Differential Systems and Isometric Embeddings.(AM-114), Volume 114 9781400882106

Table of contents :
Contents
Preface
Commonly used notations
Conventions
1 Introduction
2 Structure equations of X^n ⊂ E^N
3 Pfaffian differential systems
4 Quasi-linear Pfaffian differential systems
5 The isometric embedding systern
6 The characteristic variety
7 Isometric embeddings of space forms
8 Embedding Cauchy-Riemann structures
References
Index

Citation preview

Annals of Mathematics Studies Number 114

THE WILLIAM H. ROEVER LECTURES IN GEOMETRY The W illiam H. Roever Lectures in Geometry were established in 1982 by his sons W il­ liam A. and Frederick H. Roever, and members of their families, as a lasting memorial to their father, and as a continuing source of strength for the department of mathematics at W ashington University, which owes so much to his long career. After receiving a B .S. in M echanical Engineering from W ashington University in 1897, W illiam H. Roever studied mathematics at Harvard University, where he re­ ceived the Ph.D . in 1906. After two years of teaching at the M assachusetts Institute of Technology, he returned to W ashington University in 1908. There he spent his entire career, serving as chairman of the Department of M athematics and Astronomy from 1932 until his retirement in 1945. Professor Roever published over 40 articles and several books, nearly all in his spe­ cialty, descriptive geometry. He served on the council of the American M athematical Society and on the editorial board o f the M athematical Association of America and was a m em ber of the mathematical societies of Italy and Germany. His rich and fruitful professional life remains an important example to his Department.

DIFFERENTIAL SYSTEMS AND ISOMETRIC EMBEDDINGS

BY

PHILLIP A. GRIFFITHS

and

GARY R. JENSEN

THE WILLIAM H. ROEVER LECTURES IN GEOMETRY W A SH IN G T O N UNIVERSITY ST. LOUIS

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

1987

C opyright © 1987 by Princeton U niversity Press ALL RIGHTS RESERVED

T he A nnals o f M athem atics Studies are edited by W illiam B row der, Robert P. L anglands, John M ilnor, and Elias M . Stein Corresponding editors: Stefan Hildebrandt, H. B laine L aw son, L ouis Nirenberg, and D avid V ogan

C lothbound editions o f Princeton U niversity Press books are printed on acid-free paper, and binding m aterials are chosen for strength and durability. Paperbacks, w hile satisfactory for personal collection s, are not usually suitable for library rebinding

ISB N 0 -6 9 1 -0 8 4 2 9 -7 (cloth) ISB N 0 -6 9 1 -0 8 4 3 0 -0 (paper)

Printed in the U nited States o f A m erica by Princeton U niversity Press, 41 W illiam Street Princeton, N ew Jersey

* Library o f Congress C ataloging in Publication data w ill be found on the last printed page o f this book

William H. Roever 1874-1951

Co nte nt s

Pr efa ce

....................................................

ix

...................................

xi

C o m m o n l y u sed notat ion C o n v e nt i o n s

..............................................

1

Introduc t ion

2

S tr u e t ur e

3

Pf af f ian di f f erent ial

4

Quas i-1 inear Pf af f ian di f f erent ial

5

The

6

The c h a ract er i s t ic var ie ty

7

Is ome tri c

8

E m b e d di n g C a u c h y - R i e m a n n

Index

. . . ......................................

e q ua tio ns

n X

of

N C E

3

.................

14

...............

32

sys terns

i some tr ic e m be d d in g

Ref erenc es

xii

sys tern

sys terns

.....

...............

............

58 89 112

............

156

...........

194

...............................................

213

e mbe dd i ng s

of

space

forms

str uet ure s

.....................................

219

P R EFA CE

Thi s m o n o g r a p h Ro ev er L e c tu re s J a n u a r y , 1984,

is an ela bor at ion of

d e 1 ivered by Phi 11 ip Gr i f f i ths at W a s h i n g t o n Uni v e r s i ty in St.

11 c o n tain s an exposi tion of Pf af f ian di f f erent ial on the

theory of

theory

is

the

sys terns, wi th par t icular

the

e mb e d d i n g of a Ri e m a nn i a n mani fold the

Louis.

emphas i s The

th roughout wi th a d e t a il e d

to the p r o b 1em of

s p a c e , w her e

in

theory of q u a s i - 1 inear

the cha rac ter i s t ic var ie t y .

i 1 lus trated

a p p 1 icat ion

the W i 1 1 iam H .

local

i some tr ic

into E u c 1 idean

theory qui te d ir e ct l y and n a tu r a l l y

u nc ove r s the es sent ial po intwi se algeb rai c condi t i ons neces sary last

for

ch apter

the exi s tence of the

theory

is a p p 1ied

e m b e dd i n g C au c hy - R i emann We w is h lent

job of

in k e e p i n g

In the

to the p r o b 1em of

s true t u r e s .

to thank Micki typing and

such em be dd i n g s .

Wilderspin

for her

excel­

ef f icient o rg a niza t i on

the m a nusc r ip t in order as

bits and p ie ces be tween St.

for her

it

trave 1 led

in

Loui s and D u r h a m .

P hi lli p G r i f f it h s D u r h a m , N o r t h C a r o 1 ina G ary R . J ens en S t . Loui s , Mi ssour i N o v e m b e r , 1984

C O M M O N L Y U S ED N O T A T I O N

1)

( X ,ds^) N

is an a b s tr ac t R i e m a n n i a n manifold.

2)

E

3)

— N x :X -» X C E

4)

isE u c 1idean

is an

X.

y € X

is a poin t wi th tangent

For

space

N C E

X

tangent

space

S^ V

7)

form

space

V.

deno tes

the

A

10)

X c A *M X X

is is

e1 11)

and

space

a poin t

wi th

N^X,

and 2nd

11(x) € S ^T * X ® N X. v 1 x x

C

p ro d uct

of a

q-f orms on a mani fold

We also set

su b-b u n dl es is

thought $ C A M

( I ,J )

C°°

secti ons of by

1o cal ly g e n e r a t e d over

00 C M

by

is

is

M

is

0

I

I C T*M. s , . . . ,0

;

wp . the a l g e br a i c locally wr i te

of as

on

T M.

1o call y g e n e r a t e d over

(X) C A M

and

¥:

00 C M

0 s ,O.1

M,

C°°M = A ° M .

I C J C

the space of

some t imes we

12)

normal

Pf af f ian di f f erent ial sys tern

g iv e n by 9)

T ^X

oo

A*M = ffi A q M. q>0 8)

T^X,

x €X

d eno tes the qth s y m met ric

ve ct or q A M

space

T*X.

we denot e by

f un dam ent al 6)

i some t r i c e m be d d in g w it h

image

c o t an g e nt 5)

N-space.

ideal {X}

the c o 11e ct io n of

the di f ferent ial

g e n e r a t e d by

= {0}

wh er e

1_;

0

is

0a .

ideal

g e n e r a t e d by

X*

CONVENTIONS

1)

Summat i on conven t i o n , m e a n i ng

r ep eat ed

indices

except whe re

2)

The

in a p r o d u c t , will

sum all pai rs of be used

througho ut

e x p 1 i ci tly s tated o therwi s e .

f oil ow i ng

ranges

1
0

and

[1] has made

INT R O D U C T I O N

impor tant p r o gr e s s

(5)

T H E O RE M

on

7

the case w he n

K

chan ges

( N a s h - G r e e n e ; c f . Gr e e n e

sign.

