On Exterior Differential Systems

Citation preview

: LECTURES ON EXTERIOR DIFFERENI' IAL SYSTEMS

by

M:. Kuranishi

Notes- by · M,K. Venkat-e.aha Murthy

No part of this' book may be rep;roduced in any form by-print, microfilm or any , other means without written permission from,the Tata Institute of Fundamental Research, Colaba, Bombay 5-BR

Tata Institute of Fundamental Research Bombay_

1962

CONTENTS

Introduction

. •.

. .. .

. .•

Chapter I - Parametrization of sets of integral submanifblds

4

Chapter II - Exterior differential · systems

43

Chapter III - . Prolongation of exterior differential systems

81

References

.,,

..

148

Errata

..... .

149

' .

Intro duct ion

prob lem To begi n with , we shal l r.oug hly state the main Let D deno te a that we shal l be cons ideri ng in the ,follo wing . and let domain in the n-dimen sic,inal Eucl idean spac e. Rn form s on D ' 9 , ... , & be a syste m of homogeneous diff eren tial m 1 We adop t the conv entio n that a func which we shal l deno te by

:E

ee zero . A subm anition is a homogeneous diff eren tial form of degr ld or simp ly an inte fold M of D is calle d an inte gral subm anifo @ to M vani sh. if the rest ricti ons of gral of ,2; m

e,, ... '

prob lem: give n a sysWe will be conc erned main ly with the follo wing mine of homogeneous diff eren tial form s on D, to deter tem inte gral s of ~ suff icien t cond ition s f.or cons truct ing all the ture of ·the set of and to obta in some infor mati on rega rding the struc We shal l discu ss such cond ition s give n by integ rals of 2J cond ;i.tio ns 11 syste ms in E. Cart an. He calle d syste ms satis fyin g his ions of diffe reninvo lutio n". We shal l also disc uss the prolo ngat

Z

to him. tial syste ms, the main idea of whic h is also due a prob lem in The abov e ment ioned prob lem is esse ntia lly the theo ry of part ial diff eren tial equa tions .

This fact is made

clea r by the follo wing simp le example. Let

u(x,y )

be a func tion of two indep ende nt real vari -

able ~ defin ed in a cert ain domain

.; ,

2 D in R

and satis fy the syste m

2

.. . . of part~B_:l different ial equations '

rau

rou

·ax

cJ y

(CX.:= 1,2, •.• ;m)

(x,y,u ,-,- ) = 0

Fro v-.

u may be assumed to be once continuou s~y different iable. construct , introduci ng new variaqles

and

p

q, a system ~

homogeneous different ial forms in a suitable domain coordinat e system

{

M2

that

2

M

D1 in · R5 of

(x,y,u;p, q),

du-pdx-qd y .

be a two dimension al submanifo ld of

metricall y by

of

(x,y,u,p, q) Fe(,

Let

We will

(x,y,u(x, y), p(x,y), q(x,y)).

is an integral of the system

L

D

1

expressed para-

It can be easily seen if and only if'.' u(x,y)

is a solution of the, system of differen tial equations · 'c}u ou F"-' (x,-y,u ,-,- ) = 0 'QX 3Y tA, ·

together with p(x,y) =

0 u(x,y}

(ex'., =1, ••• ;m)

'q(x,y) =

'o u(x,y)

'o y it is convenien t approach, our in that, seems However, it () X

to handle the system of homogeneous different ial forms rather. than solving the· system of partial different ial equations ,

Moreover,

sometimes our approaoh i9 quite Uijeful for certain geometric problems also. We shall restrict our attention only to the case of systems of real anaJ.ytic differen tial forms.

... - . .

.

The extension of our

.

\

3

results to the case of

cco

forms (differentiable case) appears to

be very much more complicated and remains unsolved. confine ourselves to the so called local problem.

We shall also

4

Parametri.zation of sets of i nt egral submanifolds

In order to illustrat e the problem with which we will be

·1.1

concerned in t,l~is chapter., l et us consider an ordinary· di~feren tial equation, for instance, du --

d.x.

where

11:Z

F(x)

F is defined and real analytic in a neighbourhood of x

O.

