tha.t lnLa

CD.

2.4.

We denote by

n B =

n

A

y E M( JR )

(x; Ixl

A.

n

Sn-l;::: {x;

takes

a

into

0

and

b

the subgroups Which keep

M( 13 )

< l} invariant.

the unit sphere

(x - b)* ... (a- b)*

Because Mobius transformations are bijective

lxl ;: :

I}

and the

ext~ior of "En are also in-

variant. Lemma 2.

If'

Y EM( aD) and

yO:; 0 ,then

y

is a rotation (i. e.

yx;::: kx ,

k E O(n) }.

Proof.

for

Ixl

If = 1

y~

=

CD

we know

by Lemma 1 that

it follows that

Assume now that

'1. 100 ;::: b

In;:;;

i

Ixl;::: 1

But

and beca,use

k EO (n)

CD.

Then (18) implies

I (x - b) * + b * I ; ; for

V =!OX

const.

Imxl

= 1

22

Ixl

Ix-bllbl Ix - b I

Hence

2 .. 5.

:=

for Ixl

canst.

= 1 , and that is impossible since

We shall now determine the most general

exactly as in l.l1 for

n == 2

b

f

0 •

V E M( Bn) • This is done almost

eKcept that we do not have the complex notation.

We begin by proving a non-trivial identity: Lemma

?~oof.

3.

As it stands (19) does not make sense for

that both sides tend to

0 for

y ~ O.

y

= 0 , but it is obvious

We assume now that

y

f

0 and

write Ax ==

(x - y ** )

(20)

In other words, we trea.t

y

a.s a. constant and have to show tha.t

Ax = Bx •

It is immediate that

* ==By* = AfYJ ::;: Boo = 0

Ay

Therefore,

~onstanttl

0

and

But then

co

00

are fixed points of'

At (x) Bt (x)-l

AB-

1

and by Lemma. 1

is also constan~, for

(AB- l ) 1

is

23

-l

= AI(E (AT B'

From (20) and

-l

x}

X)· B'(B

-1

-1

=

)oBK:: canst.

(14) AT(X);:

1-2Q(X-~*)

Ix- Y*I (21) B' (x)

and.

for

Hence

x = y

lY12{1 - 2Q(y)) (I - 2Q,(x* -~)) (I - 2Q(x) ~

::

l:x:* _y1 2Ix\2

we find

AI (x) = Bt (x)

x, and since

for all

AO = BO

= -y

it follows that

Ax,=Bx.

Direct comparison of'

AI (x)

and

(I - 2Q(y» (I - 2Q(x - y*)

(22)

Bt (x)

a.s given by (21) yields

= (I - 2Q(x* - Y»)(I - 2Q(x»

•

This is an important identity.

2.6.

We repeat the construction that was given in 1.11 and. refer to the

same :figure.

* Sn-1 (a,

Given

a E~

(a

(1 a * I2 -1) l / 2 . ) 'WJ.th

intersects

n-l

S

f

0)

center

orthogonally.

we construct a*

a * and the sphere

. and radl.us

j 1- 1}2 la /

The reflection in this

sph~e

IalI

. wh~ch

is given by

24 cr x = a·* + (a. I *12 - l)(x - a * ) * a

(23 )

We let

cr

a

perpendicular to

be followed by reflection in the plane through the origin a.

It is easy to see that this second ref'lection amounts'

to multiplication with the matrix

y' and Fig. 5

=

1- 2Q(a) "

{I-2Q(a»y=y-

In fact

~

lal

shows the location of y' .

Fig. 5

We now define the canonicaJ. mapping T x

a

= (I - 2Q( a»)

x a

n y E M{lB )

and conclude that the most general kTa with

(j

with

ya

=0

is of the form

k E O(n) •

The explicit expression for identi ty ( 19 ) •

It is clear that

Tx a

be~omes

much simpler if we use the

(1-2Q(a»)a* = -a* and thus we obtain

25

I -*1 2 - 1) (1 - 2Q(a»)(x - a) **

TaX = -a* + (a

Replace

Y'

a

by

in (19) and multiply by

** (1 -2Q(a»(x- a)

:=

I - 2Q( a).

•

We attain

a + Ia 12( x * - a) *

and T x a

:=

-a* +

(I a .*1 2 .. 1)a

+ (1- 1a 2 )(x* - a) *

I

•

Fina.l.l.y,

(25 ) This is as simple as one can wish, but if cae wants to avoid the

notation it can (26)

easi~

T

[x,a] =

where

a

be brought to the form (1-

x =

la 12 )ex- a) - Ix-al 2 a [x,a]

Ixllx* .. al

Iallx - a *1

:=

2

and

We collect the various expressions for a)

T x

b)

T x

= -y

c)

T x

=

(1 -

=

(1_\y\2)(x_y) [x,y]2

y

y

T x y

Tyx

= (r-2Q(y»(y*+(jy*1

y

d)

x*

2

that we heve derived. -1)(x_y*)*)

2 + (1- lyJ ){x* - y)*

IY12)x

.. ( 1 .. 2eY + [x,y]2

Iy [ 2 U"

lx- yJ 2y

26 Recall that

[JL,y} =

Ixlly - x*J

IYlix - y*1 .

;:;:

a) and b) are easy to differentiate and yield v 12 (I-2QY» « tr'(X) = 1 _ 1 '''"2 y [x,y]

I -2Q (x-y

*»

(27) l .. =,

IYI2

2

[x,y}

(I - 2Q( x* - y) )( I - 2 Q.( x»

..

We have already remarked on the id.entity of the two matrix products. choose to introduce the notation

ll(X,y) ;; (I - 2Q.(y) )(I - 2Q(x - y *» which leads to

= ~(y,x) T

~(x,y)

•

Note that this can also be written

(28)

~(x,y)

!:l(Y,x) ;::; I

It is helpful to View the formula

(29)

)X~

Tf (x) = Y

1[x,y]

ll(X,y)

as a representation in polar coordinates: ( 30)

IT'(X)I

=

is the "absolute" value and

l ...

lyl:

[x,y]

y

a(x,y)

is the "argument" of

We make a note of the special values

Tt (x) • y

We

27

[;.rl 2

T' (0) :;; 1 -

(31 )

y

Also,

T 0 = -y .. and hence Y

~

laborious.

T

= T-y y

ITy xl

Direct computation of

1

1

.

from any of the formulas

a) - d)

is verY'

Fortunately, we can use the difference formula to obtain

ITy xl :;;Ix-yl -rx,yr

(32)

NOW a short

com~taticn

1- IT

(33)

y

leads to

x (2

=

(1Ixl 2 )(1- Iyl 2 ) . ' -- 2 . .,.... -

[x,y}

Together with (29) we have thus

-

IT'(X}! y

We have thereby proved the invariance of the Poincare llletric

(35)

ds =

2}wd 1-

Ixl 2

lt is again seen-that every orthogonal circle is a geodesic. hyperbolic distance from the

origi~

is

d( O,x) = log ~ "l=liT and more generally

The

28

I

(37)

d (x, y ) = log y , x ,

S, II

where

2.8.

g , 11 I

are the end points of the directed. geodesic from

There is a close relation between

TyX

and

rxr.

x

to

y.

To derive it we shall

first prove:

Proof: Iq

Let

= By = 0

Ix

and

be the left and right hand side of (38).

Rx

l

o.

U- ( C) ==

and hence

Clearly,

But

where the left hand equation is obvious, whereas the right hand equation

.follows by application of (34). -1 I {La )(0);

which proves that

IR-

1

= I

Thus r_1

r

L (Y)R (y) .

Apply the Lemma with

side is

and one obtains

T~

T' (y)

=-

= 1

{~(y) ,

TyX.

y == T • x

The le:rt hand

29

or T y:;:; -A(y,x)T x

x

y

(39)

is not defined for

Remark.

T y

T x = -y

~ by

y

continuity.

I y{

:;:; l

According to

T~ :;:; 8{y,X)Y for

but by formula b) we should write

l39 ) it follows that

Iyl

=

1

(40)

Tx y

One more formula.

on

=

6(x,y)x for

[xl:;:; 1

Differentia.tion of (38) with respect to

x yields

equating the arguments in (29) one obtains

~~ 6( 'VX , 'Vy) TY'1iTT ::, TY'T:YTf 8( x, y )

(41)

2.9.

Although the formulas for the selfmappings of

tF

are simple enough it

is sometimes more advantageous to concentrate on the geometric picture .. Recall that every

(42) where

(43)

'Y E M( Bn)

, Y f I , has a canonical representation

y = kT So -

-1

Y

0

o..nd

kE 00(.0) •

a

Since

Jkl = l

we have

30

tx;

The set

l v' (x) j

a*

center

= l}

Nothing prevents us from considering the full sphere rather

than only the p~t inside and we denote it by

I(Y)

is nothing e.lse than the orthogonal splu:!l'e with

K(Y).

It is called the isometric sphere of

Bn.

Moreover, we denote the tnside of

and the exterior by E('Y).

K(Y)

V, by

Observe that the identity has no isometric

sphere. The chain rule applied to

-1 y[y xl

=x

yields

and of course just as well

From this we draYT the following conclusions:

'V-1

maps

1(y.. l)

on

E( V)

Of course this does not determine

and

'Y

l E( 'V- )

on

I( y) •

completely, but only up to rotations

about the conmon axis of the unit sphere and the isometric sphere. Furthermore, the restriction

-1

Y : K( V) -. K( Y )

is an isometry and hence

a congruence mapping both with respect to the eucl:.dean and non- euclid.ean geometry.

31

III.

HYPerbolic geamet;r

This is .a. short chapter devoted to two separate tasks:

3.1. 1

0

2° 3.2.

Let

A , B ,C

Derive some formulas of hyperbolic trigonometry ..

Show how to represent Mooins groups as matrix groups. ABC

n be a non- euclidean triangle in B •

We denote the angles by

and the n.e. lengths of the opposite sides by

a, b , c.

we

shall

prove: I.

(1)

cosh c II.

(2 )

3.3.

The hyperbolic law of cosines:

= COM a

cosh b - sinh a sinh b cos C

The byperbolic la.w of siBes sinB sinh b

sinA = sinh a

PrDOf of I.

=

sine sinh c

We may assume that lies in the

Xl - axis and that

b

consider the case

n = 2

C

=0

, that

x x 2 - plane!' 1

a

falls along the positive

It is thus sufficient, to

and we may use the complex notation.

Fig. 6

The points

Hence

B

B ~~d

and

. A are at euclidean. distance

A are

~epresented

2'a

tanh

by

tanh

and

'2a

and

b tanh 2

e

iC

tanh

2'b

from

o.

The point-

pair invariant is

[tanh

B(A,B) =

(3)

~ ... tanh ~ eiC

I

11 - tanh

a;

'2

I

tanh

b

I

-iC

'2 e

tanh .£

I :;.

2

Just as in ordinary trigonometry all the hyperbolic fUnctions can be expressed rationalJ.y through

tanh

2'x .