[1]).

T here

i tera tio n

scheme

00

exi s t local

C

i some tr ic emb ed d in g s

Xn -

The m e th o d of proof a n al o g o u s

turns

s o , the

f or

f inding

root s of a

The d im e n s io n r e s t r i c t i o n comes

that at each out

is to apply an

to N e w t o n ’s me thods

polynomial. fact

EN ( n ) + n .

stage

the

1 inear p r o b 1em

from

to be

the so 1ved

to be a sys tern of al gebrai c e q u a t i o n s .

i terat ion scheme

’’loses

step,

and c o n s e q ue n t l y

a p ply

smoo thing operat or s .

f amous N a sh - M o s e r H a m i 1 ton [1]

two d e r i v a t i v e s ” at

it is n e c e ss a r y at each

theorem

each

s tage

Thi s is the genes i s of

implici t func t ion

for a recent

E ve n

to

the

(cf.

o v e r a l 1 a c c o u n t ). 00

We e m ph a s iz e

that we are

ig noring

theory of N a s h - G r o m o v ; a survey of found

in Gre e n e

[1],

these

Gromov-Rokhlin

[1],

B ef or e di scuss ing u n i q ue ne s s we t e r m i n o 1o g y .

or G r om o v

there

is def ined

[1].

introd uce a little

Xn C En + r ,

x € X

C

results may be

G iv e n a submani fold

(6)

at each poi nt

the g 1obal

the

8

CHAPTER

second

f un damental

form

II(x) V

wh ere

N X x

is

1

'

€ N X ® S 2 T*X X

the normal

X

space

is the 2nd

symmet ric power

each

there will be a Zar i ski

n,r

speci f ied

submani fold

II(x) € U v 3 n ,r thi s has

n ,r

(6)

for each

thought

the co tangent open

x € X

image mani fold of as

foilows.

be

.

It will

in the sense ac t ing on said

2 * S T X x

space.

For

D

those

n ,r

say

in case

be

seen

that

that

U is n ,r r 2 n IR ® S IR . An

to be n o n - d e g e n e r a t e

in

is n o n - d e g e n e r a t e .

Thi s may be

If we

s tand

set

(here

degenerate)

then

and

subse t (to be

is n o n - d e g e n e r a t e

0( r } x 0(n)

i some tr i c e m be d d i n g will the

x

C IRr ® S 2 IRn

intr ins i c m e a n in g

invar iant under

case

at

j* IR -v alu ed quadr at ic f orms , and we

in the space of the

X

in def ini t ion (6.64))

U

that

of

to

= IRr ® S 2 Rn \ U

n ,r

i some tr ic emb edd in g s wi th

II(x) € D v J n ,r

"D"

for

IN TR OD U C T IO N

sa tis fy an addi t ional (2).

T h u s , ro ugh ly

2nd order

these

(2) are talks we

i some tri c For rigid

imposed. shall

sys tem b e y o n d for an

that no addi t ional In the

c ons id er

first

equat i ons

six c ha p te r s

of

only n o n - d e g e n e r a t e

embeddings. these e m b e d d i n g s , we

in case

the m a pp i n g

cl ass ica l

x

—-2 —-2

result

is

( A1l en d o er f e r- B e ez )

i sometr ic e mb e d d i n g

is rigid

that

(0)

is

is u ni q u e l y de d e t e r m i n e d , up

EE ,by , by the the dsds

1ocal u n iq u e n e s s

THEOREM

s h a l 1 say

N

to a rigid mmoottiioonn of of

(7)

P.D.E.

speaking, non-degeneracy

i some trie e mb e d d i n g means beyond

9

—

onon X X.The . The m ainm ain

the

f o 1 lowing

A g e n e ri c

local

in case

r < [n/3].

Here gener

ic m e a n s •1)

the

submanf old

has dimens ion

submani fold has maximal ref e r e n c e s ).

A proof

Chern-Osserman [1];

r e la te d

Kaneda-Tanaka In these

of

first n o r m a 1 space rat every point;

t y p e , (see

d e no te s

and

the 2)

the

the f o 1 lowing

thi s r e s u 11 may be

[1] or K o b a y a s h i - N o m i z u

[1],

f ound

in

or S p iv a k

r igidi ty ques t ions are di scuss ed

in

[1]. talks we will

us ing exter i or di f f erent ial ^(X)

of

the b u nd l e

s tudy

i sometr ic e m be d d in g s

sys terns ( E . D . S . ’s).

of o rt h o nor mal

frames

If on the

10

CH A P T E R

Ri e m a n n i a n mani fold

X,

1

then we

s h a l 1 set up an E.D.S.

on

3f(X) X gc(EN )

whos e all

1ocal

integral

mani folds

Darboux

f ramings

of

The a p p r o a c h

is b a s i c a l ly

of Car tan [1], di f f e r e n t l y . of

except

the

i sometr ic e m b ed di n gs

(0).

same as

one

that we will

Our main

technique

the c la ssi cal

pro 1ong

to the

to the

f oilo win g

system

the

(cf.

a)

theory

(c f . Brya nt

i some trie e m be d d i ng problem, B e rg e r - B r y a n t - G r i f f i ths

B ry an t -G r i f f i th s-Yang

THEOREM:

the

is to ap p l y

the char ac ter i s t i c var i e ty of an E.D.S.

e t a l . [1])

(8)

local

(i.e. , ’’s o l u t i o n s ” ) are

1eadi ng

[2] and

[1]):

A 1ocal

i some tr i c emb edd ing

is rigid

if r < n, r < n

-

1,

r < 4,

b b)) ex xp pl la a i ne d below) be low)

when w hen when

It depen ds

n > 8; n=4,

5 or

7

n = 6.

only 11 on depend co ns tasn tonly s (toonbecons tant s

if

r < (n - 1)(n - 2)/2 = n(n - l)/2 - (n - 1).

IN T R OD U C T IO N

c) *

If

11

d e t ( R . .) £ 0, v iJ J

then

local

C

also give a p re t t y g ood p i ct u re

of

embe dd ings

X3 C E 6

exi s t in the case

Our

results will

n = 3.

char ac ter i s t i c var i e ty for arbi trary

n,

so

that

the

the

00

genera l some

C

1ocal

sense

red uce d

i some trie embedd ing p r o b 1em is

in

to a q u e s t i o n

theory

(in fact,

to so 1v ing wi th

1st order

d e t e r m i n e d 1 inear

Br y a n t - G r i f f i t hs-Yang As a fur ther di f f erent ial

in case

cons tant

sui table es t imates

systems and

in m ind we

k = 0,

N = 2n,

sys tern; c f .

the use of e xt eri or

iso metric

first

k < 0.

inves t igate

the e m b ed di n g The

W it h

N = 2n - 1

B ot h

is r eq ui re d

systerns.

( thi s turns out

h y p e rb o l i e

to be

re 1evant a l g e b r a , w hi c h

s i tuat ions are beauti ful

o v er d e ter min ed,

the case

the si tuat ion whe n

to E . Car t a n , is then eas ily a d a p t a b l e

k = -1,

e m b ed d i n gs

is a R i e m a n n i a n mani fold of

sui tably n o n - d e g e n e r a t e .

N ).