=

Then there exists a unique function u(x,w) , real analytic in x, depending real analytica lly on a parameter w such that for sufficiently Email fixed

w, u(x,w) is a solution of the different ial

equation and · u(o,w)

=

w.

Thus the solutions are parametri zed

by

the parameter w. More generally , in. order to consider the situation ·independent of the coordinat e systems, we shall use the following terminol9gy.

We s~y that _a real analytic function v(x, w1 , ••• , wh)· is a

parametr isation of solutions of the equation, when, for any fixed

(~~,, •• _,w~), with w~

small,

w ,~ v(x, 1 • 0

0

•• ,

wh )

is a solution of the

differen tial equation and conversel y any solution which is suffi.

·

0

0

ciently small at the origin is obtained by choosing ( w1, • ~ • , wh) suitably. Then; fpr any parametri zation:, the number of parameter s is ·the same and is a constant determine d by the equation equal to 1 in the above· instance) .

(being

5

In the case of partial differential equations, the solutions are often parametrized by arbitrary functions.

Take, as~

simple example, the partial differential equation

where

is an unknown function of the variables

u

any real analytic function

f(y) , u

~

f(y)

(x,y).

Then, for

is a solution of the

above differential equation and any. real analytic solution is so obtained-.

In such a case the solutions of the partial differential

equation are said to d•pend on one arbitrary function in two variables.

However, .no strict definition of this notion is known.

As

a consequence, the number of arbitrary functions on which the solutions of the equation depend may not be an invariant of the equation.

For instance, we can give another parametrizatlon of the

solution of the above partial differential equation, in which the Namely, for any two n n n L.-an y , g = .Z,b y , we associate

solutions depend on two arbitrary functions. real analytic functions a solution u

=

f

~ (an y 2 n

~

=

+ bnY 2n+ 1) .

The main purpose of this

chapter is to introduce a notion of parametrization of a set of submanifolds by arbitrary functions.

This notion will be used to de-

fine systems of part.i al differential equations or an exterior differential system, the solutions of which depend on certain number of arbitrary functiqns.

In this definition the number of arbitrary

functions and the number of · variables will be invariants of the system. ·

6

varia bles

p

x , ••• ,xp , which are conve rgent op. a neigh bourh ood of the 1

C of compl ex numbe rs.

origi n and with coeff icien ts in the field We set

p

denot e the vecto r space of power serie s,in

H

Let

1.2

are real numb ers, let Hp (u,v)

If

H0 == \," •

f

On a polyd isc [ x ; f xr .0

By a system - S

Defin ition.

t.

denot e the

Hs

p

H • p

copie s of · the vecto r space s

s

E:},

depen ding on

Let direc t sum of

5 satis-

consi sting of all power serie s

Hp

denot e the subse t of

of chara cters we mean an order ed set

H(S) the of non-n egativ e integ ers s 0 , s 1 , ••• ,s . Denot e by .S p p Sp s1 S0 q can be natur ally v)) u, (Hq( . ,H ••• , H , direc t sum of H s p "ep'6 P 1 o We denot e q ident ified with a subse t of • , H (S) =(±)L.; (H )

>,

q

q=o

(2)

It is clear that

H(S;u ,v).

the subse t by

1 H(S; u,,v) C. H(S; u ,v)

H(S; u~v) C

and theref ore- (3)

if

v~ v

if

u~u

1

I

V ~ V

H(S; u' ,v') if

I

and

I

U · 3-U . , I

K(a)

Let

denot e the open disc in

C

of radiu s a about

the origi n. Defin ition.

A mappi ng

an analy tic funct ion in SI

of

H(S; u,v)

regul ar curve in

Let

_,f/;

'

=· ( s 0I

K(a)

into

H(S; u, v)

if each compo nent

I

id;

is called ·a

(z)A. (x 1, ... ,xq) is

>

u. z I ~ a ; \xi\ (z,x , ... ,x ) for q 1 11 ) I be anoth er system of chara cters. , s 1, ••• ,sp 1 .