The formuJ..as are

2 x

cosh

X

:;.

J..+tanh

'2

2 1- tanh

!

2 tanh

sinh X :;

2

!2

From (3) 2 a 2 b ( .L + th 2') (1 + th '2)... cosh c

;;

2 {l- th

~)

4

th

'2a

(1 ... th2

th

~

b 2'

cos C

)

::: cosh a cosh b ... sinh a sinh b cos C

3.4.

Proof of II. cos C

. 2 S1n

C

:;

2

8in C 2

=

From (1) : ;: ;

cosh a cosh b - cosh c sinh a sinh b

2 2 2 (ch a. ... 1) (ch b ... 1) - (ch a ch b ... ch c) 2 2 ' sh a sh b

222 1 ... ch a ... ch b ... cit c + 2ch a ch b ch c

sh c

The synmetry prove s (II),

222

sh a sh b sh c

33

3.5. In this section we want to show that MO~.n-l) is isomorphic with M(B n ). For n = 2 this was already done in Ch.I by observing (tacitly) that

M(R1 )

2 M(H )

on the other hand

M(H 2 )

and

and

a conformal mapping from

H2

are both equal to PSL 2 (R), while 2 M(B ) are isomorphic because there is

to

B2

In the general case we begin by extending an arbitrary • • to a mapp1ng 1n

ya

n

=a

, yb

=

•

For t h·1S purpose we ~"dent1. f y

Rn • Recall (Corollary 2.5)

{x =o} of

subspace

with

M(-'rTIl) n

ro

,

b F

~

~as

Rn- 1

that any

with the ye:M(R

n-1

~

a unique representation of the form

yx = rorex-b) * - (a-b) *' ]

(4)

,....

=

Ak, A > 0, kE O(n-l). If we replace m by m we obtain a n mapping of R whose restrictio'n to Hn is an extension of y • where

m

should be replaced by yx = m(x-a». This construction shows that M(Rn - 1 ) is isomorphic to M(H n ) • It remains to prove that M(Hn ) is isomorphic to M(B n ). For this it suffices to find a standard Mobius transformation 0: Hn+Bn which (In case

b =

00

we choose so that

(4)

x

= (0 , en ' (0) carre spo·nd to y = ox = (-e n ,O,e) n

where e n is the last coordinate vector. The restriction of Rn - 1 is the usual stereographic projection.

to

The following pictures illustrate the choice better than words:

x

LII,I

x-en

- e", '.

.

I

I / /' I

34

e + 2(x ... e )* n

n

I

e

n

y

= aX = [e

n

+ 2(x - e ) *] *

n

.... e

Fig. 7

n

The correspondence is thus given by

(5 )

**

y = (e + 2(x - e ) )

n

X :=.

n

e + 2 (y* - e )*

n

n

In terms of coordinates one finds y

*= 1

2x.].

(6) if-

Y :;;: n

and

Ixl 2 -1

Ix - en l2

(i =

~, ••• , n -

l)

35 2yi

(i

2

\y- e n I

~ 1 , ••• , n ... 1)

(71 1-

xn =

When x n = 0

Iy\ 2

ly- e n I2

lyl2

one verif':'es that

= 1

Yi

Iyl ;: :

and for

J.

~

Ixl 2

Y' = n

+ 1

*

=y , and (6) reduces to

Ixl 2 - 1 Ixl 2 + I

2x. (8)

Y

1

y.

(9)

J.

X.

l-y n

J

one recognizes these

3.6.

X

::; 0

n

f'o~mulas.

n-l { 1 w~.th B = x l ' · · · ' ~-l ,0

The stereographic projection (8) mapa

2

2

xl + ••• +x _ < I n 1

on the lower hemisphere of'

(IO)

B _ n 1

n 1 "'-ER -

mapping

,

t \ IX

< 1 .

The

on itself' defined by 2x

y :;;;

4\e

II

{ n yElR; y =1,yn >O}

Geometricg.lly, the lightcone {<x,x>=O}

boloid

Recall that the group

and leaves the quadratic and bilinear forms

invariant. In ma.trix language

is denoted by

~ loglx,y,t.ni.

n 1 R +

<x,y> = xoYo

(11)

• Thus the n.e.

=

I}

A E O(n,l)

_ The same is true of the hyper-

with two mantles, one with

x o >0

and one with

x >0,

Xl

o

=

II

are in bijective correspondence through the central projection Xl

= x/x. o

line through

x

and

and

x',y'

tions

x,y

shows

Fig.40

y

~I,n'

E U

and the intersections

of the

with the lightcone together with the projec-

•

By elementary geometry the cross-ratios Ix,y,~/nl and Ix',yt,~I,ntl

are equal. To determine their common value we observe that are both of the form

x+ty 1+ t This leads to the equation (12)

with real

t

<x,x>+2t<x,y>+t 2

=

which consequently has two real roots

V

0

can be identified with the Klein model of

Bn.The

= ~Ilog

t1/t21.

d(x',y'}

By conjugation with the central projection each

of

(12)

~,n.

t

straight line segments are the geodesics and

a bijective mapping

AI

of

V

A

E

SO(n,1)

deter.rnines

on itself. Because the coefficients

are invariant the same is true of the cross-ratios and hence

also of the n.e. distances

d(xl,y'). This means that each

is a

At

n.c. motion, and it is obviously sense-preserving. Conversely, if mapping

n

<x+ty,x+ty> =0.

1 ,t 2 corresponding to tl/t2 which is positive.

The cross-ratio turns out to be The section

satisfying

and

~

A

of

collinear points

U

A'

is a n.e. motion it can be lifted to a

which carries collinear points (AX/Ay,A~,An)

of this the coefficients of

(12)

(x,y,s,n)

into

with the same cross-ratio. Because will remain proportional to each

39

other, and since a~

~

<x,x >

=1

it follows that =<x,y>

well.

Fig.I0

40

The mapping COne by defining subsequently

Ax

A

can be extended to all

Ax = <XtX>%

=

-A (-x)

extension still satisfies

A«X,X>-~ x)

A

= <x,y>

= nAx

A{ax)

inside the double

in the upper cone and

in the lower cone. One verifies that this

It follows that

x

and

=0

as well as

•

A (x+y) = Ax + Ay • In other words,

is linear, at least inside the double cone. Because every

1 X E Rn +

is a linear combination of vectors in the cone it is clear that A n extends to a linear mapping of R +1 on itself, and becauSe = <x,y>

A E DCntl) ; since the upper cone

it is represented by a matrix

maps on it,self it is even in

SO (n, 1) •

It is evident from the construction that the correspondence between

A

and

AI

is bijective and product preserving. We have thus

proved, so far, that showed that

M (B) n

M eE) n

is iso:norphic to

is isomorphic to

SO(n,l). In (3.:5)

( n-1,) ran'd t h·1S completes the M.R

proof of Theorem 1. The passage from

M(Rn - l )

to

O(n,l)

means a l,QSS of two

dimensions. For instance, the classical Mobius group Rented by the

Lorent~

group of

4x4

matrices. For

M(C) is repre-

practi~al purpose~

this is much too complicated, but for theoretical reasons it is imporn tant to know that every M(R ) can be written as a matrix group.

41

IV.

4.1.

Elements of differential geometg.

I want to take a very 1lllsophisticated look at d.ifferential geometry and

regard it simply as a set of rules for changing coordinates. A differentiable n-manifold is a set covered by

coordinat~

patches such

that over lapping parts are connected by coordinate changes of the form

- x l... ;:

(1)

X. ( Xl ' ••• l.. _

,xn ) ,

i;;;l, .•. ,n

The change of coordinates shall be reversible, and this means that the Jacobian matrix

ax. II

ox~J " ex)

is non- singular.

The coordinate changes will alwB¥s be

C

The typical contravariant vector is a differential

. d? =

(2)

L: j

oX..J

oXJ.j

dx

Observe that we are using upper indices for the differentials, but not for the variables. An arbitrary contravariant vector is a s.ystem of vector-valued functions,

one· for each coordinate system, connected by the same relation as in (2),

namely (3 )

-i a =

aXi

ax:-J

a

j

where we are using the summation convention, as always :from now on. Similarly, a covariant vector is typified by a derivative (or gradient)

42

of'

(4)

'::."'1r"

AX.

:;:::

~

We use lower indices for covariant vectors, and the general rule is OX. b. ~

b.

J

~

uX.

~

The idea carries over to tensors of higher order.

Here are tm rules for

contravariant, covariant and mixed tensors of second order:

;J-j

-i a. J

=

;;;

aX.

hk

--2:.

OX

a

h

OX.l. Ox

h

It makes sense to speak of a

hk

= a

kh

, then

'aij

The Kronecker coordinate systems.

=

a:ji

ax.

o~ o~ l. = 8~ = OX. 3i":" J o~ J J

-

Ox.l-

OX.

and

dX,

J

ij

all

In fact,

because the matrices

k

•

J

8k

a

and. skew-symmetric tensors, for if

8~ is a mixed tensor with the same components in

h

4.2.

~etric

~

OX.1.

are inverses.

An important way to form new tensors is by CO:ltraction.

is a twice covaria.nt and once contravaria.nt te.:lsor, then

a. ;:: a. j•• ~

l.J

(summation! )

Example:

if

is a cO'JaX'lant vector, for -j a ij

4.3.

Ox,k 'Ox. h o~ Oxh h -.-J. == = ~ OX.). o~ ~ Nf" oX.

---

J

l.

A differentiable manifold becomes a Riemannian space by the choice of a

metric tensor 'Whose matrix is positive definite.

1.

It defines arc length acd volume.

2.

It serves to push indices up and. down.

3.

It defines covariant differentiation.

4.

It defines paraliel displacement •

It serves several purposes,

.Arc length is defined by set..tiIlg 1

as

= (g .. l.J

and. dcf'ining the .length of an

t(y)

=

r

~

i

.

~c

'V

ax rucJ)

'2 by

ds

y

is covariant of" ord,er two and the differentials

B~ca.use

variant,

ds

axi

are cont.rR.-

has a meaning independent of the local coordinates.

The determinant of g.. l.J

is denoted by

g.

fi

transforms like a

density in the sense that

R

= (det

This follows by the determinant multiplication rule. volnme by

-rr::

V- tJ...;g

dx1 ••• dxn

This defines an invariant

44

(we consider only sense preserving changes of coordinates.)

4.4.

The inverse of the matrix g..

is denoted by gij

l.J

contravariant tensor of order two.

Multiplication with

by contraction pushes indices up and down.

g. . a

j

gij

followed

For instance

are =

aaf oc

or

l.

The mixed components of gij

The gradient

gij

= a.

l.J

4.5.

It is clearly a

8~ J

of a scalar function is a covariant vector.

For

i

other tensors differentiation of the components does not by itself lead to a new tensor.

It has to be replaced by covariant differentiation, which we pro-

c eed. to d.efine. The Christoffel symbols are

(6) a.."1.d

(7)

rl:j l.