1ocal

c u r vatu re

and

the g ene ri c

the i r ch arac ter i Stic v a r i e t i e s ,

study

— 2 ( X ,ds )

k = -1

due

P.D.E.

i 1 lus trat i on of

sectional

P.D.E.

[1]) .

in S e c t i o n 7 we will (0)

in 1 inear

to be

to

the case

the mini mal

e x a m p 1es of

exter ior di f f erent ial

is

12

CHAPTER

As

indicated,

em b e d di n g s v ia

1

our phi 1osophy

the general

var ie t i es of di f f erent ial i n c re a s in g l y c 1e a r , this

is

systems.

As

b e en done told by

so far

these

11 is our

is the

i)

Use

of

ii)

(c f . the examp 1es feeling

in

that what has

i c e b e r g ; the

story

in an extens ive d e v e 1o p m e n t .

that

seem

to ment i on the

to us

r ipe for

the

In

two m ain

study*

the comp 1 ex charac ter i s t i c var i e ty to s tudy

n a t u r a l l y ar i s ing ell ipt ic fini te

sys terns that arise

tip of an

par t i c u l a r , we w o u 1d like areas

for

talks h o p e f u 1 ly may be v ie w ed as only

b e g i n n i n g chap ter

g e neral

is b e co m i n g

is a p o w e r f u 1 m et h o d

in g e om e t r i c p r o b 1ems

B r y an t - e t a l . [ 1 ] ) .

i some tr i c

theory of charac ter i s t i c

s tudy ing cer tain n o n - 1 inear P.D.E. naturally

to s tudy

sys terns (o ther

than

those

of

t y p e ).

Use

of

the charac ter i s t i c var i e ty in g 1o b a 1

questions. Regarding I. _3

(X

Are —

,ds

2

the

latter,

we ment i on

the foil owing p r o b 1e m s :

there curva ture as sump t i ons

)

that g u a r an t e e

on a compac t

r ig id ity of an

i some tr i c

embe dd ing

X -> E 6 ?

Here we have Theorem.

in mind g e n e r a l i z i n g

(See Che rn

[1] or S p ivak

the famous C o h n - V o s s e n [1]).

IN T R O D U C T I O N

Th er e

13

is the f oll o w i ng result of C h e r n - K u iper ^ 2 (X ,ds ) wi th non-p osi tive sectional

A c om pac t

[1]:

^

2 ^

c u rv at u re

canno t be

i s o m et r i ca l l y

(See K o ba y a s h i - No m i za [1]). ^

a flat

(X

immer sed

into

E

In Chap ter 7 we pr o v e

that

i s o m e t r i c al l y e m be d d ed

in a

2

,ds

)

canno t be

2n

2

n o n - d e g e n e r a t e fash io n even 1o c a 1 ly in E , w h i 1e a ^ 2 space (X ,ds ) wi th cons tant negat ive sectional cu rv a t ur e

can be 2n

em be d d e d

1o cally

i some tr ically n o n - d e g e n e r a t e l y

^

in E

.

The

f o 1 lowing p r o b l e m

is open

for

n > 3: II.

Does

there exi s t a global

i sometr ic e m b ed d i n g

Hn - E 2n~1 ,

wh ere

Hn

is the

n -d i m en s ional h y pe r b o l i e

Here we o b v io u s l y have

in mind

the class ica l

Theorem.

(See Do C arm o

III.

there cur va t ur e a ss u m p t i o n s

Are

p r ev en t

space

f orm?

Hil be r t

[1]) .

the exi s tence of a global

on

_3 2 (X ,ds )

that

i some tr i c e m b e d d i n g

X -* E 6 ?

H e r e , we a g a i n h ave

in mi nd

extens i on by Ef imov

[1].

the H ilb ert T h e o r e m and

its

CHAPTER 2 S T R U C T U R E E Q U A T I O N S OF

T h r ou g h o u t the c alc u l us

these

talks we

of d if f er e n ti a l

of you know,

p ic tu r e

then

in m ind

they are defined.

convention,

and

shall

We

we

shall

shall

emplo y

as

the g e o m e t r i c

to use d i f f e re n t ia l

a c c u r a t e l y and e f f e c t i v e l y wi tho ut wh ere

Moreover,

c omment

r es t r i c t i o n notation;

if one keeps

it is p o ss i b le

s h a l 1 use wi thout

forms.

f r eq ue n tl y omit p u l l b a c k and most

Xn C EN

forms

u n d ul y w o r r y i n g about also use

the f ol l o win g

s u m ma t i o n ranges

of

indi ces

1 < a , b , c < N = n + r

(i)

1 < i ,j ,k < n n + 1 < p ,d < n + r .

Or thonormal fx,e,,...,e„), v 1 N set of all

f rames

on

E

will

some tim es a b b r e v ia t e d

f rames

cons t i tu tes

bund 1 e

14

be deno ted by to

(x;e

the or thonormal

). a' f rame

The

S T R U C T U R E E Q U A T I O N S OF

wi th

f ibre

N ^ (E )

O(N).

may be

m o 1 1 ons

ofr

11

is

expl ici t .

Xn C EN

U pon cho ice of a ref erence

ident i f ied wi th the gro up

E(N)

15

fr a m e , of rigid

T E7N . impor tant To do

to make

this we

thi s identi f icat ion

repr ese nt

E(N)

in

GL(N+1;IR)

by

E(»>

The

= < (o *

s tandard ac t ion of

a c t i o n of

E(N)

:A e 0(N) , a € IRN }

GL(N+1 ;IR)

on

simplici ty we writ e

multiplication

rule

in

( a ,A) E(N)

( a , A ) ( b ,B)

and

N+1

induces

on

E N = { [ * ] : * e E 1*} C

For

E

the act ion of

E(N)

e

for

"*1.

^

.

T he n

is

= (Ab + a , A B ) ,

on

E

N

is

(a ,A)x = a + A x .

For a ref erence (0,e ^ , . . . , ) 1

N

is

,

frame of

w her e

0 € E

E N

the s tandard bas is

N

we choose

is of

the o ri g i n and N IR .

The

the

the

16

CHAPTER 2

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

E (N )

where

3t

£

(a,A)

is then g i v en by

-> ( a,A )x (0,fcl

is the

The Lie a l g e b r a

w r i t e

i t^1 c ol umn of

£ (N)

of

■X

(loo

( x , X)

X

f o r

E(N)

The

(X y

form

M au r e r - C a r tan form of €

o ( N ) ,

x

=

The M a u r e r - C a r t a n

( a , A )

e

0 0

[ ( x . X ) . ( y . Y ) ]

T h u s , at

i s

Y X ) .

r es t r i c ted

; I R)

A f y a . A )

=

eN )

by

H N)

We

i s o m or p h is m

(E N ) ,

e. = A e. 1 * l

. - R ljk«»"k“ e '

modulo

nothing.

Consequently, equating to

1o o k i n g

where

the

at

the

subset

^(X)

sat i s f ie d .

Gauss

if

by r

a

are

smooth not

d i m e n s ion is

M

let

l a t e r , of

equations

M = |N = °-

At

the o ther e x t r e m e , we may

non-involutive funct ions

say

that

N

is s trongly

if the set of c o mmon zeroes

{ f ,g}

wher e

f|^ = g J^ = 0

wo u l d conj ec ture

t h a t , for a general

Xn C £ n (n + l

the s ubv ari eti es

of all

the

is e m p t y .