) (p ' may be diffe rent frqm p.

7

of

F

De!1nition~ A mapping

H(S

H(S; u,if)

into

,g ·_ in

H(S; u,v)

1

u' ;V 1 )

;

is said

to be regular if -

(i)

f(O) = 0

£ H(S 1 )

for any regular owve

(ii)

gular curve in 'H(S Propositi on 1. 1

I

1

H(S; u ,v) any£

,.

1 ~'

;

1 ) _.

H(S; u,v) . into

is a regular mapping of

F

If

,v

then for any real number

b

-

with · 0O

5 in

a

Take

Proof.

H( S; u, v) .

curve in in H(S

1

for any fixed

f (z).

function

I zl 6- 1 + c,',

f(z)

1

Hence

,

being regular,

€

F( z~ )

1 ,

z~

a regular

z ➔ z~ . is

is a regular curve

F(zE;)A (x 1,.\nxq)

The component

u',v').

;

F

~ 1 + E,

and for

For sufficientl y small

H(S; u, v).

jzj

if ·

is in H(S; u,v)

is a re-

fo,/j;

is an analytic

\~I"'-

(x , ." ., ,x ) with q .1

u '. and

satisfies the follow:Lng two conditions: f(O) = 0

since

F(o), = 0

and Hence by Schwarz 1 s lemma it follows that jr(z)j L... J

z j )

=

= (t 0 ,

~

r"'f O ... ,

t ( f (1) r r

tq)

1:vith

sp

f

O , tq

f

0.

be the germs of linear infinite analytic maps defin-

ing the isomorphism .

rJ

is

· . p . sr fr (1) _ and Then dim (H(S)/H(S)( l)) = r~ 0 q

.r

of degree

( r+lr-1)

the dimension of

1r + ( lower powers ..r rl a ) ( + ➔~ Xr-1 + ••• -, polynomial where f X denotes the r · r (1)

Hr } Hr

the quotient space of

1 .> 0

k

We can, without loss of generality assume that

> O.

The map

[H(S)(l+k)]

C.

j-

of

H(S)

into

H(S

H(S 1 )(l) (Propositio n 5).

1 )

is surHence there

1

17

exists an induced surjectiv e map of

l+k . onto

(1 \

.

.

)

(

H(S)/H(S)

H(S 1 )/H(S 1 )

I

and therefore

l ,. the compariso n of dominant factors on either side of

For large

the inequalit y shows that

~ r

q.

q~p and

Similarly , we show that

s .• q

Then_, by the same reasoning we show that

p = q.

hence ,

p

The converse is proved in tvm steps. Ca se_ (J,t. f,or

+ ••• +s t_, = s_.,. + s v+1 1 p . U V

Consider the particula r case where

')) = O,;., ,p.

Then .~ r

has

s

S~

Let

ft

.be the component of

r

s~

€ H(S)

(x , •••

c o,nponents which are functions of

r

1

,x)

S:ilnilarly '>'1_, ~

by

and we denote them

Sr

Hr •

in

I

(for()'--'= 1, ••. ,sr+, .• +sn) .

in

H(S

denote the component s of an element

11

~

1 ),

Now we define a. linear map follows: for any

5€ H(S)

F

of

H(S) . into

H(S

1 )

as

set

~oso + •• "+sq-('·O,:-~.c:r-:(0) for q=O~ • .,,p ,0"'=1, •.•• ,sq, if sqfo '.:>q

F( s; where

. s 0 +. " .+s q-J. + O . means

where

s + .•. +s

~

r

+(F" means

q- 1

analytic (see ex.1 of

.

§ 1.2)

~

when

q = 0

·when

q = r.

and

Clearly

F

is infinite

and linear because of the linearity of

tB

defined as follows: f ,or a~y"l,_ f H( S

(l(Yif,: •

'(:o~ ·: X

+

J 0

for

r~1, .•• ,p;

that

F

and that

1

0

+ ••• +

f

?C,r

is

sr

f

0,

11. ·

'YJ dx r ·tr

0

=1, •.. ,sr if

H(S)

into

)

:I+ .. • ••r-1 +,l dxl

1

.