=:

ghk

r.l.J. , k

•

In order to find out how they transform we start froll

and d.ifferentiate using the chain rule

ogab OXa

ag •.

.....:2:J.

::

o~

-OXi

~ c

o~

oXc

OX.

~

J

o~

+

a gab ( Oxidx k

~ ax.

o2;t

+

J

a Oxj~

oX

b ax.- ) , :1

where we have interchanged a, b in the last term. Permutation of the subscripts leads to (8)

r l.J, .. k

=T

oXa ~ oXc ab,c ax. ax:- ax; + J.

a~a gab

J

The extra term on the right shows that

r l.J, .. k

~

Ox Ox i

a~

j

is not a tensor.

Multiply both sides of' (8) by g

hk

o~ ~ = ox axc d

-. -cd g

This gives

d

r:'-. l.J

(9)

= rab

oxa

o~

ax.- rx:-J l.

a~

oXd

~

2-

+

'0 xd

Ox

h ~ OXiOxj d

Again, there is an extra term. We use (9) to find the mixed derivatives

~

multiply (9) by

~

oa-

to obtain

ax.Ox. :1

(11)

For that purpose

~

(10) 4.6.

•

If

yj

J

is a contravariant vector we shall show that

is a mixed tensor.

From

we obtain

Use (10) to eliminate the cross-derivative:

Tte very last term sim:plifies to

side Which

T-a r . v a:L

•

We :pull it over to the left hand

became~

On the rig'ht we are left with

-oJ1

Ox.,lt

Ir!terchange

h

and.

aXk

-

Ox~

a

:L

ax.

-l. +

oxh

~k

oX.

-..J.

OXa

~ --ax.

in tl1e second. term.

and vre have shown tha.t

which is the rule for a mixed tensor.

V

h

J.

It becoltes

The formula for the covariant d.erivative of a covariant vector is s:.milar:

We skip the invariance proof. ~eneral

The

rule is to add one term for each contravariant index and

subtract one term for each covariant index. Example: k

avo "~ -Y + r=-h

k V V." h l.J

~

a ·.v ., _ a. 1J

raih

k ... aj

v

rajh

r

h

ai

.

One verifies t...hat the covariant differentiation of a product f'ollows the usual

rules.

Another practical rule is that the metric tensors For instance

behave like constants.

_ Ogij

V~ gij -

a

~

fk" g . 1.

aJ

ag ..

~ - f1.-- j v~

Q..~,

... rkj .

It is customary to write

4.7. If

~

is a scalar, then V V f k j

is symmetric in

j

== V

and

k

~,

i:Jf

Oxj so that

}l.

= 0

48

For other tensors this is no longer true.

For instance, operating on

a covariant vector one obtains

where R"h

is a tensor, the curvature tensor of the metric k the somewhat complicated formula

~J

given

by

arh;;

h R.~J'k

(J2 )

~+

~

We make no attempt to carry out the computation. h

and

It is

Contraction with respect to

leads to the Ricci curvature

k

(l3 ) and double contraction to the scalar curvature a

(14)

R - R

a

= gab R ab

4.8. Vectors and tensors are sitting in the tangent

s~ce

attached to each

point of a differentiable manifold, but there is no automatic way of comparing vectors and tensors at different points.

or connection, serves this purpose.

The notion of parallel displa.cem.ent,

We shall use only the stmplest connection,

known as the Riemannian connection. Let

tensor

x = x (t )

repres ent an arc

g .. ; we assume that ~J

at x(t) •

y

x(t) ECOl.

in a Riema..nnian space with the metric Let

set)

be a contravariant vector

49 Definition.

We say that

~(t)

remains parallel along y it

~si(t)~(t)

(15)

= 0

for all t .

More

explicit~,

(15) reads

(16) or =

0

•

This is a system of first order linear differential equations. Because the and the ~ are C- it has a unique so1ution with giVen initial i dt values ~ (t~) . In other words, ars vector ~(to) determines a vector ~(t) obtained by parallel displacement along V •

r!j

4.9.

The length of a vector

~

(;,~)1/2

, measured in the Riemannian metric, is . . 1/2 = (i~j gij '~CJ)

Similarly, the angle between two vectors

cos

.5

~

on is given by

e

Theorem. The inner product (~'11)' and hence length and angle, remain constant under parallel displacement. The proof is a straight forward cow;putation making use of (16) and tte

identity

50

4.10. ,Definiti~n.

x' (t)

An arc x(t) ,

.:5 to ' is a geodesic

0':: t

remains parallel to itself.

if the tangent vector

. dx. t;l. =

dt

If we put

in (1.6) we obtain

the condition i

(18)

fkj

x(t)

for

~ d.t

dx.

..-J. = dt

0

to be a geodesic.

It is not only the shape of the arc but also the parametrization that I!1akes an arc geodesic. along a geodesic. mUlt:!.plled by linear:

t =

dt d5 a7

By the theorem the length of

t

If we replace the parameter and

by

x' (t)

remains constant

or, the length would be

would no longer be constant w:less the change were

+ b •

We can regard (18) as a system of differential equations for the functions According to the general theory we can prescribe the initial values

x. (t). 1.

x.(O) 1.

and the initial derivatives x!(O) . ~

In other words, there is a geodesic

tram every point in every airection.

We ma¥ even nor.malize so that the tangent

vector has length one, in which case

t

by

becomes arc length, usually denoted

s.

The existence theorem produces but we can start anew from the point

x(t)

only in some small interval

x(t). o

[0, t ) , o

Thus a geodesic can be continued

forever unless it "tends to the ideal boundary". A geodesic arc is, at least locally, the shortest arc between its end points. 4.11. ball.

This is proved in calculus of variations. We shall now applY these notions of differential geometry to the unit

n B

with the Poincare metric

51

(19) We are in the unusual situation where we can use the same coordinate system n B , namely by identifying each point

in the whole space coordinates

(xl' ••• ' x ) . n

Any diffeomorphism of

n B

JC

E BO with its

would lead to

another coordinate system, but we choose to consider only coordinate changes

of the form (20)

x = yx

where

o£

n y E M(B ) .

This means that we are interested in the co-O.rormal structure

Bn. The metric

(19) corresponds to the metric tensor 48 .. ~J

(21)

8 .. ~J

A little more

general~

we consider an arbitrary conformal metric

and have then g

(in this connection

5 .. ~J

ij

-2 = P 8 .. ~J

is an element of the unit matrix and not a tensor.)

We wish to compute the curvature tensor. notation u

=

log p

,

It is convenient to use the

52

On.e

obtains

+ 5ij(~ - ~~) - ~(uik - u i '\.) 2

+ (8ij \k - 3ik \j)l vu l R .. = 8 •. bou ~J

~J

+ (n - 2) (u •. - u. U. + B. j 1J

1

J

~

For the Poinca;r9 metric

I 12 )

u = log P == log 2 - log (1 _ x u

i

u ..

1J

=

=

2B .. 1J

1-

(xl

28 .• 1.]

1-

(xl

2

2

Ivul 2 )

•

53

Rh. 'k ;; ~J

~ijk

4

2 2 (B.. 5. . - 8. kB .) nk l.J ~ nJ

(1 _ Ix I)

16 = (l-

Ix12)4 (\k5ij - 5 ik \j)

=

~gij - gikgjh

4(n - 1}8.. l.J

j

R = gi R .. = n(n- 1) ~J

The scalar curvature is constant) but

~

= -1.

However, the formula for

~ijk

shows that the sectional curvature is constantly equal to

4.12.

The Beltrami parameter

-1.

There are three important invariant differential operators which generalize the grad.ient, the divergence and tile Laplacian.

a)

The gradient maps functions on vectors. V.f

is a scalar

: :; of

Ox.1.

1.

is a covariant vector.

If f

The corresponding contravariant vector is ii' of gJ Vf = g J _

Oxj

j

The square

le~th can be written either as a i j gij V f V :f

JVfl2 =

(22)

= gij Vir

I: g i,j

ij

v j:f

d.ot product

Vif· V.f :L

:::;

...2!. of ox.J. Oxj

ThiS is the first Beltrami parameter and it used to be called

V:f. 1

It

obviously generalizes the square of the gradient.

*See L.P.

Eisenhart:

1960, p.Bl, (25·9).

Riemannian Geometry, Princeton Uhiversity Press t

or as

b)

The d.ivergence leads from. vectors to scalars.

If

v

is a vector-

valued function we define i

divv = V.v

(23)

~

i

= V v.

~

These expressions are identical for

g

ij

behaves like a constant and we

obtain == 9.g

ij

;;: g

v.

J

1.

iJ'

V.v, ;;: J

~

From j

;;: -av

ox.J..

j

+ fik v

k

we have

(24 ) There is a sim?le formula for

By

the well known rule for dif'f'erentia-

tion, of a determinant.

~

log g

~ij

=

= tr (g-l ~)

=-

g

iJ'

ag ..

-3J.

~

Btlt

'::I.x.. u

f k ,· ~,J

It

+

rkj .

,,~

and hence ij

g

and thus

(25 )

Ogij _

o~

-

g

ij f k1 ,j

+ ij r g

= 2 ri kj,i

ki

55

From (24) and (25) we get divy ::::

(26)

c)

0

1

~

Ji

(jg

~

.

yl. )

When both operators are combined we obtain the second. Beltrami

P8:!ameter ~f

= d.iv gradf ::::

1

.fi

This is a direct generalization of the Laplacian and it is frequently referred to as the Laplace-Beltram:.. operator. I-ts importance lies in the i'act that it is independent of the local

coordinates.

For instance, if

Poincare metric, then for

f

is d.efined on

For short, a. solution of

harmonic

(h .. h)..

a.ad if we use the

n yEM(B ) ~

~(foy) = (~f)OY

(28)

n B

~f = 0

• n B

on

TNe see from (24) that if

will be called hyperbolically f

is

h .. h , so is

fO y

This is not true for ordinary harmonic i"unctions, except, as we shall see, when

n;;:; 2 • On a Riemannian space a solution of'

~f

will be called harmonic si.'lce

this does not make sense in aQy other meaning.

4.13. (29)

We compute

~f I'i... f

""'2

for a conformal metric =p -n

0

oXi

(1'..I 11- 2 _C>f' )

oXi

ds::::

pldxl .

One gets simply

2

P ;::

In the special case

D

1 ...

( 30)

~f ~

(1-

Jxlc

~j222

one finds

[ Ar + 2{n -

2~

xi : : 1 ,

l-Ixj WIth the cusLolll.ary I,lOtation

of

x.J- ~ J.. (31)

A.2f' ==

For the half-space

( 32 )

:::;

IxJ::;

r

of

r '-"-

ar

~l- r2}2 [llf + 4

1 P :;:-x n

1.

and

~{n-2)

l ... r

2

r

M] or

57

v. 5.1.

;Hyperharmonic :functions

We shall now study hyperharmonic functions on Bn.

will be to find all solutions

Fo!' a f\mction

u(x)

of

~u = 0

our

first task

that depend. only on

uCr) one obtains x.

oU

= u ' (r) ~ .....