We

submani fold

I k c T«X\{0}

are

s tro ngl y n o n -i n v o l u t i v e .

h om o g e n e o u s is

cone

lying over

Here,

is

C PT^X,

the

and

{0} C T

the z er o -sec t i o n .

From

( i ) in

(27)

COROLLARY:

(25)

is of

finite

(26) and

(7) we have

the foil owing

The i some tr i c emb e d d in g sys tern type

for

if

r < (n - 1 ) (n - 2 )/ 2 .

Thi s is par t ( b ) of T h e o re m

(1.8).

The

r igidi ty result

is p rov ed by a f u r t h e r , n o n - 1 i n e a r , c o mm ut a ti v e a l g e b r a ar gu me n t

that we

will not give here

B r y a n t , and Gr i f f i ths

[2]) .

(c f . §111

of B e r g e r ,

THE C H A R A C T E R I S T I C V A R I E T Y

F rom

(28)

(ii)

in (26)

COROLLARY:

mani fold w it h

If

together wit h

X

is a

de t ( R . .) ^ 0, v ij'

131

(23)

we have

3 - d i me n s io n a l then

1o c a 1

C

the

Riemannian

00

i some trie

e m bedd ings

X3

exist.

Of c o u r s e , it is c o n j e c tu r e d true

for all

n

c u r v a t u r e ), but

(assuming even

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

§ IId of B r y a n t , Griffiths,

in the

this result

some n o n - d e g e n e r a c y

qui te surpr izingly no t kno wn

di scuss i o n ) .

that

to be

and Y ang

Thi s c o r o 1 lary

the

1 inear P.D.E.

locally [1]

f or

is

so 1 v a b 1 e (c f . fur ther (1.8)

introduction.

above.

In order

to prove

the s y m b o 1 matr ix f or

(5.4)

on

is par t c ) of T h e o r e m

We have n ow reduc ed every thing

(I ,J )

remains

given as

in (5.4).

foilows:

{!_}

the

11

to T h e o r em

(26)

thi s theorem we mus t c o mput e isometric

e m b ed d i ng

is c onve ni ent

is g e n e r a t e d by

sys tern

to rewrite

CHAPTER 6

132

(29)

— 1 W

0)

(ii)

co

(iii)

Cl) . -

fi

0

=

i J 0ni

(iv)

The

i

(i)

— i (j .

= 0

J

1J

s tru ctu re

Cl)1 ) = 0

d(w

mod{ !_} (o1.) = 0 J3

(vii) d G *1 1

(viii) v 3

symbol

mod{ !_}

= 0

(vi)

The

h?. = h*t. 1J J1

equat i ons are

(v)

(29)

= o,

mod{ I_} mod{ !_} .

= -ir^ i J.

re lat ions are

( ix)

t

( H str) = 0

mod{J)

(30) (x)

Our

first

type of e xam ple ignored

p = iJ

tr . .

ji

tr . ..

Ji

observa t i on is that (10).

in compu t ing

Thus

this

sys tem

( v ) , (vi) and

the essent ial par t of

(vii) the

is of

may be symbo1

matrix. To get w ar me d up we n = 2 , r = 1 :

s h a l 1 b egin w it h

the

the case

THE C H A R A C T E R I S T I C V A R I E T Y

(31)

Iso met ri c e m b e d d i n g of a surface

In thi s case (31),

one each

Tf(H,TT) 1 2 1 2

there are

in ( ix) and

- h 1 1 ir2 2

w h ere we h ave d ro p pe d v a lue

is now

3.

Fr o m

(x).

the

11

\i

in

E

3

.

two s y m b o 1 relat ions From

+ h 2 2 Trn

From

r 212 - ,h 11>

(32)

just

133

in

( ix)

2 h i2 ir1 2

index here

~ r j '"’ji

s ince

its only

this we have

u

rx

- h 22>

rio 12

21

11

rg

,

h 12

-

-

12

.

(x)

2i 1 — r. 7T . . j

ji

we have

21

r2

(33)

,

22 , rj21

= 1 = Tj

?e s h a l 1 deno te a point

H =

(x ,e ,x ,e 1(e

b e c a u s e , as we will on

the

H

, H)

€

in

n

22 .

= 0 = r2

M

by

M C £(X)

x ?(En )

x 3f,

s e e , the s y m b o 1 m a tr i x d e pe nd s

coordinate.

From

(32) and

(33)

only

CHAPTER 6

134

(34) h 22^1

V H)

= (rjk f k }

C o ns e q u e n t l y

(35)

a ^ ( H)

is not

+ h n ff2 f 2 - 2

The charac ter i s t i c var i e ty is

O b s e r v i n g that

h 12f l

injective when

S H = {[f] e p 2 :det a f (H)

(35)

h l 1^2

-s.

det a f (H) = h ^ f ^

is z e r o .

h 12f 2

the di scr iminant

of

^

^

the quadr i c

= 0}.

the qua dra t i c

form

is

h ll h 2 2

wher e

K

is

If

co n j u ga t e this case the point

= K’

the Gaus s ian cu r v a tu r e

Gaus s equat i on

(36.1)

~ (h 1 2 ^

(2.19)

K > 0,

of

X

in this c a s e ) , we

then

^

infer

cons i s t s of

pure ly imag ina ry points whi 1e the

(this

is

that

two

H x = .

In

i some trie embedd ing sys tern is e l li p t ic

x € M) .

the

(at

THE C H A R A C T E R I S T I C V A R I E T Y

(36.2) y

If

J

K < 0,

di s t inc t real d i rec t i ons e m be dd i ng

(36.3) v J one

poin ts

on

E

x

= E^ C ,x

co nsi sts

( c or r e s po n d in g

X ).

In this case

of

two

to the a s y m p t o t ic

the

i some trie

sys tem is hyperbolie.

F i n a l l y , if

real

R em a r k

then

135

point

that

assumption

K = 0

c ou nt e d

to the Gauss

M

S

= E^ C ,x

x

consi s ts of

twice.

the poss i b i 1 i ty that

then

lie

H = 0

is ruled out by our

in the set of o rd i na r y

sol utions

equations.

To c ompu te the charac ter i Stic var i e ty in g e n e r a l , the s y m b o 1 m a t r i x

it will

he lp

to give

and

this

some addi t i onal no tat i on will

We

for

intrinsically, be usef u 1 .

let

r W = IR ,

wi th or thonormal

be a E u c 1idean vect or normal

space

a v ec to r We

w

U

repres ent ing a typi cal

s p a c e , and

V* = lRn ,

Xn .

bas i s

space set

wi th bas i s

r epr es ent ing a typi cal

w*

co tangent

space

to

CH A P T E R 6

136

S qV* = S y m q V*,

For

f ,r\ € V

product.

As

&

we deno te by

q = 1,2,...

2 ^

frj 6 S V

.

the i r symmetri c

In terms of c o m p o n e n t s ,

3

n > 3

there

that will

to

E

n+ 1

Xn

n > 3

we

all be is no

guarantee

the

of an

thi s rea son

sys tern for for

for

can never

cur v a tu r e a s s u m p t i o n

X n C E n (n+ ^ ^^

t e r m s ).

j*

x

convexi ty of all p r o j e c t i o n s

fails

1 inear

c on t a in i n g

x

(ii)

from

t e r m s ).

that

the

£ n (n+ ^ ^

al ways

(c f . the c omment be 1ow

(18)).