7l 2

Again

G is. infinite

f 1.2) and linear because of the linearity of inte-

Obviously and

\:'rl

1

set

)

2 s 2+., , +sr_,+A

i\.

analytic (ex. 2 of gration.

: +~r-1+

1

H{S

of

The inverse map . G

partial derivation.

F and G are surjective,

It is easy to verify

G

define germs of infinite analytic maps

g'

(re~p;

:J fj) O

is identity on

H(S)

ig:

and

1,

(resp. H(S

1

)).

The assertion follows in this special case, Case 2. H(S) .

To prove the assertion in the general case we shall write

explici tely a s s

H(s 0 , ,

,.•

It is sufficient to prove that

,sp) . I

H(S)~ H p. p

In v.iew of the Case 1 proved above we can without any

loss of generality assume that isomorphic t o

s0

/.

O,,,,, spfo

because

H(S)

is

Then it can be seen

H( s 0 -1- ••• +sp,. , ., sp- I + sp, sp).

-•.

of the without muc ~ difficulty that our assertion is a consequence . . following statement : if (

s-

r

> o, . •. ,sp > O, )

.

r-,.J sp ) .:;:=:;;:- H,O, ,,o ,O,s 0,-1,sr+l''' ' 'sp.

Hr

H(o, • •• ,o,s r ,s r+ ,, •• ,

Now since

1

f'!",!_

(i:'i

r.:..

Hi= Ho±,H 1 ~••• ~

we can write·

.

l':"l

.

~ H(o ; ... ,o,sr~1,sr +l'"•,sp)G,)(H 0 (±) ••• (±)Hr)

19

H( O, ••• , 0, s r -1 , s,r+l -1 , ••• , sP )

® Hr+ 1

H( 0, • ~ • , 0, s r -1 , s r+ 1, ••• , Sp) .

1. This is obta ined by succ essiv e usag e of Case 1.4

the case of Hith erto we had rest ricte d our atten tion to

the field of comp lex conv erge nt powe r serie s with coef ficie nts in d aJ.l the notio ns numb ers. We can, with out much diff icul ty, exten to the c~se of conv erand resu lts prov ed in the prev ious sect ions ch we call , here after , gent powe r serie s with real coef ficie nts (whi sp) be a syste m of as the real anal ytic case )_. Let S = ( s 0 , , •• , l) subv ector spac e of H(S) char acte rs. Let HR(S) deno te the (rea · )l r to be real ts onen comp its all with H(S) in cons istin g of ~

?

-

R

R . A mapp ing FR of anal ytic. Set H (S; u;v) = H (s)n H(S; u,v) R( 1 , 1) is said to be a (rea l) regu lar map R H (S; u,v) into H S; u ,v H(S ; u,v ) i·nto H(s', · u',v ') · ther e · exis ts a regu lar map FC of if C

such that

FR

-

rest ricti on of is the , -

it cah be shown that such an

Fe

to

R H (S; u,v) .

is

and by ' the theor em of ·iden tity

FR

calle d the com plex ifica tion of

Fe

is uniq ue.

Two (rea l) regu lar

l'exi ficat ions are maps are said to be equi vale nt when thei r _c omp l) regu lar maps of · equi vale nt. Thus we can -defi ne germ s of (rea HR(S)

into

J\s').

Sim ilarl y we can defin e 1the notio n of germ s

of infi nite anal ytic maps of

HR(S)

germ s of infin ite anal ytic map of

into H(S)

HR(S into

1 )

when ther e exis t

H(S' )

mapp ing

~(S)

20

rfcs').

into 1• 5

Exampl e s_,

( 1)

Consider a system of functions

(A.=1, .•• ,s')

•

j~j &

defined and analytic in the domain (0"=1, ••. , s ;

f

v·,

1~+/4~ u➔i­

satisfying the following conditions:

=1, ... ,1)

A/\.(o, ... ,o) = o

(i)

j AA (y 1, •.• , ,Y 8

(ii)

xp+ 1' ... ,xft

;

)i .e:. v' -8 for small 8,)0 in

the above domain, . SI

Then

we

every

define a regular map 1 O ..,t,, u 4 u f

for every

A.