Ox

i

x.x. -¥ r

ull(r)

8;;

x.x.

r

r3

+ u l (r) (~- .2:.....sl)

and thus 00.

t

;;:>

ull(r) + (n -1) u (r) r

According to equation (3 2 ) in 4.l3 (1)

u"(r) + (n-I)

If ut(r}

f

ur~r) +

u(r)

will satisf'y

2(n- 2) rul(l") 2 l ... r

=

0

.

0 this can be written as unCr) u t (r) +

en -1) + 2(n - 2)r = 0 r

1

-r

2

or d dX [ logu'(r) + (n-l) logr - (n- 2) log (1- r 2 )

from which we conclude that r

u l (r)

.0-1 =

const.

This lea:ls to the general solution r

(2 )

u(r) = aJ

2 n-2 ( 1- t ) dt + b t

n-l

] = 0

r:=

Ix] •

We see at once that no solution can stay finite for

o.

r::

As a

normalized solution we introduce

For

n =2

g(r) = O«l_r)n-l)

5.2.

%.

g(r};:: log

Toget.her wiLh

r _1 •

for

g(r)

because by (29) in 4.13

any y

g(r)......

n>2

For

g(

For

11'x\)

n

=3 with

2 n 12 r n-

for T:::'

g(r)=C"r n

l' EM(B)

-

r ~0

and

1 2 1 - ) ::-+r-2.

F.

is again

r

h - h

comnmtes with the Laplace-Beltrami operator.

In

particular

(4) has a. singularity at

pole at

5.3.

and Will be regard.ed as the Green' s function with

y

y.

The fact that a

h -h

constant or has a rather singula.ri ty of a

h - b

function that depends only on

stro~~

r

is either a

singularity suggests that any isolated

function. should be equally strong.

In :f'unction theory

we are used to study such questions by use of integral formulas, especially Cauchy t S integra.l formula and for ordinary harmonic functions Green t s formula. For h - h

functions there are two ways of approaching the question.

Ei ther one can derive Green t s formula in the general setting of Riemannian spaces or, in the hyperbolic case, one can apply the classical Green's formula to suitabJ..y chosen functions that a.r e naturally connected with the hyperbolic metric.

59

We shall use both methods, but we begin with the second which is by

far the simplest.

5.4.

As a preparation it is useful to collect a few facts about multiple

integrals especially these connected with spheres.

r - function

Recall Euler's

to

(5 )

r(a):::;

(6)

B(a,b)

a-l

Jo t 1 = J o

They are closely related.

and

t

= sin2~

(8 )

in

-t

dt,

Re a> 0

ta-l(2._t)b-l dt , The substitution

= 2

t:;: x

Re a 2

> 0 ,

Re

b

>

0

gives

2

OCt

rea)

(7)

e

B - function

and.

I

x

2a l - e-x dx

o

B(a,b)

gives

B( a, b) = 2

'JC/2

S

sin2a-1. cp cos2b-l cp d cp

o

~"\rom

(7)

and in polar coordinates

rea) reb) : :;

2

to

4

So

r

= r(a+b) B(a,b) and.

(9)

~j:2

2a+2b - 1 e -r dr J~ 0

thus B(a,b) :;: rCa) reb) r(a+b)

cos

2a-l cp sin2b-l r.pdcp

60

and

In particular, since

r(l):= 1 ,

(10) We shall denote the "surface area·' or (n - 1) - dimensiona.l euclidean n 1 m.easure of 8 by 0,) (some authors prefer c.o 1.). The area of Sn-l(r) n nn-l n 1s then (J)n.itl. r and the volume of B (r) is r

J

n 1

V (r) := co r nOn

dr :=

To compute the area of a spherical cap of radius unit sphere) we project on the equatorial plane

x.

n

=0

cp

(mea.sured alopg the and note tha.t the area.

element on the sphere is dx dO' =

1

... dx _

n 1

x

n

Fig. 11.

The area of the cap is thus

A(cp) -- enn-1

sin cp r n-2

S o

dr

61

With

r ;; sin

e ,

ax

(11)

= cos e de

, x

= cos 8

n

we find

S sinn-2 e de ij)

A(cp) = {Dn_l

o

In particular,

ill

n

1(/2

S n-1 0

= 2A ( !.) 2

n-2 sin

:;; 2m

a de

and thus

w n (D

= B( n;l , ~) =

r

n-1.

(E. ) ~

independent o£

This makes

n and since ill2 = 2n it

follows that (J2)

(l)

n

=

For comparison we shall also compute the area of' a sphere and volume of

Recall that a hyperbolic radius

a ball in hyperbolic space. to the euclidean radius

r = tanh

2s

of a ball with center

s

o.

corresponds Tbe n.e.

area of' the sphere is tben n-1 n-1

ill

2

r

=

n

and the voiume

(Vh

n

.

n-l

S1M

s

is

s

(13)

ill

V. (8) = h

ill

stands for n. e. vOlume).

n

J0

n 1 sinh - t dt

The formula is thus the same as be£cre with

sinh instead of sin and of course

n+1

in the place of

n.

62

5.5 We return to h - h functions.

su ~vdx:

(14) Here

D

=r

&!

U

~D~

D is a region in

Rec~ll

S (Vu .. V'v)dx

dO' -

bn

the classical Gree.o, I s formula

D

JRn,

dx = dx

l

...... dx , n

~ is the derivative in

the direction of the outer :!l.ormal of the smooth boundary

~D

and

(VU" vv)

is the inner product of the gradients .. Let us now assume that

D lies in Bn.

its hY.Perbolic counterpart, using the

We shall repJace everything by

f'ol~ow1ng

notations:

for the hyperbolic

volume element: 2 n- l d g

the ar ea element:

=

(1-

txI2)ll-I = 1-

the normal derivative:

Ixl 2 Q! 2

bn

= 1- t x I2 Vu

the gradient:

2

the 1aplac1an:

~v

..

With these notations we clatm that Lemma 1:

It is ea.sy to chec.k that this is exactly the same as (14) with 2 -u = 2 0 - 2 (1_ 1.A. .... 1 )2-n u

..

u

replaced by

As customary, one can eliminate the ItDirichlet, integral tl

on the right by interchanging

S eu l\v -

(16)

u

and

v

and subtracting.

v~ u) ~

The resulting formula

- v

D

is the one in most common use.

The special case v = 1

r

(17)

.J D

leads to

t\ udx -- J

- bu

~

bD

and it s:3.ows above ali that an

h- h

d~

function has vanishing flux

(18) In euclidean terms (18) reads

r

(19)

"bD If D = B{r)

we obtain

~ n v

d

g~

(1-

Ixl )n-

bu dO' br

=0

2

= 0 •

*)

SS (r) In other words, the mean value m{r) ==

1:..

(On

is constant.

In particular, if

u

JUdo-

s(r) is

h - h

in a full neighborhood of the

origin, then

uCo)

(20)

*)

From now on

n

=

I

will be fixed and we w.rite

B for

n B

and

S

for

8n - 1 .

for all sufficiently small r .

This is the mean-value property.

The mean-

value can also be taken with respect to volume, and because of the :cotational symmetry it does not matter whether we use euclidean or non-euclidean measure. However, the n.e. average is invariant and we obtain; I.ennna.

The val·ole of a

h- h

function at the center of a n. e. sphere or ball

is equal to the n.e. average. Remark.

The mean-value formula can also be proved without 8.J:lY computatiou

whatsoever.

In fact, it is possible to write

r

m( Ix!) =

u(kx)d.k

"o(n)

where k

dk

is the Haar' measure of

it follOWS that

m(lxl)

is a

and hence must be a constant,.

O(n) h-b.

Because

u(kx)

is

h- h

for every

f'tulctionwhict depends only on

As a result of the mean-value property

Ixl h .. h

functions satisfy the maximum-minimum principle: A non-constant

h - h

fwlction cannot have a relative maximum or minimum

on an open connected set. The proof is the same as for

5.6.

n = 2 •

The question of removable singularities has the :following answer,

Theorem.. D \ ra}

Suppose that

DcB is open

a.nd

a ED.

If u(x)

and. if'

limu{x)

Jx-a

In - 2

::::

0

if

n>2

if

n=2

x-ta,

lim. u(x) x-+a. log

1 1

lx-at

::::

0

,

is

h-h

in

then u

has an h - h

Proof ~

extension to

D.

Tliithout loss of generality we may assume that

p

> o. We choose a fixed y E B(p) , y

V

= g(x,y-) ~u

Since

~v :=

=0

and

a.r:.d apply Lemma 1 to with r

D

= B(p)

,

u:= u(x) ,

< min (iyl

, p - lYl> •

0 we obtain

r

(21)

0

n(p) \ (n(r) U D(y,r»

in the region =

I

a

bg~Y)

(u(x)

- g(x.y)

b~) ) ~ (x) ~

0

•

S(p) +S(r) + S(y,r) We look for the limit as s (y ,r).

for

r .... 0

and start by considering the integral

It is evident that the integral of the second term will tend to

g(x,y) = 0 (---1..-2)

r

n-

contains the factor

-s

(22)

r

(or

n-1

(1-

•

1:r ) )

>.'Ou is bounded and dah un n The integral of the first term can be witten as O(log

Ixl 2 )2-n u(x)

Ix- Yl =r where

di.D

is the element of

We recall that -

"solid

g(x,y):;: g( ~'X

( bgb~~Y)

0,

y

,

I

xl)

= -g < TyXJ )

while

bg(x,y) rn-1dro ~r

angle" (or the measure an and

ITy xl

blTxl

J

=

= Jx -yJ ~

Sn-1).

so that

66

Here

[x,y] -+ \1-

ly(2 as x-+y and

r ~ log

l'ly x I

Or

= r

-2.. ~r

log

It follows a.t once that (22) tends to

ill

n

r ... I . (x,y)

u(y) •

We have now to investigate the part of (21) that

re~ers

to

S(r) •

We

make first the observation that by Green's formula the integral

I

(23 )

S(r)

is in fact independent of

y.

b~,

and hence defined for all

r

y E B(p J \ (o} .

it is;evidently a h - h

g(x,y) = g(y,x}

Moreover, because

~) d'b (x)

(u(x) ?$(X>Y) - g(x,y) ~

function of

We wish to show that the boundedness condition of Theorem 1 makes it

identically zero.

Let us rewrite (23) as (24)

Observe that n-2

u(x) Ix I

- 0

~

and

g

remain bounded when

r - O.

Hence 'l:ihe condition

makes the limit of the first term zero.