(74)

REMARK:

e m b ed d i ng

Sup pos e we

X n -» X C £ n (n + l

try

to f ind a ^

where

1ocal

i some tr i c

in a sui table

THE C H A R A C T E R I S T I C V A R I E T Y ,. , c oo rd i n at e ( y 1 ;z*1)

f 1

^ sys tem

the

(y

image

, . . . ,y

X

,z

n+1

n(n+l)/2 . , . . . ,z v ' }

is g iv e n as a g r a ph by

= z>*(y).

^

T h e n , a p p r o x i m a t e l y , the imply

n

153

isometric e mb e dd i n g e qu a ti ons

that

o2 p

d

(75)

„ i~ k

dy

z^

„ i„ 2

dy

dy

dy

R ijk«(y >

6 y Jdy

d y Jdy

(The ac t u a 1 Gauss (75),

but

the

equat ions are a s 1ight modi f icat ion of

s y m b o 1 p roper ties are

the same as

those

of

(75)). Wh e n

n = 2

equat i on for one

e qu a t io n

(75)

is one M o n g e - A m p er e

f unc t i o n ; these have b e en extens ively

studi e d . Wh e n

n = 3

the

sys tem

(75)

is

6

equat i ons

u nk no w ns , whi ch is ’’a pp ar en t ly ” overde termined . ent i rely cons i s tent w ith s y m b o 1 matr ices are 3 in the case

(47),

l x l

n = 3) .

s y m b o 1 1 eve 1” w h en e v er

sys t e m , w h i c h

e mb ed d i n g d i m e n s i o n .

imp lies

in the case

(They are n > 3).

charac ter i s t i c var i e ty has de te rmined

whi ch

it s h o u 1 d be

6 x

and on

the other h an d

the same dimens i on as

is as

the

’’o v e r d e t e r m i n e d On

3

Thi s is

that

n = 2

in

the

the

for a

in the

154

CHAPTER 6

One

of

the early at t e m pt s at

e mb e d d i ng p r o b l e m

in the case

sys terns (75).

feel

advantage

We

that

n = 3

there

in k ee p i ng e ve r yt h i n g in re 1 egat ing all

as possible,

the

C

00

was

i some tr i c to study

is an obvious

intr ins i c a n d , insofar the anal ys i s to the

charac ter i s t i c var i e t y .

(76)

REMARK:

G iv e n

R € K,

(77)

the equat i ons

'r(H.H) = R

have been

the object

is invar iant under

of cons i d e r a b 1e s t u d y . Since

0 (r)

the g r oup

nr

of normal ro tat i ons

we have

(78)

dim

image nr < rn(n + 1 ) / 2

H o w e v e r , the equat i ons symme tr i e s .

For

( c o r r e s p on d i ng

to

(77) may have addi t i o n a 1 "hidden"

e x a m p 1e , wh e n X

B r y a n t ,and Gr i f f i ths

4

- r (r - 1 ) / 2 .

C E [2]

5

)

n = 4,

r = 2

it was p r ov e d

in B e r g e r ,

tha t

dim K = 20 right hand dim

side of

(image nr) = 18.

(78)

= 19

THE C H A R A C T E R I S T I C V A R I E T Y

T h u s , there of

is a n o n -o b v i o u s

extr a

f ibre d i m e n s i o n

'Y . The A l le n d o e r f er- Bee z

theorem ment ioned

C ha pt e r H,H'

1 is b as e d on the fact t h a t , if o ^ C W ® S V are n o n - d e g e n e r a t e wi th

'y (H.H)

then

155

H = A ♦H '

study of and V ilms

for

some

the eq u a ti o n s [ 1 ].

in

r < [n/4]

and

= nr( H ' ,H' ) ,

A € 0(r)

(77)

.

A recent a lg e b r a i c

is g iv e n

in S t e i n e r , Teufel,

CHAPTER 7 IS OM E TR I C E M B E D D I N G S OF SP AC E FORM S

Two of

the salient

di f f erent ial

features

sys terns are

of

de te rmined

sys terns (it

o v er d e t e r m i n e d important

i ) that overde te rmined

is pr et ty c 1 ear

sys terns will

bec ome

in g e o m e t r y ) , and

ii)

ques t i ons

two g eom e t r i cally

overde termine d h yp e r b o l i e ad dre ss

the

the

of

the

theory g 1 obal In this

sec t i on we

interest ing

sys terns.

N a m e l y , we

shal1

f o ilow ing

PROBLEM:

of cons tant

that

increas ingly

shouId b ecom e a ccess i b 1e .

di scuss

sys terns

f oo t ing wit h

that b ec a us e

intr ins ic geo me tr i c charac ter of

(1)

theory of

1 eas t f o r m a l l y , pl ac e d on an equal

a r e , at

will

the

Let

(X n ,d s ^ )

sect i onal

be a Ri e m a n n i a n mani fold

c urva tur e

k.

De termine all

1o c a 1

i some trie em b e d di n g s

(2)

where

X * X C E n+r

r

the sense

is m i n i m a l , and whi ch are n o n - d e g e n e r a t e that

their Gauss m a pp i n g

156

is an

immersion.

in

ISO M E T R I C E M B E D D I N G S OF SPAC E FORMS

Thi s p r o b 1em was po s e d and __

Sinc e

—

-2

( X ,ds

)

so 1ved by E . Car tan

a n aly t i c ) so

[3].

is real analyt ic the Car t an -Kahler

is a p p 1 i c a b 1 e (when

th eorem

157

that

the

the f ac t that

be h y p e r b o l i e

is not

modi f icat ions

of

initial

d ata

the p r o b 1 em

so re 1e v a n t .

is turns

out

to

However, 00

this p r o b 1em

ques t i o n s , and

for

as

We

impor t a n t .

this

to p roper

its h y p e r b o 1 i c i ty

reaso n we vie w

s h a 11 restrict

C

our at tent i on

to

the

k < 0.

case

Case

1:

k = 0.

equat i ons

We be g i n by

s tudying

the Gauss

of

(3)

X n C E n+r

wher e

X

is flat.

E u c 1 idean v ect or pr od uc t vect or of

1ead

• , space

W ® S ^V*

For

thi s , let

space of dimens ion

and

let

V

be a real

(wi thout a g i ve n

be a real r

w it h

inner

n - d im e n s i o n a l

inner p r o d u c t ).

will be call ed 2nd

2 nd f u nd ame nta l

W

fun damental

E 1 ements forms.

form

H € W ® S 2 V*

is said

to be n o n - d e g e n e r a t e

the var i a b 1 es of

V

;

i.e.,

in case

H

invoIves

all

A

158

CHAPTER 7

H

for any pro per

€ W ® S2U

subsp ace

U C V

.

(Note:

this use

n o n - d e g e n e r a t e , whi ch is the s tandard one a l ge b r a wh e n these

talks

di ffers (3)

dim W = 1,

is spec ial

is n o n - d e g e n e r a t e

y J

2 ^

x

1 inear

from

to Sect i on 7 of

- it agr ees wi th Car t a n 's t e r m i n o 1ogy but

from our prev i ous

II(x) € N

of

® S T

x

if

t e r m i n o 1o g y .)