Fu

of

into

The germ of

Fu

I

Hp+l(u,v) for

by setting

with

can be verified

to be a germ of infinite analytic maps. (2)

· Consider a system of functions

. (

,

AIL x 1 ' •• •. '~+ 1 ' y 1 ' • • ' 'ys' ' ' '

defined and analytic in the domain (i=1, .•• ~p+1; r=1, .• , ,P ;

f-

tial differentia l equa.ti'ons

r

'Yµ ' ' . '

)

(A

=1, ...,,s)

~xij L. /~, J ~ ...i:::: v~i-,

=1, •.• , s).

t~ \c:::: v*

Consider the system of · par-

21

Such a system of partial differe ntial equation s is called a system of partial differe ntial equation s of Cauch.Y-Kowalewaki type. Now we make the followin g definiti ons. \ ~IL(o>l L..

If ,~€ H; such that

Defirri& Qn.

) for l\. =1, •••., s

and

then an element

r=1, ••. ,P

(Jrl )

a solution of Cauchy-K owalews ki system

dition

t

(i) (ii) Let

911A

(x1, ... ,x .

YA = 71,A is

~ij.ni tion.

is called

with the initial con-

p

,o)

=

.

(/\.=t, ..• ,s)

' s/L(xp ... ,x) p

a solution of

( ,{Jl).

be strictly positive real numbers,

I

A mapping F of

v,u- 1v 4 v➔r)

(,{f[)

nI .€, Hps +l

if

I u,v,u ,v

system

v➔(-

s

H (u, v) p

into

S

Hp+ (u 1

I

I

,v)

(where

is called a solution mapping of the Caucey-K owalews ki

(J'{, )

s

is a solution of

if, for every~ 6 Hp (u,v) .

with the initial conditio n

t

We remark ' that there is no ambigui ty in this defi•n i'tion ➔~

Lv

since the conditio ns

and

Now the classica l theorem of Cauchy-K owalewsk i on the existenc e. and uniquen ess of solut~on s of Cauchy problems can be generali sed as -follows . Tbeore~ _l.

and

\-~ -~

(i) . Given an element :;, in

(0

Hs p

with

)I ,: __ v* , the solution of the system

(,l[)

initial conditio n .; is (locally )uniq~e if it exists; (ii)

there exists a solution mapping ;

with the

22

1\~ (x,O)

when

and (iii)

A

= 0 for all

=1, ••• ,s, any solution

mapping is a regular map and all solution mappings are equivalen t t o each other. '{

AA(x.;o) = 0

Assume

tion mappings of the syste~

for a11·

(,et)

The 13olution germ of ' s s analytic maps of H into ,,H

Then the solu-

define' a germ of regular maps

called the s oluti on germ of the system Theorem 2.

A =1,. ~., s.

C.

V ,

(

system every

v'i')

into

bl.'(I )

if

V- J.

df , L

0

=

Then we have the follo wing prop ositi on: of Er (r~ q-1 ), rega rded as a cont. s.ct elem ent

13,

Prop ositi on

is a regu lar inte gral elem ent o"f

(

2J 1) •

D , 1

More over, we have

for r = -1,, ,.,q- 2;

Firs t·, we show that ther e is a neigh bour hood

) 1rl 0 r( D ' such that for any t(Er ,

Z ).

for any bour hood

E

E

11

in

Take a neigh bour hood 1

Ur

in of

U, df E

r

jE' IO.

Ur

n r b 1 t ( E11 '~1 ) ur n .r U of

Eq

in

!JJ q

of

E

r

in

=

D such that ,

By Prop ositi on 8 ther e is a neig h-

such that for any

E

11

in

urn ~ r

\

2-J ,

ther e

64

E

is

1

u n.95.i

in

urn. J)._ r~ Lt

is in

II

1

'

then

t(E

E

'X;)

,

D

. .of is a subma mfold

and

0

dim }

1

2.J)

1

+ 1

(

integ ral point of /case of all

r 14'.. r

4

E0

in

SL L

)

and

s 0 ( z,

1

integ ral eleme nt of

t(E

2.; 1'

s

(E

q- 1 . q- 1

s_ 1 (z,

,

=

2./1)

1

,

Z, 1 )

LI 1)

==

is in

s

O

t(w,,6 )=

)

for

(

z, Z,,).