For the second term we write

~(r)

=

r

g(r~,y) bu(r') ameS) br

.. 8(1) ",

and obtain

2r

S

jJJ(t) dt ==

f

.. S(l)

2r

den

JS(I)

r

=

r

~

g(ts,y) n(tg)

f

'r

g(ts,y) bn(t~) dt ~t

I2r - S2ru(tt:) r

r

ag(t~2Y) bt

J

dt dm(g)

67

This is with

o~

1

o( -.n:2)

the order

•

Hence there is a

r and since

~(t):: o( ~l)

lim r n-l..-L (r )

between

t

. t s l.. t mus t

ex~s

r

and

2r

b e zero.

t

We can now conclude from (2l) that u(y) = -

1.

a;-

S

n

The right hand side is a

extension of u

5.7.

h- h

function in

U

S(p)

B,

(25)

If

u

is

t S

u( 0) =

is again

h .. h

in

B and has a. continuous extension

u(x)dro.

n-l. S

b- h

u(VO) = ron

We specialize to

'V == T-

1

Y

uaY

S

for a fixed

y EB .

Because

u(y) and replace

x

by

T x y

u (y) ::

or explicitly (27)

u(y)

1:.. ron

in the integral.

r

.~

S

where

y EM(Bn )

and extends continuously to the boundary.

sn-l u( y x) CIm(x)

1

(26)

•

and. defines the desired

We obtain. a more general formula if we appl¥ (25) to uoy

dO"h

then the mean value property implies

n

Indeed,

}

n

.

Bo".mdary values.

to the closure

:o.~

(u{x) eg(x,y) - g(x,y) ~~

S(p )

This leads to

u (x) IT 1 (x) In-l dill ( x) Y

•

Hence

68

where 'We have used the simplification

the

[x,y}

= Ix - YI

Ix]

when

= 1 •

This is the Poisson formula far the ball.

Note that the kernel is just

(n - l)st

n

power of the Poisson kernel for

2 .

=

5.8. We shall use the notation k(X,y) =

1-

IYI~

Ix-yl with the understanding that

suspect that

k(x,y)

Ixl

= 1

is always a

Iyl

and

< 1.

Formula (27) lets us

function of y.

h - h

The direct verification of this is not difficult, but it is much quicker to pass to the half'- space

Jf-.

Reca1.1 that the laplacian fo,.. the half' space

is

It follows without computation that a A x :: 0:(0: - n + 1) xo. h n n x

ThLlJ::l e ver"y

°.

n

~l

Is all E::dgerl...f'unctlon and

oX.

n

is

We recall (see 3.5) that the canonical mapping

x

n

where ~

n x E H , y E Bn.

for

for using

n

B

and that

=

n B

-+

tf

is such that

k(e , y) n

We conclude that k (e ,y) ex is an eigen:f"unction of n

k( en ' y)

n-l

is

h - h.

x

twiee) there exists a rotation ~ -1 is obvious that k(x,y) = k(f3e ,y) = k(e ,t3y)

n

we conclude that

h ... h •

n

Given any xES such that

px

n-l

= en

(apologies ,

But it

Because of' the invariance

~ k(X,y)O:

(28) and

~ k(x,y)

5.9.

n .. l

= a(o:- n + 1) k(x,y)CX

= 0 •

Suppose now that

f(x)

is of class

J If(x)r da>(x) s

cos cp

over the polar -cap of' radius

tp

•

Clearly, the

u(re ) can be w-ri tten in the form n

ra K(r,cp) Ft (cp) dqJ -:rr

u(re ) ;;:; n

(31) Ft (cp)

where

exists a.e. and the measure F t (cp)dtp

is a.bsolutely continuous.

Integration by parts yield.s

We wish to show that

assume that

u(re)

Without loss of' generality we may

fee ) .

n

f(e):;;::: 0 , for otherwise we need only replace

n

It is :tmmediate that any interval

-+

n

8.:s tp

.:s 1C

5

> 0.. Hence u(re) has the same n

8

u{5(r) :; -

Jo F( r.p) fro K(:r, cp)d-C9

•

I

Because of (30) we can choose

8 so small that

= I~

:for

e c.o

n-

1

Jq> sinh - 2 e de 0

follows that

M

bcp

by

f - fCe ) ..

~ tend uniform1¥ to zero for r - 1

K and

,

f

> 0 • Another integration by parts leads to

limit as

n

in

71

lua(r)1 ~ - eA(8) K(5)+' e

:n:

3

it is still referred

to as the quotient-manifold. but it probably need not be a mapifold. The di:fficulty comes from the fixed points.

Suppose a is

a fixed

r

.

80 Then r'Y I (a)f = 1

point of Y

lv' (a) I

l-I'Va/

as seen from

1

2

•

In other words, a lies on the isometric sphere K(V)

ala

that

If

a =0

y E soC n)

then

k -dimensional. plane with

(see 2.9), provided

.

and the fixed points of V fill some Hence an arbitrary

0:5k.:5n-2

V fixes some

As is seen by the above equation we have

k-dimensional geodesic subSJ;lace.

I y' (a) I -0 uniformly for all a from

n B

any compact set inside

these subspaces tend to the boundary.

Therefore

We conclude that every point in B

has a neighbourhood which meets onlY a finite number of these fixed subspaces.

7.3. We adopt the followipg definition: Derini tion:

A point

infini te sequence of

b E:B Vv

er

is called a limit point of a E B such

and a point

r

The set of all limit points of set of accumulation points of ra

r

the. t

Y,'V a"'b

A = A(ro)

is the limit Bet

1s denoted by

if there exists an

The

Clearly,

A(a)

A =UA( a)

The following is true:

Theo1"em. Proof.

A=A(a)

for all

(with a few trivia~ exceptions);

If a,b E B it is trivial that

d( 'Va, 'Yb) = d(a, b)

is fixed.

Consider now the case

a

aEB

In particular

A(a) =A(b)

because the distance

A(a) = A( 0)

We shall assume that

a ES

is not a fixed point for the whole group.

ra ~ (a}

There is then a

, i."e.

b =y a~ a o

'I E r o

1)

We pr'ove first that

interim point of the geodesic every

e E 1$ 0)

FaT this purpose let

A(O) C A(a)

there is thus

(a,b)

.

We know that

c

A( c) = A{ 0)

be an For

_ -___b

81

Fig. 14

a seq:uence Y'J~

0

(a I , b') . but

2) [OY'$ }

a'

a~so

to

and b l

S

belong to A(a)

the

'tv!! at

suchthat

'YV,a.-d

while

Because

lies in If

I(y ) v a~ e' then

d=eEA(o) A(O)

I,,' (x) I = 1

its image

OEE(y.)

It follows that

-1)

a=e' EA(o)

v

A(a)

The proof fails when

we have

a

and hence

proved that

a

( ) K'Y,

r y' (x)! > 1

and

1 I (''J- )

and 'V'J -10 and I (y\» to e'

e

yaEI(y-l) .... e 'V

\I

then Y'Ja E A(O) dE A(O)

b =Y a

and a sequence

Recall that

lies in tends to

is closed and we find again that

arbitrary point in

dE A(a)

I'VI (x) I < 1

'V'\) 0

for large

aEE(y\) ) But if

Ih'"

Because

and we have :proved that

and. ~v -1O-e l

YvO-e

(see 2.9).

b'

Or

e E A(a)

Thus

it is clear that

at =b'

must tend to the geodesic

For the opposite conclusion choose an arbitrary

are characterized by

But

fbi

If

, and benet! ei the.!· to

E( 'V), I( 'V)

i.e.

and b '

at

a'

On the other hand, if

e=a'

We can pass to a subsequence for which

with Y'J c-e

converge to limits

Y'Jb

and

both

Yv E r

f

b·~ca.use

Since

'Vvt\.(O)

=:

A(O)

was an

d

A(a)cA(O)

is a fixed point for all of

r

There are at

most two such points and all such groups can be classified (the elementar,y

groups) .

7.4.

We list the following ~roperties of L)

A

is closed because

A(O)

A

is closed.

open and is called the set of discontinuity.

The complement

0

=:8 \ It is

82 2.)

A is invariant

unde~

r

, and every invariant closed set contains

A

r

3. ) Either A = S or A is nowhere dense on S the' first kind if

A =8

or

, otherwise of the second kind.

4~)

A has no isolated points; it is a perfect

5. )

Let A and

number of

is said to. be

set~

Then A meet s only a finite

B be compact sets in 0

yB,V£r

The proOfs of these statements are all very easy and are left to the reader.

Assume tha.t the identity is the only element of

1.5. words,

0

is not a fixed point.

sphere

K(~)

With correspottding

Then every Y Er\I E(Y)

and

I(V)

r nSO(rt);

has an isometric

In what follows these

notations will refer onlY to the part contained in B a non-euclidean hyperplane, and E{Y)

I (y )

in other

Thus

lC('V)

is

are hyperbolic half-spaces"

We shall consider the set

=n

p

E{'V)

'Y~r\I

and prove that it has the following properties:

(i) ( i1) (iii)

P

is open

bPc u te( 'V ) every

x

(here

tB

bP

and

(ii)

B

).

is equivalent to either a single point in

at least one and at most finitely many (i)

is the boundary in

poin~s

on

P

or to

~p

follow from the fact that every intersection pnB(r)

automatically contained in all but a finite member of the E(Y)

is

and is thus a

tini te intersec"tion.

Recall that

Iy , he) I < 1 and = 1/(1-lxI 2 }.

K(y),E(y)

IY'(~) I

> 1

and

I(y) are characterized by

Iyl (x) 1=

1, 2 respectively, and that [yJ(x) 1/(1-[yxI )

8~

It follows that if and only if

x E E(Y}

is equivalent to

> r xl

jv x , =

while

In other words, a point is in P

x E K(V)

it is strictly closer to

'vxl

0

than all its

images~

if and

and it lies an

Ixl on~

rY if there

are several closest points (including the case of a closest fixed point). the

~oints

if

Since

in an orbit are isolated the existence of at least one and at most

finitelY many c10sest points is obvious, and property (iii) follOWS. the rather remarkable fact that equivalent points on

~p

Observe

are all equidistant

f"rom 0 Property (iii) characterizes

P as a fundamental set.

of half-spaces it is convex in the n.e. sense.

P is a n.e.

As an intersection po~hedron

and

it is referred to as the Poincare'(or Dirichlet) fundamental polyhedron of r with respect to the origin. intersections

?)p nK('V )

The faces of P are the The faces on K(y )

(n-I)-dimensional

and K(Y

-1

)

are equivalent

and congruent in the euclidean and non-euclidean sense.

7.6.

Convergence and divergence.

points in an orbit tend to

We are interested to know how fast the

S or, which is the same thing, how fast

the orbits tend to infinity in the hyperbolic sense. The first observation is that any two orbits

fa

in the sense that the ratios

lie between finite limdts. In fact, from

d(Ya,Vb) ==d(a,b}

it follows that

and

rb

are comparable

8L

or log ,1 +

iybl

l_jybj

< log .!..t IvaI = l~-Iyal

+ d(a"b)

1 + Iyb(

l-/Yb/ from which we deduce that

for all 'V

A good

w~

to study the density of an orbit is to investigate the divergence

or convergence of series of the

~ (1- ·';lal

yEr

for different

po~ers

~orm

fi a

We :prove first:

Every discrete group

Lemma 1. (1)

~

(1-

yEr

r

satisfies

-

IvaI )a n-l .. Proof. It is sufficient to consider the case denotes the Poincars polyhedron of

Because a>n-l

J (1_ fxJ2) a--n dx = c
v0

v

There is then an

a

with

\I> \)0 inside the

'\)

Stolz cone with angle between

g and a

\I

trigonometry to 2 sin

0:'V

2

1-1 ~'V ,.