A submani fold

its 2 nd fund ame nta l

is n o n - d e g e n e r a t e at each

form x € X.

Thi s is equ i v a 1en t to the Gaus s m ap p in g

X -» G( n, n + r) x

h a v in g maxi mal We

T X x

rank.

s h a 11 restrict

our at tent i on

to n o n - d e g e n e r a t e

H ’s .

DEFINITION

(CARTAN):

or thogonal

if

(4)

where

H € W ® S 2 V*

'y ( H . H )

nr

is def ined

the Gaus s equat i ons

is e x te r i o rl y

= 0,

in (2.21) . for a flat

Cl ear l y , these are submani fold

(3).

The

jus

I S O M E T R I C E M B E D D I N G S OF S PAC E FORM S

m ai n a l g e b r a i c Car tan [3],

(5)

fact,

S pi v a k

THEOREM

s h a l 1 n o t pr o v e

w h ic h we

[1],

or Moo re

(Cartan):

[1]

is

(cf.

the

H e W ® S 2 V*

If

159

is

n o n - d e g e n e r a t e and exter iorly or t h o g o n a l , then

r > n.

If

1

r =

f or

n

W

then

there

a nd bas i s

is an or thonormal

1

i

® o 1 g . .o)J = e ® & ij p p

p v

p j

i

,

(13).

(16) is

P

p

PROPOS IT ION :

The P f a f fi a n d i ff e r en t i al

involut ive and h yperbolie.

s' = n 2 , 1

Thus

s'

2

Moreover,

= n (n - l)/2, K

the n o n - d e g e n e r a t e

system

3

s' =...= 3

flat

submani folds

s' = 0 . n

(15)

166

CHAPTER 7

X n c E 2n

1 oc al ly d epe nd on

n(n - 1 ) / 2

func t i ons of

two

variables.

PROOF :

Dur ing

convent i o n .

this proof

we will no t use

The M a u re r - C a r tan equat ions

the s true ture equat ions of

n+i

_

the P.D.S.

the (14)

summa t i on give

for

(15)

0

(17)

where

=

den ote s

c o n gr u e nc e modu lo

equat ion as

n+i i f II d(w . 3

(18)

so

that

the

t a b 1 eau m a tr i x

is

{I J .

Wri te

the 2nd

167

IS O M E TRIC E M B E D D I N G S OF SPA CE FORMS

11

In

.

nl

7r

11

In

nl

As a lw ays we may kth col u m n

*

ignore

is an

the block s

of z e r o e s . T he n

the

1 -f orm

(n x n ) - m a t r i x - v a l u e d

T k = ^ j k L < i ,j < n ’ At

this

junc ture an

N a m e l y , by def ini t ion

1

< k < n - 1

{J} and

k

to the

that

s£

is for

in dep end ent of

the

ir

A gener i c flag

1 -forms m o d u 1 o

for a gener i c f l a g ,

f lag g i ve n by

is n o n -g en e r i c (in fact,

i th ver tex of

J/I,

of

co lumns

it is p re t ty c 1 ear

1 2

of

s ' +...+

the n umber

in the first

{to ,to } , . . .

interes t ing point ar i s e s .

i to

{to^ } ,

c o r re s p o nd s

the charac ter i s t i c var i e t y ). is obt a i n ed

whi ch is re lated

to

f rom a gener i c bas i s to1

to1

by

to* = 1 t to^ , j J

wher e of

(18)

( t ^ ) € GL(n;ER) b ecom es

is gener i c .

T he n

the r ight

s ide

168

CHAPTER 7 „

i

-2

f.,

k

J

A

2

« +.k~£ t^G)

x ~i

=

“2

IT .0 AG) ,

where

Hence “ ? t«1rk' k

Thus, e.g.,

1 -forms

in the

s|

is

n x n

the numb er

A k = tk , A = ( A 1

= TTk ^ k

A n ).

tt(A)J' = 2 w J'aJ + 3

7r(A) v j 3. We claim

We have

2

i

^ 0,

are

1 inearly

are

1 inear ly i nde pendent modu 1o

by



in (17),

comput ing

the c hara cte ri stic var iety as

the s y m b o 1 m a t r i x H o w e v e r , us ing

and

oc

£

the

f rom

fails

these

symbo1

a p pear ing

to be

relat i ons the

1 ocus where

injective.

involut ivi ty of

an a l t e r n a t e a p p r o a c h b ased on a general

(15),

there

is

result g iv e n

in

172

CHAPTER 7

Br ya nt

et a l . [1].

deeper

results

As

this general

it will

also a pp ly

the charac ter i s t ic var i e ty of

show how

sys tem

the

to

shal1

the

involut ive

the c o mpu tat i on of

the h y pe r b o l i e

be 1 o w , we

(32)

it may be a p pl i ed

T h u s s we give

is one of

on charac ter i s t i c var i e t i es of

sys terns, and as

e mb ed d in g

result

state

space

f orm

it here and

to the p r o b 1 em at h a n d .

fo ilo wi ng

DIGRESSION:

Let

( I ,J )

be a quas i-1 inear

di f f erent ial

sys tern on a mani fold

M

Pf af f ian

g iv e n by

(c f .

( 6 . 2 ))

e“

= 0

a =

d 0 a = ~irpaw^

1

(0 A . . . ACJ

D

mod{0}

p =

0.

/

summat i on co nvent i on is bac k

use

the no tat i on i nt rod uce d at

6.

T h u s , for any

Ix C Jx C T x M .

1 , . . . ,p

~

The

C h apt er

1 , . .. , s

the b e g i n n i n g

x € M,

Us ing general

in effect,

and we of

= Jx / I x *

theorems about

shal1

whe re

dua1

s p a c e s , we ob tai n

v

where in

I^~ X

TxM .

and For

J"*" X

are

e x a m p 1e ,

X

= iVj1, X

X

the annihi la tor s of

I

X

and

J

X

173

I S O M E T R I C E M B E D D I N G S OF S P AC E FORMS

l£ - {v € T x M:0(v)

T h u s , the e 1ements m o d u 1 o v e c to r s For a class

in V*

^ V V v

x x p

, and

1-f orm by

x

V

are

x

tangent v e ct ors

a n n i h i l a t e d by co € J a).

J

x

Thus

in

T M x

.

we deno te its equ iva 1 ence co^ , . . . ,

is a bas i s of

deno te the

dual bas i s of

e q u i v al e n c e

class

_

v 1 , . . . ,v l p

we let

, w he r e

of

= 0 V 0 € I }.

vde not es p

the

of

€ I1 , x If

the s y m b o 1

relat i ons

for our

sys tem are

(c f .

(6 .2 ) iii)

a

(x)ira \ J pV(x)J = 0

then f or each

x € M

mod{vJ x } j ,

A = 1 , . . . ,Q ,

the charac ter i s t i c var i e ty

5 C P V* x

is def ined by

(cf.

(6.4))

~ x = {[? ] C P V * :r ^ p (x)f pT)a = 0

We

s h a l 1 w or k over a f ixed point

refe nce

for

rj ^ 0 } .

some

x € M,

and

s h a l 1 drop

to it.

S up po s e n ow char ac ter

%,

that

( I ,J )

is

involut ive and has

(see def inti on above

(4.26))

~ i.e.,

174

CHAPTER 7

s' * 0 , s' + 1

For

e x a m p 1e , (15) has

integer

= n(n - 1 ) / 2 .

