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English Pages 155 Year 1962
Table of contents :
Contents
Introduction
Chapter I. Parametrization of sets of int egral submanifolds
1.1
1.2
1.3 Regular linear maps
1.4
1.5 Examples
1.6
1.7
1.8
1.9
1.10 Differentials of regular maps
1.11 Germs of submanifolds of a manifold
1.12
Chapter II. Exterior differential systems
2.1
2.2
2.3
2.4
2.5
2.6
2.7 Differential systems with independent variables
2.8
2.9
2.10
2.11
Chapter III. Prolongation of Exterior Differential Systems
3.1
3.2
3.3
3.4 Admissible Restriction
3.5
3.6
3.8
3.9
3.10 Some results from the theory of ideals in polynomial rings
3.11
3.12
3.13 Definition
3.14
3.15
3.16
3.17
3.18
References
ERRATA
: LECTURES ON EXTERIOR DIFFERENI' IAL SYSTEMS
by
M:. Kuranishi
Notes by · M,K. Venkate.aha Murthy
No part of this' book may be rep;roduced in any form byprint, microfilm or any , other means without written permission from,the Tata Institute of Fundamental Research, Colaba, Bombay 5BR
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 ndimen 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),
dupdxqd 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 nonn 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 · 3U . , 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+lr1)
the dimension of
1r + ( lower powers ..r rl a ) ( + ➔~ Xr1 + ••• , 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 qJ. + 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+ .. • ••r1 +,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.
: +~r1+
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.YKowalewaki 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 CauchyK 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 CauceyK 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 CauchyK 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~ q1 ), 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
\
2J ,
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 ~ q2.
= dim D 1 (q1 ).t(Eq 2 , ~ I
'Li ).
= dim' Dqt (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 ...9o..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 crosssection
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 , .(dw")
E
z
in
denote the set of all rdimensional 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 qdimensional 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 fjr(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 1forms 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 qdimensional 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>fL) 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,t1,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 sectionwe establish an inequality regarding t 1 (E). 1 If Eis a qdimensionaJ. 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
q1
. 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
t1 (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