~o

arbitrari~

The fact that

close to a

'\)

Let

be the angle v is in the Stolz cone translates by ~

Fig. 16

In the limit _

lim ctv

1-

T.he point

a

radius

~

tan CPo

'V

is contained in the gpherical cap ~ith cente~

S is covered by all these caps

This

'V

V o

'V

o

by (11), and consequently equal to zero.

'V

to a n-l which is majorized

Hence the measure of L(B)

by

~th

and

avl1av'

r

For if

S on a set contained in a

y;

r is of divergence type we conclude that either m ( L) =0 or Thus, for !BZ grQup

We shall eventually prove that

r

onl.:y the extreme cases can occur. m ( L) -= ill

n

for all

r

of divergence type

so that the distinction between convergence and divergence is the same as betw~en

to say

m( L) = 0 and m( L) = (Jjn

However~

it is appropriate at this time

something about the history qf the problem.

OriginallY the problem dealt with the geodesics on a closed Riemann surface. In the 1930' S important groundwork was done by Morse,

Hedlun~

Myrberg and

95

many others. that r

A very decisive step was taken by Eberhard Hopf (1936) who proved

acL1:l e.rgodlcally on

i~

Sx S

tbe case of' a compact Riemann surface

with or without ptmctures • The action is the one that takes This is of course very much stronger than ergodicity on

(;tt})

to

(y ~ ~y 11

S

It did no'L take long for Hop£' to realize that his theorem can be extended to several

d~mensions

in a much more general form, and that finite volume is In 1939 he proved the following:

not the right condition.

r

(E. Hop£')

Theorem.

acts ergodic ally on

Sr S

if ann only if

m(l,)

=mn

The proof makes essential use of Birkhoff's individual ergodic theorems and cannot be considered elementary. The most recent development of the theo:t"Y is due to Dennis Sullivan (1978). Theorem. (D. Sullivan)

r

acts ergodically on

S Y. S if and only if f'

is of divergence type. His proof uses Markov chains, and I must confess.,that I do not understand it.

However, there are now in existence relatively elementary proofs of the

fact that every

r

of divergence type satisfies

have been given both by Sullivan and Thurston.

meL) = (1)

n

Such proofs

In what follows we shall give

a rather detai1ed version of Thurston's proof. 7·14"

The action of r

on

Sx S

is defined by y( s, 11) = ('VS' "(1)

is said to be dissipative if there exists a measurable An 'Vll == fJ

for all

m(lJ , ..,1 A) = m(S y6"

Briefly,

l'

S)

'V E r \ I

=of-n

A is a measurable fundamental set.

set

~~S

This action xS

such that

)

Lemma 4, Proof..

If meL) = 0

I I

Ir

V-tct\

.>

by

-on

(~'T!) ~ (8: L) x (9 - L)

Suppose tha.t

. 1- a 11m V

r

then the action of

.... 0

- a'V f

I I I" - a:v J

' 1-, a \J l l.m

,

V .... ea

\

Sx S

is dissipative.

diagonal..

Then

.... 0

the definition of L As before

Either

y

-1 0=a

'\)

~ .1

II! - Eov'

2

and hence

v

I:: - il'

I'T\ -

or

av I ;;-

1. 2

'g . 111

This implies

Theref"ore, by (13),

1"10

for Which

I

:: *f; 0

(14)

for all

,

(~i' . r'V" '(11) r -0 and it ,follows that there is VoE r

(~) J .. ('Vo ' (nH

this also means that

o

I

is a maximum.

~.ince

Equivalently t

~

rY"

11 = Y 0

0

r'V

g -' ry

o

0

nJ

is: a maximum.

11 then

I~ o -11'0I ~- r'V~ 0 -''Y1l 0 I 'V E

r

,an4 this is equivalent to

In other words, if

97 For simplicity, let us assume that the origin is not a fixed point.

We claim

that the set

has properties 1) and 2). they cannot both be in

(~,~)

In the first place if

rSo-11

A for that would imply

0

(~o'~o)

and 1

>, g-1l1

are equivalent

and

Secondly, our construction has shown that every

a)

S =11

c)

IVi (;)

J

.

I'V' ('Tl)'

=1

for some

y Er \ I

It is clear that a) and c) are true only on a null-set, and if m(L):=: 0 same is the case of b).

the

The proof is complete.

(The idea of this proof is due to Sullivan) 7.15.

We embark now on Thurston's proof of the following fact: Theorem 4.

Remark.

If'

m(L) = 0

r

, then

is of conyergence type.

This is a consequence of Sullivan's

~heorem.

On the other band,

Theorem L makes Sullivan's theorem a consequence of Hopfts theorem. Proof.

We consider again a ball Bo = B(p,o)

(Thurston).

origin and we denote its images B'

\)

The measure of a set

y-~

Ec S

~

0

with B

v

and the shadow of

will now be denoted by

m(E)

The condition m(L) lim

I

'Vo-ttIJ'

=0

II B f I = 0 ~. \)

\)""'-v

o

centered at the

can be written in the form

IE/

B

V

with

rather than

Indeed, let

denote the characteristic function of

~o

B' V>V v

Then (16)

a

"'neans that

lim Va-IXI

f \, (~ )dro( ~) .:: 0 S

IL I = 0

whi le'

U

0

means that

I

lim

S

'VO-M

X'J (~)~(~),; 0 0

These conditions are simultaneous~ fulfilled by virtuE of Fatou's lemma. On the other hand. the

asse~tion

that

r

is of convergence type is. as

we have seen, equivalent to

p ) >P

for

Then .choose i

'V

k +1

as the smallest

'V

= 0, ... "k.

other a

v

IS

Fig. 17

such that

99

This choice is always possible and we see that the following is true: a)

d(a, a \/h

>

)

P

for all

h

'-'k

f:

k

,
d(O,a.) +p - 4p j

1.

0

+lJp

0

101

and 1+

/80.1 J

1+

>

Ia.1 1.

e

p -

4p

0

1-1 a.1

1.'

which implies

By

use of' formula (4) we final4r find

r~

(18)

....Ii.

r

< 4e

_ +4 P

i

Po cos h

2Po

--

2

The main thing is that this quotient can be made arbitrarily small by choosing p

large enough.

Consider now all the Bj Elm +1. Wh:i:ch are partially ecl.ipsed Their shadows Bj from the rim of

by

Bi Elm

P

are disjoint and the,y 2ie asymptotically within distance B!

1.

Their total area is therefore

asymptotical~

at

most

(We have again used the notation A(r) radius

r

).

for the area of a s~herical ca~ with

In view of the estimate we can thus choose p

su:fficiently large m ,the total area o:f the shadows

Bj

so that, for of aJ,.l Bj E 1m + I

that are partially' but not totalJ..y eclipsed by some B1 Elm will be less than

102

Step. h.

We pa.ss now to the

Letnma. 5. -

Y

o

which are totally ec lipsed by a

We need an auxiliary lemma.

B.1. € I m

(ai'S)

Bj E Im + 1

~

If

':.

IE B"i

n Bj

with

I

intersects the ball ~j

is the radius of" B

Proof.

goes to . en geodesic

m , Bj E Im + l. ' then the geodesic with center a. and n. e. radius 20 ,where 1.

J

'0

0

We map the unit ball eonformally on the half-space

and

(O~S)

through at

B. EI

~

to

We keep the names 07*

~

becomes a vertical. line through

th~

en

If1

so that

points

and

a ,a The 1 j (ai'S) a vertical

whose intersection With Jan-l we shall denQ~e by c

o

a Fig. 19

Let bi,b j

be the closest points to

ai,a

j

on the vertical through

0

0

and let

c

j

be the closest point to b j

picture is misleading in two verticals).

a~

much as

a

j

on the vertical through need not be in the

S~e

c

(the

plane

~s

the

The non-euclidean distances are computed by "Jt

2 ~i d(a.. ,b.) -J~ -logcot2 ~ ,J sin (0 ct)i

n =

J ~ '"

CPj

log cot

sin q>

.f

But

cos

so that

to. >cP i 'J

and

It follows that d(aj,C .) < d(aj,bJ +d(b ,c.) < J j J = J =

2p 0

and the lemma is proved. The next picture shows on the left a together with their shadows.

transformed by Vi

B.

~

and

so~e

totally eclipsed B.

On the right the whole configuration has been

J

104

y.B~ 1. J

The image

'V.B~, ~

J

is not the same as

-,

is contained in

(ViBj )

However, the lemma tells .

It is clear that condition (16) remains true when

Therefore, as soon as

t j

1'V. B ~ J 1.

J

:5 ~ I ('Yo; B.) - j

... J

On. the other hand, if

area of the shadow

a

o

In other worJa,

But

p

I

r

r

Fig. 20

'A

o

is Tf?placed by

R0

m is big enough,

< e

()I)

then y.O approaches the boundar,y while the 1.

of Bo viewed trom 'ViO decreases· to a

positiv~

limit

~05

where the minimum and may.imum

m~

be taken with respect to B!

1.

From (19), (20) and (21) it follows that

Here the maximum of

The ratio tends to the same

is taken at the center and the minimum on the rim. ri/(l- I air)

limit as

and we know this limit to be finite

There is thus

a constant K such that

~ 1Bj' I < K £

j

We choose

I B~ r

~n

E

1

so small that the factor on the right is

1

Proof of the Lemma.

r:

\I(Y) :::

f

Ix-y!

We assume first that

Vij(x)

, eo

'V

EC

and

~(x-y)dx

>p

S Vij(X+Y) Y~j(X)dx

=

Ixl >P for the translated Dhik(y) p

For

p - 0

~otential.

=S

DhV

'1 X >p

ij

It follows at once that

(x+y)

l'~.(x)d.y. = l.J

the surface integral tends to

The evaluation of this integral requires a formula for

Jxi Xj~Xk S(l)

dm(x)

write

(26) exists

118 The easiest way to get this formula is to substitute (27) in

S XiXjXh~ dill =

(28).

One obtains

ill

n(~+2) (oij~ +5ih5jk +oUBjh)

8(1)

and finally

It now follows =rom (29) that

It is known from the Calderon-Zygmund theory of singu1ar integral operators that the truncated integral converges to its pt'. v. in

rJ

for every p> 1

From this it is trivial to conclude that the limit of DhI: derivative

DbIk

,a.nd the lemma. is proved for all

To prove it for general \}

~ e:

CCIO

is the distributional •

we pass a.gain to a convolution

'V

'*5e

where

J 5e: dx = 1

•

From what we have proved

The pr. v. is a bounded 1inear opera tor on LP

"

~>1

oo

8 EC e

is supported in

B(e)

band side of(30)tends in side has a limit in LP

LP

and

.