Car tan -Ka hle r

1 ocal

f ind

integra1

charac ter

two and Car tan

A c c o r d in g

to the proof

integral

mani folds

submani fold on

admi s s i b 1e

of

s^

N

£

arbi trary

^-d i m e ns i o na l

admi s s i b 1 e requ ires

CM,

where

that

Pi

E fl

initial

data

N.

e 1ement s in

1^, x

E C T^M. and

to be

X

E,

el em ent s at

x € M

under

V.

In this w a y , we

^ -d i me nsi on al

as be ing c on t ai n e d of

the p ro jec ti on of

X

subspace of

the admi ssible

s h a l 1 def ine

Hence

-» i1 / /

may cons ider

(V)

^ J* 0 ,

to deno te

X

^ - d im e n si o n a l

£

= (0).

i1

We

the

that

whi ch we cont inue

Grassmannian

^- di me n s i on a l

" g e n e r a l ", we cons ider

integral

1

is an

the

by pos ing an

f unc t i ons g i ve n on

to A . . . A to

E,

( I ,J )

^-p lan e mus t be c on t a in e d

whi ch means

of

t h e o r e m , in the real a nalyt ic case we may

To clar i fy the m e a n i n g of

Such an

s^ = 0 .

val ue p r o b 1 em a lon g a " g e n e r a l ”

initial

depe nds

=...=

^-pla nes

in

integral

in the V.

the Car tan charac ter i s t i c var i e ty

IS O M E T R I C E M B E D D I N G S OF SPACE FORMS

A C G^(V),

and

then

" g e n e r a l " in case that

T N £ A x

it

the

initial

mani fold

will

is n o n - c h a r a c ter i s t i c in the

for all

x € N.

be sense

T h u s , the Car tan

ch arac ter i s t i c var i e ty addres ses the word

N

175

the P.D.E.

"charac ter i s t i c ” , w here as

m e a n i n g of

the usual

charac ter i s t i c var i e ty is def ined by prope r t i es of symbo1 m a p . ca lied

s u b 1 1ety

The

the o v er d e te r m i n e d

is

the

t h a t , in what migh t be £ < p -1 ,

case w hen

Car tan chara c ter i s t ic var i e ty

A C G^(V)

and

the the usual

x

char ac ter i s t i c var i e ty

H C PV

= G

. (V) p— 1

1ive

in

di f f erent p l a c e s . To def ine bas i s tt(E)

{el, 1 tJ be

K J

A

we cons ider

1 < t < £.

Writ e

~

the co 11 ec t ion

5-planes

of

e ^ = t^v t t p

1-f orms

consi s t s

1 -f orms

in the first

ofthe

set up w ith

respect

ojP = taP 0.

T h e n , for

if j > 1, by

(26)

0 = b-2 1

i

X.b. = X , l b , | 2 - X. i x 1 » 1 ' j

0 = b .•2 X.b. J t 1 1

= -X,

+ X . |b . |2 1 J1J1

2 j

2

X. i X.. i

CHAPTER 7

184

Su btr ac t ing , we have

X x (l +

\b1 \2 ) = X j (1 +

from w hi c h we c one lud e thus may

set

that

= 1 /B^

A^

A . > 0

J

> 0

i

for all

and n o r ma l i z e

2 b./B.

(i) v J

|bjI2 )

i

i

j.

so as

We to have

=0

(28) (ii) v 3

C om bi n i n g

(26)

and

(28) we get

(29)

B.

For shown

future

(29) ho Id for any (26) and

Jus t as

= 1 +

|b.|2 .

ref erence we no te here

that a u niqu e

satisfy

2 1/B. = 1. i i

set of re lations

set of ve cto rs span

charac ter i s t i c var i e ty H

g i ve n by

(30)

P RO P O S I T I O N :

(ii)

d i s tine t r e a 1 points

(28)

just

together w ith

b ^ , . . . ,b^ € W

whi ch

W.

in Proposi t ion

form

that we have

(7) above we may comp ute

H u C PV Jri

of a 2nd

the

fund am e nt a l

in (27) wi th (28) h o l d i n g .

H jj =

^

cons i s t s of

the

n(n ~ 1 )

IS OM ETRIC E M B E D D I N G S OF SPACE FORMS + [ f . .] = [V b T ca. + VB . co .] u ijJ L i i -

PROOF:

As

in the proof

[f ] €

is, by 6 .57,

0 / 7] € V

sat i sf y ing

of

that

w H

N o w , for an rank

H

< 2

g i v en by

if,

(7)

( i j* j ) .

1

the condi t ion that

there exi s t w €

W

and

= frj.

(ii)

and only

185

if,

in (27), for

w #H

some

i and

has j

di s t inc t

we have

(31)

w - b k = 0,

W he n

n = 2

w € W = IR1 .

(31)

V k * i ,j .

imposes no condi t ions on

In this case

(28)

says

that

B2 b 2 = “i f b l

and

thus

H = if ®

Up

(B i(u 1 )2 - b 2 ("2 )2 ]

to scalar m u 1 1 i p 1e , for any

w € W,

w •H

is g i v e n by

186

CHAPTER 7

B ^ ( ( j^

Thus

)2

-

B g ( c

2 }2

=

( +

V S ^

the only poss i b i 1i ty for

factors

on

Wh en

the right n > 2

or thogonal

side of

then

c om ple me nt

(31) of

Us ing

(26) a ga i n we

(b.

-

bj)*H

(29).

(6.23)

an al og o u s

If

the

imp lies

that

two

w

lies But

in the then ,

to scalar multiples,

(1

=

B .t o ,1 ) 2

(30) now foilows

this

we may draw

+ b j - b jK u h 2

-

result

immediately.

together wi t h

□

(6.15)

the foil owing conelus i on whi ch

is

( 1 0 ).

i some trie embedd ing sys tem for

involu t i v e , then n(n - 1)

the

this e q u a t i o n .

+ b . - b . H w 1 )2 - (1

=

Proposi t ion

to

is e i ther of

f ind that

As b e f o r e , from and

f

^ - >/B^ co2) .

s pan{b^ * •k ^ i ,j } .

by reason of dimens i o n , up

by

c o ^ J f V ^ Y

the

local

integral

^

is

mani folds depe nd

f unc t ions of one var i a b 1e .

sys tem is hyperbolie.

X n C E 2n

Mo r e o ve r

the

on

I SO M E T R I C E M B E D D I N G S OF SPAC E FORMS

Therefore, the

to pro ve

i some tr i c e mb ed d in g

we will

theorem

sys tern is involu t i v e .

a g a i n use Car t a n ’s test.

# C W x ...x W

be

the

set of

Remark

that:

a)

%

by

(ii)

(29)

relat ion in (28),

giv es

We

show

that

For

thi s

let

factors)

sat i s fy ing

= -i.

i * j.

is a submanf old of dimens ion

n (n - l)/ 2 , b ) that each un iq ue

(n

( b ^ , . . . ,b^)

b.-b.