CJ?:2..1 1

Assuming the

; it is only

remai ns const ant when

r ~ q-2.

= dim D 1 -(q-1 ).-t(Eq- 2 , ~ I

'Li ).

= dim' D-q-t (Eq- 1

1

is a regul ar ·

E0

E

I

is an

Er . We have alread y

t(E',2 _11 ) = t(Er, ;&).

sr(Eq , 6

LJ 1 ) =

in a suffi cient ly

Hence

1

Er

LJ 1 )

suf~i cient ly near

shqwn this and moreo ver,

z..r 1 )

O

to show that

neces sary to show that

(

w of

for any integ ral point

small neigh bourh ood of

sr(Eq _ 1,

is ordin ary

E0

Thus

• In parti cular ,

= dim ...9-o..E,

impli es that

On the other hand we have alread y shown that

s_ (z,Z:, ). t(z,

L

IO

on a neigh bourh ood of

1

' (0) -- O. havin g regul ar local equat ion LI 1

z

suffi cient ly small ,

df'Eq

r = 0, the condi tion

When

-s;:- n D1 Clo L.J = v-

ur

If we take

Now the proof can be compl eted by in-

= t(Er' ~).

r.

ducti on on C'i o"' LJ 1 ..y...

II

and hence it follow s that t(E , ~ ) =

11

. • f rom p ropos 1t1on 12 • 11

E

in the secon d part of Propo si, the condi tions

tion 12 are satis fied for Z ) t(E II' LI

Then if

as a subsp ace.

E"

conta ining

So we have

By defin ition ,

1

)

'L.J

+ ( t(E q- 1

This comp letes the proof of the propo sj_tio n.

)-t(Eq- 2'

b ))

65

2, 7 Diff er ential syst ems with independ ent variables. (D,D 1 ,'?;J) be a fibred manifold where D and D1 are q+m . and Rq respectively. Let· ('lj ) domainsin Euclidean spaces R Let

b e a differential system defined on

A pair consisting of a

(Zi ) ' on D

and a differential system

(D,D 1 ,f'"(;J"')

fibred manifold

D.

is called a diff er ential szst em with inde12endent variables and is

1.

denoted by

~ ; · (o,D•~

Definition.

A cross-section

of

f

(D,D 1 ;7v)

.over an open set

f old of

>]

U

into

D defines a submani-

(Z:, ).

which is an integral submanifold of

D

tact elements

E , .(d--w")

E

z

in

denote the set of all r-dimensional con-

f/ r (.D)

which are such that if

is injective on

_Jr [~,(D,D

1

E.

is the origin

We set

8

,~~

=

(P2;

(! f/r(D,D 1 ,'CJ);

n ~r(D,D

CR r

~,(D,D'

Let

E be a q-dimensional ordinary integr3:l element of

=(R,r?.J

1

,"Ci:f).

this system (an element of (9. q ~ , (D,D 1 ,7.;J' )] ) (x , ... ,xq , y ,,. .• ,Ym) 1 1

around

z

,'m ~ - :; ~r,Zn fj-r(D,D',r;:)');

_@ r [6,(D,D 1 , ~ and

of

f

if the map

Let ~r(.D,D 1 ,7:J')

Let

u

D1 is said to be an integra l of the differential system

[ Z, , (D, D1 , x

and

clTc,.(,) )x

do not change when

n o

moved in a suffic i ently small nei gh bourhood of_ .Y.. direct sum decompo sition

J(X) = (G(l))

X

+

·;,:. ~-

X

14J

The

PS

(TI~-l 1 )X

(t)

+

is

(A0 )x shows

10s

J(X)

that any change in the dimension of

(.A~))X'

in the dimension of If

if

is due only to the change

is any .q-.. dimensional integral element of

E

Pt ZI

X is the origin of .E , it has already been proved that

is generated by

I;q 1

. f (j .:q : ; 7

q1 Ii Iiq 1 w. dy - w dx~ J.;q O" (7"',q ].

on 1!J.q Ji(dx , .•• ,dxq), I 1

Remark. Eq

of

For any

Eq

J, 1(Ti') .i.:, ,

onto

iJ,.