Therefore the right

to the right band side of (26).

Hence the lett hand

and that limit is the distributional derivative.

The lemma is p:roved.

8.9. I

k

'V

•

We regard

Iv as a vector-valued function with the components

It makes sense to form

S (I \2)hlt and by Lemma 2 we find a.t once!

where

'V~j

in the sense of

sta.nd.ing for a vector with the components

k ij

V

The explicit expre,ssion is n+2

S'ij,hk(x)

+ n(5ij x h%1t + \.It Xi!jJ

Observe that

r!r

n+2 ...

(6ikXj~,+ 8ihXj~ + °jhXiXk + BjkXi~) I!I n+2

It is also a C. - Z

S'Vij , hk = Syhk, ij

Suppos e that we apply (31) to SIv = -c Sf = -c" n n

8ij ohk} I~

= (oik8 jb +8 JkBib - 2

'V :::

Sf

.

Becaus e

• kernel. ISf = -c f n

we have

and thus

with

a

n

=c

Conversely, if

n ~

-b = 2(n+l)(n-2) -po

CI)

n(n+2)

(with cQmpaet

sup.p~rt)

n

satisfies (32) then (31) implies

120

so that

is satisfied by

Sf = \i

Theorem. 2.

If

~

1

Iv .

We have proved:

n

\) € Let:> has compact support a necessary and sufficient

condition that the equation is that

-r = - ·c-

Sf:= 'J

have a solution

with compact support

f

satisfies (32).

As for the sufficiency we are not claiming that Sf" =0

support, but since

="

= --1-

I~

bas compact

c in a neighbourhood of infinity

form (15) and by subtractir.g that trivial solution of solution of' Sf

f

f1

Sf = 0

is trivial of the

we obtain a

with compact support.

8.10 We sha11 now applY Theorem 1 to show that every q.c. deformation satisfies a near-Lipschitz condition. Lemma 3. Every q.c. deformation

f

Proof.

From Theorem 1 we obtain an obvious

We may choose

y' = 0

satisfies a condition of the

ro~

estimate of the form

+ C{R) Repla.ce

of degree

x

Iyl by

171 x

I-n we obtain

in the integral.

Because

'Vk(x)

is hcmogeneous

If

B(2)

r

yl' >R

the integral on the right is smaller than the same integral over

which is a finite constant by rotational symmetry.

Ixl ~ ~I ~ f Ivk(x -rtF) _...,.k(x) II = o(fxf- n )

also estimate the integral over

It is readily seen that

2

Iyl < R

If

we must

~

the integral is bounded by a constant times

l~

as

Therefore

x'" QO

and we conclude that

(33) is true for a suitable A(R)

We shall now prove that every q.c. deformation

8.11.

r(x)

support gives rise to a one'·para;nleter group of conformal mappings such that FsOFt = Fs+t - and Theorem 3. If

f'

is a

F{~, 0) = f(x.) k - q.,c.

with compact F (x) = F(x~ t) t

. .,More precisely:

deformation then

F t

is a

ekit! -g.c.

mapping. (~his t..heorem

Proof.

was firsi. proved

Assume first that

f'

by M. Reimann).

is

co

C

For fixed x we consider the

differential equation (3~)

.

F(X, t) = f(F(x~ t) }

with initial condition

F(x~O)

and the uniqueness fo~lows because Condi tionll •

Moreover,

The eJl"ist'ence of a solution is classica~

=X

r

satisfies (33) wnich is an "Osgood

Actually, the solution wi~l e""ist for all

f

because

f

is bOl,lllded.

1.22

F(X,s +t) =F(F(x,s) "t)

for both sides are solutions of means that

FtoF s = Fs+t

continuous iD

x

.

(3~)

with initial value

In particular,

FtoF:-t

=x

it is a homeomorphism.

Differentiation of (3'+) with respect to

x yields

.

(DF) (x,t)=(DfoF)DF(x,t)

Also

· =tr(Df'oF) (log JF)· = tr(DF)-1 (DF) -1 We o.ppq this to

XF = (JF) n: DF

to obtain

. 1 (XF) = ( - - tr(DfoF) +DfoF)XF n

and T

•

(['

T

2 n

[eXF) (D)] = (XFT[DfoF +Df of-- tr(DfoF) 1XF from which it follows that

By the Cauchy-Schwarz inequality

Together with

II sf'.1

< k we now obtain

F(x,s) and since

.

This F(x,t)

is

1?3

and by integration

where we have used the initia1 condition 11 XF(x,O) I r

If

and

is not differentiable we form the convolution

f

use it to generate Fe (x, t)

=.jn

is

Fe

K - q.c.

We write the

lls~el

Because

I

~ k

it follows as before that

with K = ekl tl

diffe~ential

equation for

F

E!

in integrated form

t

:re (x,t)=x+S o

Because the

Fe

compact set.

f

e

(Fe (x,t))dt

are K - q. c.

with a fixed It they are equicontinuous on every

Hence one can choose a sequence

converges to a limit

F (x,t) o

e(N) - 0

such that

Fe(N)

which satisfies

t

F (x, t) o

=x + S r(F 0 ex, t) )dt

This means that

o

F (x,t) o

is the unique solution of (3 4 ) and hence equal to

Since it is a homeomorphism. and limit of K - q.c.

it is itself K - q.c. 8.12. When F

The quantity

mappings with

and the theorem is proved.

MF undergoes a very simple and predictable change

is composed with a Mobius

transformation.

The corresponding rules

124

for the operators

Sand S *

are not quite as simple, but of vital importance.

Because we have used the letter

capital letters

~

in an important role we shall now use

A, B, etc. for Mobius transformations.

Recall that a Mobius

transformation A defines a change of coordinates that we prefer to write as x =Ai

(rather than

x=Ax

A contravariant vector

)

the i-coordinates acquires the co~onents

which we can also write as

We shall denote the function

f

by

fA

Explicitly,

The formula (35) defines a groUJ) repres'entation t for

as seen

by

the computation

The basic transformation formula for

Sf reads:

r(x) expressed in

125

Lemma. 4.

We remark first that differentiation of the identi ty X-~ = I

Proof. yie'l.d:s

and hence

D .X-1 = - X-l( D.X )-1 X # J

J

Therefore, differentiation of'

leads to (DrA) ij = - ((DA) - \ DA] ik(foA)k

(38)

+( (DA) -l(DfoA)DA)ij

In the first term on the right -2

where we have used the identity 2

(39)

(DA)TDA = (JA)ii I

.

]26

Differentiation of (39) gives

g

g

(DkDA)TDA + (DA) TDkDA =~ (JA)n (nit log JA)r =~ (JA)n =

~

-1

(tr(DA)-~kDA) I

2

(JA)

n (tr(DA) TDkDA) I

This shows that the symmetrization and normalization of the first term in

(38) contributes nothing.

On account of (39)

while tr[ (DA) -~foA DA~ = tr DfoA

Therefore the second term of (38) contributes exactly

(DA)-l(SfoA)DA

•

Observe that this computation has made essential use of the fact that is a conformal matrix. with values in

Note that

8.13.

VA

More generally, we shall say that a function

s~f transformB like a mixed tensor

has again values in

vex)

if

mf1 .

In order to continue this discussion of the invariance properties

we want to take a good look at the relation (41)

DA

SSf'. cp dx

=-

Jf. S*cp dx

We would like these inner products to be invariant when

f

is replaced

127

by

fA

and

~

transformed by

by

$A - aut this is not so if

(40)

and the integral in

Sf

and

~

are both

_ Indeed, we would have

(41)

would not

~tay

invariant. TO remedy the

situation we regard the integral as an inner product between the function

Sf

~dx,

and the measure

and We transform the measure

according to the rule

This amounts to transforming

$

as a mixed teqsor density according

to the rule

(42)

Similarly, the right hand side of regard

S* ~

(41)

makes sense only if we

as a contravariant vector density. In order to derive

the transformation rule for

S *4>

let

f

be a test-function and con-

sider that on one hand

and on the other hand

-f Sf·

dx ;;; -

J

(SfoA) (<poA) 1A

I

(x) t n dx =

128

= -

J At (x) -l(SfoA)A' (x)

• tfIA (x)dx

The comparison shows that

Here A1 (x)T == /A'(x)/2 A,(x)-1

and if we were to regard

S~ as a. contravariant

vector, then the transformation rule would read (43)

S *(~A) = ,At (x)

In +2 (s *-..p) A

•

However, it is more natural to regard

v=S *(CPA)

as a covariant vector density

which transforms according to

To sum up: The pairings

are defined and Mobius invariant when

f

is a contravariant vector,

v

is a

129

covariant vector density,"\J is a mixed tensor.

cp is a mixed. tens'or density

which transform according to the following rules

~

v A(x)= A' (x) -1 ,,(Ax)A r (x) a>A (x) ;:: 'A I (xl InA' (x) -lq>(Ax)A I (x} •

The invariance is in the sense that

The operators

S

s*

satisfy

S(f ) ;:: (Sf) A A

S {CPA' =

Sf'

,A' (x) In+2 (8' 'cp) A

is a rUxed tensor, "S*~ is a covariant vector density.

Sv or

S*"

8.14.

The fact that

Sf

densities makes the operator

is not a density while

* 8-)(8

S* 'V is defined only for

meaningless except in the euc/lidean case.

To rescue the situation we need an invariant density. such a density is given by the PoincarE! metric. with

We do not define

In hyperbolic space

We use the notation

ds

=

p' dxl

130

2

the density itself being given by

n

p

Now we can pass from tensors and vectors to densities simply multiplication by S*"n ~ Sf

For instance

pnSf

is a coyar i an t vector density, an d

is a mixed tensor density~

p -n-2 S *PnSf

is a contravarian t

To account for these changes it is convenient to define two new operators)

vector. namelY

pn

through

p

=p nSand

P* :;: p -n-2P

They satisfy the relations

and thus also

The basic invariant second order differential operator is this connection we shall say that a vector-valued function

P*Pf=O

(44) or

f

P*P

and in

is harmonic if

It is convenient to split this into two pieces, namely c:p=Ff, P

S *tp = 0

For

*~=O n =2

it turns out that

cp is a holomorphic quadratic

differential. How does ~

*

F P

compare with the Laplace-.Beltrami

applies to scalars and P*P

to vectors.

~.

In the first place

But in the Hodge-DeRham theory

131

there are two invariant operators

dB and

8d which apply to differential

forms of any order and in particular to first order differentials

which may be identified with whereas

in our termdnology

A differential is harmonic if (do + od)a= a

f f

is harmonic if

1 ) 1 (n - 1 d 60: - -2 8~ - ID = 0 Here

i ID=R. f.dx 04 J

3.

c..

R~ is the Ricci curvature tensor.

where

We sball now study the theory of functions

8.15.

in some detail. LeD1tJ19. 5.

a)

If

S SfijX j

P '*P.r = 0

then

dm(x) = 0

B(r)

r Sfl.J.• x. Xjdlc(x) = 0

..