(24) we must

187

(b^ , . . . ,b^)

€ Ik

sat i s f i es a

( i ) in (28) wi th the n o r m a l i z a t i o n g iv e n and c ) that

the ar gum en t

leading

that

On

M = £(X)

we cons ider e m be dd i ng

the

x 3T(E2 n _ 1 ) x *

f oil o w in g var iant of

sys tern of Sect i on 5

the

i some tr i c

to

188

CH A P T E R 7 1

to

(i)

(iii)

toi. - —toi. = 0

3

J , jit i - b .to ^ 0 i i

(iv) (V ) V J

where

b.

1

A to . / 1

=

EXPLANATION:

0

=

= 0

(ii) (32)

— I to

-

A (i) V }JL\V

(b'f) .

are m o t i v a t e d

(27).

Given

a point

def ine

a 2nd

fu n d a m e n t a l

H =

a

consequence

of

sat i s f i e d .

(32) X

give E^n

normal not. the

par t ially

\

where

frame The

b^

equations

The

is

2 b^e

. 1 u

(26)

reason are

for

®

summary

following M

we

equation

= -ds'2

n-dimensional framed

to

of

( w 1 )2 . v J

the G a u s s

c o m p o n e n t s of

is a

that

the

that

tangent

for

tensor

mani folds

of

e m b e d d ings

fr a m e d ” means

spin but

this

integral

i s o m e trie

"partially

free

the

f o r m by

'r(H.H)

a re

by

((y.e.J. fx .e ., e^),(b.))

11 .1

As

i)

0

^

( b 1 ..... b ” _ 1 ) = V 1 1 J

We

(no s u m mati on on

in

b . =

l

ERn

the

frame (b^), v lJ 0 IRn ,

is where the

I S O M E T R I C E M B E D D I N G S OF SPACE FORMS

= -1 ,

*b.

are

invar ian t

ro tat i ons

of

e m b e d d i ng

X

u n iq u e

under IRn .

frame

g e n e r a t e d by

*

de termines

but no t under an i some tr ic

(up to p er mu t a t ion)

charac ter i s t i c sys tem of f ieIds

(32)

As a c on s e q ue n c e

(jjl < v ) ;

d/d(J^

of

co1 = J

0,

i.e.

the Gauss

co* = 0 , co*f = J J

(i)

d (co1

(ii)

dco^ s 0

(iii)

d (co* - co1.) = 0 v J 3

(iv)

d(co^ - b^co1 ) = -2 (5. v i l 3 . v iJJ

0}.

equat i ons be ing

sati sf i e d , the s true ture equat ions of

(32) are

- co1 ) = 0

-

(b? v i

-

b*f Jco* )a co j 3 3

where

P»f = db»t + 2 u » b v J

and w her e that

=

d eno tes

in d er i v i n g

a

is

sys tem g iv e n by

co* = 0 , (0^ = 0 ,

1

ofIR

X.

the ve ctor

the F ro b en i u s

(co1 = 0 ,

ro tations

field on

The C au c h y

i / j,

Put di f f erent ly ,

E 2n

189

3

v

V

3

c o n gr u e n ce m o dul o

{ I_} .

(iv) we h a v e , as u s u a l , u sed

Remark the

is

190

CHAPTER 7

st ruc tur e

equ at i on s

s u mm a ti o n on reasons

i.

(23),

and

that

in (i v ) there

R e m a rk also

that

since,

e x pl a i ne d a b o v e , the

in dep en d en c e

condi t ion

(v)

( b^

v i

-

. For

the

ite

will

appea r

as

of

J

Thus we

where

(33 )

The

the

the f orms

co* J

theory

in the reduce d

is now c o nven ien t

in (4.1).

tab 1eau m a t r i x

torsi o n ” of and

are

J

Thus

the

(36)

of

co* (32)

{J } .

to use v e ct o r - v a l u e d

forms.

set

n U. = i

t (fc o1

P . = J

t ( p 1...... /31?- 1 ) = db . + tob .. v J J J J

v

co = (co**) ,

n - K) i '

co .

l

and we write

v

d (Q . - b .co* ) = -2

fact

am ong

terms

J

’’par t of

they are not zero modu lo

11

the

in the

b^)co*Acov

the sys tern (32) the g eneral

the

do not appea r

in (32),

should no t be cons idered as (32)

co*

for

is no

i

that

( i ) in (28)

the

of v e ctor s

, J

(6. .p. ij J

i

'

b ^ ’s imp lies

(iv ) as

(b . - b .)co*. )aco J

v i

is the only

that

{ b . - b^ |i ^ j }

for each

1 inear

re lat i on

f ixed

i

is a bas i s for

W.

the set Thus,

ISO M E T R I C E M B E D D I N G S OF SPACE FORMS

m o ti v a t e d by in terms

of

the

term

(26)

(34)

= 2 (b. j

1 -f orms

{tt ^ ^ |i & j } .

Differentiation

(26) and

(29) , imp lies

that

J

ij

i

there are at

that

from am ong

1 eas t

The conelus ion is that

The

(35) y

'

the

n(n - 1 ) / 2

m o r e o v e r , (34) gives

TTiJ

Ji

the def ining equat i ons

is c 1 ear

the

B.

i

- b .)tt. ., J 1J

B .7r. . + B .ir . . = 0,

'

Recal 1 ing it

we expre ss

of

gives

whi c h , w ith

v

,

this bas i s ,

P. i

for uni q u e

I U VJ .. lw # u' . 1 AW A'■■' -( b ,. - b v i 1 j' J J

191

n(n - 1 ) / 2 (34) are

i ^ j.

for

#

and

n(n - 1 )

the

ir. .

ind ependent

(i & j )

fo r m s ;

ind ependent the only

p..,

relations.

relat ions am on g

(i / j). s true ture equat i on

d(fi. - b . n 1 ) = -2 (b. i i j i

(no summat ion on

i ).

(33) now

is

- b.)(ir..Aw1 - m ! a J ) J i J J

The n on - ze r o b 1ock of

t a b 1eau matr ix is the matr ix of

W -v a l u e d

the r e duce d

f orms

192

CHAPTER 7

2 (b. - b.Jw..

-(bl - b 2 ) ^ . . . - ( bl

- ( b 2 - b l)w^

2 (b 2 - b j ) ir2 j

(36)

2

(b n - b l K

The

symbol

(Oj + (o^ = 0. X

n

C E

2n

rel ations are giv en by

As

the

j

tab 1eau m a t ri x

this

(36)

To compute

mus t take a general Whe n

(b n

(34)

- b ,)ir . J nJ

plus

in the p r e c e e d i n g case of a flat

n o n -g e n er i c f l a g .

(36).

b )(o^ nJ n

is with respect

to a

the Car tan charac ter s we

1 inear c ombinat ion of the c olumns of

is done we ob ta in

for

the

first

colu mn

£ i^1(b i" b j)'u ‘ M W "? -•••- MW"i -M W " ? +

■•••■ M V b„>"?

J

- M b

From

the

1st entry we kno w

^ l ir1 2

and

f rom

. .+ t

-

2 (b -b .) ir .

njVn

the

n

1-f orms

^ 2 W2 ’

the 2 nd entry we k now

the

1 -f orms

J

nj

I S O M E T R I C E M B E D D I N G S OF SPA CE FOR MS

T a ki n g

1 inear com bin at ions we kno w

Cont inuing the

in this way we

an^

^ \2

f ind that

193

•

thi s vect or

n(n - 1 )/2 + n(n - 1 )/2 = n(n - 1)

conta in s

inde pen den t

f orms

*ir

From

this we c on el ude

(i