(i _. .•

a::

1

q 1 .,L q 1 .,i,: q I·\,,_

with

q,I

spanned 1-., uy

jection of

J(E)

J(X): together with I

~

and

••

Tqi(E1)

c ,..,

Eq 1

•

•

1

).

let

be the subspace

"'-~1 J.ei., ~ ·'· IY1 mi-.. · L be the natural pro-

Clearly we have I

and henco we e;an simpl.y write with out any ambiguity.

l;;t:,

lil.so,

2 . we have

.

.

s1.nce i,j.

For any

instead of

1-·.1e

Therefore

J(E)

j,31

- i .... i i 1ij B

b

gener·ated by

j

J(X)

l.-

.

are symmetric in

and

i/J denote the subspace of 1-forms q . ' 1 generated by i.f ), S~; ,., . , (' ~,q (I=(i , •• , ,~ _ ) 1 1 Let

/l = \

A.

1, ... ,m 1 ) .

/L

;

109

Then we prove the following: Proposition 11.

E be a q-dimensional integral element of the

Let

standard prolongation to the space of

[z,, (D,D

1,

'0

D (0

I ' / dx. , Lq (E) · '-].

we

E dual to

being a basis of

=

. (i~ q) ~ qi

()].

so that the Pfaffian

t

forms (;q'(I = (i 1, ... ,i,e_ )) have the reduced form ~;!q' 1 / (f,) Ii;q' TA_+u qI Iq' the subspace of A rh·. • Denote b-.f - w dy........, + wq+u dy'J q -"'J. 0-' O'"' w Pfaffian forms generated by the following set: I~

kj;I

":icp

t

= Ak Adyjl - Aj

.

'I -y, I; q , __

½

-(Cf>f-L) 2 )

,

(dx)

t

'(i/

a

q+u

1

If we denote by

=

Ad

kl yA

'

r A.+u dyil ~

.;.,i,{_ 1) 1, ... ,q; I= (i 1,_ denote the dimensio,n of

.0, the space generated by (dx 1)X, ••• ,

D/D, ) .

+

1

t 1 (E)

Let

X being the orifin of dim (( A~ ' \

+ w

;\ov= 1, ••• ,m ;q

withe 1 ~ i1, ... ,i,t-1,j,k ~ p.

cJ A~ ~E.

q,

Iq ,

a,J

g-/

where

dy

rp

,{

cp

E, thert the definition shows that

Thus

t' (E)

is defined independent

of the choice of the coordinate system. (because of Proposition 11). In this section-we establish an inequality regarding t 1 (E). 1 If Eis a q-dimensionaJ. integral element of pslz and if 1 (E),~··,

3.8

Lq(E)

is a basis of E dual to

cix. /E, ••• ,dx /E, denote by 1 q

E 1 the

110

q-1

. 1

spanned by L (E), ... ,L J, natura l projec tion .o f J"tD, D1 ,'UJ"") onto

E

subspac e of

clear that

E

()

11

(E). J

t-1 (D, D , Q(¢ ({) ,p)

for

l {

such

q_fk,p)

is called Macauly functi ~:m.

Q(k,p)

Definitio n.

k , there exists an integer

p. and

Given

Theorem 1.

(l) ,p)

0

O

(1) ,

(I)

118

-Let us write the symnetric algebra of the 5Ubmodules

A

li(l,h)(z)

A

. (0, h)

(degree Jl in

f 2.; li( ,h\z)

g(z) =

Definit.ion. ideal

and degree

i

B:.( )( z )

defined to be

h

in

Y , .. . ,Y 1 ) . 1 m

=l ~=l g