3.

S(r) c)

r Sf .. X.X.x. d:'Jl{x) =0 l.J 3. J K S(r) J

Proof.

By

Green's formula

and this im,plies (45a).

Similarly~

f

that satisfy P*'Pf = 0

S(r

:x r nSf' x ~ JP i'J" '1 r =

J'

SDj (n P Sf".x. , dx = J pnSf .. 0 .. dY. = 0 ~J 1.

B(r)

I

l.J l.J

B(r)

and

r. nSf'1j JP

xxx. i;i a: r

A_ '-q

r DJ. ( P

I1-.p

=~

~.LijXi '~

) dx

.

B(r) =

J pnSfij(8ijXk +\:jXi)dx

B(r) =

S p~fikxi

dx

=C

by (45a)

B(r) Cor.

:[=*Pf = 0

S Sf •• 1.;]

implies

Y~j (x) 1-

dO' (x) = 0

S(r) In fa::t,

2Sf.k" Sf .k (x) = . J iJ :Vij r n .

x + (n-2) j

r

n+2

Now Theorem 1 (8.2) leads at once to the Center formula. c f(O) = r (I + (n-2)Q(x»f(rx)dro(x) n Btl')

(46)

Recall that If

c

n

2(n-1) n

c :reO) = n

r

J

S

$

r < 1

.

n

limi ts a. e", the formula ramains true for (46' )

0

ill

has a continuous extension to

f

,

(I + (n-2)Q(x) ):r(x)dm(x)

S(l) r

=1:

,or even if it has radial

-133

There is a simplification if

f(x)

is ta.ngential for

J

xl =1

for then

(Q(X) f(x)) i ~ X.X .f.. (x) = a ~

J J

and the formula reduces to (~6 ft )

:r(o)

Sf(x)dm

=2c

n

8.16.

.

S

Just as in the case of Poisson's formula for harmonic functions we

can use (1.6') to express

f'(Y)

in terms of the boundary values.

Clearly,

we need or..ly apply (46') to the function

fT-l(x) = (T -1), (x)-lr(T -\r) y y y Because

Ty-I( 0)

=y

we obtain

On the left

and on the right we replace

x

by

T x to obtain y

r (I + (n...2)Q,(Ty x)TY• (x)f(x) ITY

cnr(y) = (1- ry( 2)

f

(x)J n-ldm(x)

t,I

S

Recall that

Ty I ex)

'T '(x) Y or

1. -

ly(2

Ix-yJ2

=I. Ty '(x)j

A(x,y)

and

I = 1- Iyf 2 [x,y]2

because

ix' =1

.

We have also shown (see

2.8~

(40») that

lah Ty x =A(X,y)x when

Ixl =1

.

Thus

or

Q(T x) ~(x,y) = ~(x,y)Q(x) y

Taking these simplir1cat1ons into account we end up with Theorem

4..

c fey)

=j

n

S

I [2 n+l (1-;,) A(x,y) (I + (n-2)Q(x) )f(x)dc.o(x)

1x-yI-n

There is again a simplification if .6(x,y) = (I - 2Q(x,y» (I - 2Q,(x»)

Moreover,

is tangential, for then Q(x)f(x) = 0

f

•

so that, if we -prefer, (47) can be

written as

(47)'

'bf(Y)

=.r (1- 111:!n+l(I - 2Q(x-y: )f(x)dm{x). s

8.17.

Ix -y I

Evidently we can get a corresponding formula for

erentiating(47) or (l7').

For arbitrary

y

12

( l -Jy ) n+1

2n

J

and

x-y1

f'>J

,1 0'" 1+2n(xy) ( 1 - 2xy )

by diff-

the campu~ation becomes almost

prohibitive, but we can use the device of ccmputing the Ignoring quadratic and higher terms in y

Sf(y)

we have

d~rivRtives

only at

y=O

.

135

Q(X-Y\j

= (Xt -'1i )(:fY}

'" (xi"'j - .xil'j - XjYi){l +2xyl

\](" -yl rv

xixj +2xi Xj(XY):'XiYj-XjYi

so that (1._ly!2,n+l

(I - 2Q(x .. ,y)\j '"

Ix _yf2n (I - 2Q.(x)) .• +2n(I - 2Q(x)). j (x.y) - 4x. x. (xy) +2{x' Yj + x.y.) ~J ~ 1. J ~ J J. It follows that

For simplicity we shall do only the Xj f j

=0

tan,gentia~

on the boundary the result reduces cnDkfi (0) =

S (2nfi~ +

ca.&;le.

By virtue of

~o

2f x i )din(x) k

S Symmetriz~tion

leads to

n (Dk f i (O)"+Di f'k(O»)=2(n+l)

C

S (fj~+f1tXj)dm

.

S

The trace on the right is already zero a.nd we find (48)

CnSf( 0) ik = (n+1)

J (fi~ + fkX i ' dro

,

S

This is already quite neat, but we get an even nicer formula if we replace

the integral

by

a volume integral.

By

the Green-Stokes formula

I S

and

j

(fi%k +f'kxi)Cko=

B

(Dkf'i +D :f'k)dx 1

~

B imi laxly

0==

!

fhXhda> =

S

I J)hfh dx B

Accordingly, we cO::lclude from. (liS) that (49)

cnSf(O) ::; 2(n+l)

J Sf(X)d,...

.

B

It is now very easy to pass to the general formula.

(49)

to

iT -1

Sf(X}

~

before, we applY

is then replaced by

if

(T -1) '(x)-lSf(T- 1 x)(T -1) '(x) Y' y Y For x

=0

factor is

the left and right factors cancel against each other and the middle

Srey)

variab Ie by T x y

In the integral on the right we replace the integration If we observe that

the integral

becomes

S'fy' (x}S:f(X)l'y' (x) -1

.

B

The scalar factors in (50)

Ty'(x)

cngrey) ==2{n+l)

and

T' (x )

-1

y

cancel, and we obtain

r b(Ji:,y)Sf(x)A(y,x)

u

•

B

It is again preferable to

f'OCUB

the attention on

We obtain the reproducing formula for Tkeorem '5.

If'

cp =p nSf

~

in the form:

wiiih tang-cntinl

f

and

S*cp= 0

, then

It is instructive to compare (51) with the Bergma'I,l kernel formula t'~

analytic functions in the unit disk.

Izl

E LI(I)

then

converges absolute1y a.e. and belongs to Proof.

Let

P

be a fundamental set :for

J II cpr IelY =E SII cpll dx < B

1 L (r)

AEr AP

00

r

By assumption

~40

13ut

p

and CPA {x} = rA· (X), nA t (x) -lcp(Ax) A' (x)

f f q>A (x) , , = 'A' (x) ,n, , tp(~) r ,

From this we conclude that

r

(53)

L;

"qJAfI dx
where

Jx = x

cp(x*)

Ixl

and for

=

=1

*

~

would be defined by

This condition reads

Ixl2n(I - 2Q(x) hp(x~ (I - 2Q(x) ) it reduces to

Q.(x;cp(x) =q>(x)Q(x}

The following is an analYtic finiteness· theorem whose topological and

geometric consequences for the quotient manifold Theorem 7space

Q(r)

r

If

is finitely generated, then the dimension of the linear

As already mentioned (see 8.18) if

is a trivial vector function.

quadratic polYnomial in the Suppose that o.f

Q(r)

are not clear.

is finite.

Sketch of proof. fA-f

M(r)

r

Xi

CPA =cp then PAf=

As such each component of PAr is a

determined by a finite number of coefficients.

is generated by Al , ... ,.AN .

There is a linear map

on a l'inite dimensional vector space which takes each cpt:: Q(r)

into

the coefficients of P f"."PA f The kernel. of this homomorphism consists A1 n of all ~ such that the corresponding f is automorphic with respect to r The thp.orem wi1.1 be provpd if we show tha.t

For this purpose we study the

~ntegral

t=1U~h

s.

(,fI

must be idfmtjcaJly l7.ero.

~46

which can also be written as

.r Djfi

-.r fiDjqlijdy.+! fi(Oijnjdcr

CPijdX=

P

P

bP

is the outer normal of oP The boundary

oP consists

Plrt of n and points on A . faces.

* .)

of pairwise congruent faces of the polyhedron, Let

A and

A~

be a pair of corresponding

Then

r

.1 AA

foi tn~J.nJ.d a =

. . . -1.

f f.1 (Ax)cplJ •• (Ax)n.J (Ax)' A

•.

I

(x)

In - Ida

A

On the right we substitute

f.(Ax) = (A'(x)f(x»). ::;A'(x). f (x) 1

1

The minus sign is because

at Ax

A maps the outer normal at

x

on the inner normal

.

The integrand

*)

1a a

~ecom~s

lie are temporarily assumming that

Green's formula is applicable.

147

-

f

a

ti) 4> ac (x) n c

(x)

and thus

S'

A6

f.

~

4l • •n . dO' 1J

J

f

=-

A

f. 1

~ . . n . dO' 1)

•

)

The integrals cancel against each other.

Suppose now that

of

A

in

(56)

ap.

belong to

P

is a finite polyhedron and that no points

The remaining part of the boundary integral

is then

S

f . cp •• x. do •

nnP

1

1J

)

Now we make use of Condition IV

~

= 0

2

f.~ .. x.;dw 1. .l..J J

=

f. '. • x . x~ dw .l.. 1.J J K

f.x. 1. ].

=

0

which is zero because and bence

to write

=

by assumption.This proves

($,~)

=

0

under strong regularity assumptions.

In order to get rid of the extra assumptions we use the same mollifier technique as in the standard proof of the finiteness theorem 00

support on of

ap.

P \

~he

A and satisfies

-

C (P)

for Kleinian groups. Suppose that

A(Ax)

I

= A(X)

A

~

0

I

has compact

at equivalent points

earlier reasoning is sufficient to prove that

148

(57)

= -

(CP , ).

Möbius transformations in several dimensions

Möbius transformations in several dimensions

Miracle in Three Dimensions

Phase Transformations in Materials

Dynamics in Infinite Dimensions

Dynamics in several complex variables

Art in Three Dimensions

Functional Equations in Several Variables

Miracle In Three Dimensions

Several Devils

Functional equations in several variables

Stochastic equations in infinite dimensions

Graphene: Carbon in Two Dimensions

Complex Numbers in N Dimensions

Stochastic equations in infinite dimensions

New Dimensions in Regional Integration

Wave Equations in Higher Dimensions

New Dimensions in Body Psychotherapy

Lie-Bacjlund transformations in applocations

Conformal transformations in electrical engineering

Cosmic explosions in three dimensions

Lie-Backlund transformations in applications

Conformal Transformations in Electrical Engineering

Dark Dimensions

Wave Equations in Higher Dimensions

Complex numbers in N dimensions

Death's Dimensions

Geometric Transformations, Volume 2: Projective Transformations

Lie-Bäcklund transformations in applications

Dark Dimensions

Möbius transformations in several dimensions