DEFORMATIONS OF INTEGRALS OF EXTERIOR DIFFERENTIAL SYSTEMS

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 170, August 1972

DEFORMATIONS OF INTEGRALS OF EXTERIOR

DIFFERENTIAL SYSTEMS

BY

DOMINIC S. P. LEUNG (!)

ABSTRACT. On any general solution of an exterior differential system 7, a

system of linear differential equations, called the equations of variation of 7, is

defined. Let v be a vector field defined on a general solution of 7 such that

it satisfies the equations of variation and wherever it is defined, v is either

the zero vector or it is not tangential to the general solution. By means of

some associated differential systems and the fundamental theorem of Cartan-

Kahler theory, it is proved that, under the assumption of real analyticity, v is

locally the deformation vector field of a one-parameter family of general solu-

tions of 7. As an application, it is proved that, under the assumption of real

analyticity, every Jacobi field on a minimal submanifold of a Riemannian

manifold is locally the deformation vector field of a one-parameter family of

minimal submanifolds.

0. Introduction. The theory of analytic exterior differential systems was

developed by E. Cartan for the study of infinite pseudo-groups. E. Kahler com-

pleted the theory and generalized it. It reduces the existence of solutions

(called general) to a purely algebraic problem. Such a general solution depends

on the initial data which in turn depends on arbitrary functions. So it is of

interest to study a family of solutions of an exterior differential system, in par-

ticular, a one-parameter family of general solutions. For a compatible system 7

of ordinary differential equations, the following facts are well known (see for

example [2]). Along any solution TV of 7, a system of linear differential equa-

tions, called the equations of variation of /, is defined. Any vector field v

defined on N which satisfies the equations of variation is the deformation vector

field of a one-parameter family of solutions of / on N. In particular, if 7 is the

system which defines geodesies on a Riemannian manifold, we have that every

Jacobi field along a geodesic may be obtained by a variation through geodesies.

The purpose of this paper is to generalize the above to an arbitrary general

solution of an exterior differential system. We will prove the following (see the

Main Theorem):

Received by the editors September 17, 1971.

AMS 1970 subject classifications. Primary 58A15; Secondary 53A10, 53B25.

Key words and phrases. Exterior differential system, Cartan-Kahler theory, general

solution, regular integral chain, one-parameter family of integral manifolds, deformation

vector, equations of variation, /-field, normal system, normal solution, normal data,

associated differential systems, minimal submanifolds, Jacobi field.

(^This research was supported in part by NSF GP 9613.

Copyright O 1972, American Mathematical Society

333

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

334 D.S. P. LEUNG [August

On any general solution N of an exterior differential system I on a manifold

M, the equations of variation are defined. Let v be a vector field defined on N

which satisfies the equations of variation and such that for m £ N, \(m) is either

the zero vector or it is not in T (N). Then under the assumption of real analytic-ZZZ . t j

ity v is locally the deformation vector field of a one-parameter family of integral

manifolds of I on N. As an application, we will prove that under the assumption

of real analyticity every Jacobi field defined on a minimal submanifold can be

obtained (locally) as the deformation vector field of a one-parameter family of

minimal submanifolds. This solves the problem as posed in [12] in the real

analytic case.

In §1, we will give a brief summary of Cartan-Kä'hler theory of exterior

differential systems. Regular integral elements and involutiveness will be de-

fined in terms of the necessary and sufficient conditions for the existence of a

regular integral chain as proved in [7, p. 40]. This approach may not be the most

natural, but it immediately reduces the problem of finding general solutions

essentially to that of checking the compatibility and ranks of systems of linear

equations. In §§2 — 5, the Main Theorem will be formulated and proved. We will

begin by establishing that it is sufficient to consider a class of Pfaffian systems

called normal systems. Associated with a normal system, two other differential

systems are defined. By means of the fundamental theorem of Cartan-Ká'hler

theory, they will be used to construct the required one-parameter family of inte-

gral manifolds and also to prove that its deformation vector field coincides with

the given vector field. In §6, after some computations we will apply the Main

Theorem to Jacobi fields defined on minimal submanifolds of a Riemannian mani-

fold.

This paper is a continuation of research done at University of California,

in a doctoral thesis under the direction of Professor S. S. Chern. I wish to

thank him for his advice and encouragement.

Throughout this paper all functions, manifolds, submanifolds and associated

differential geometric structures will be assumed to be real analytic. When no

confusion is likely, we will simply regard an immersed submanifold of a manifold

M defined by f: N —* M as a subset f(N) of M and its tangent space T (N) at

x £ N as a subspace of the tangent space T,, AM) of M. To avoid repetition,

we will fix the ranges of the following indices, 1 < i, j, k < n; n + 1 < a, ß, y <

72 + p; n + p+l<s<n+p+q; n + I < p, o < n + p + q; 1 < A, B, C, D < n + p.

Ranges of other indices will vary and will be defined accordingly.

1. Exterior differential systems. In this section, we will give a résume of

Cartan-Kähler theory of exterior differential systems, mainly to establish some

notations and to state the existence theorems which will be used later. Details


1972] DEFORMATION OF INTEGRALS 335

and proofs can be found in [3] and [7]. For an interesting modern survey of the

theory, one can refer to [5].

An exterior differential system (or simply differential system) 7 on a manifold

M is an ideal (finitely generated) in the ring of analytic differential forms on M which is

also closed under exterior differentiation, i.e. dl C 7. We shall denote by 7 the set of

all forms of degree v in /. Points of M, at which

(Ll) '0=0,

constitute an analytic subvariety fe (7) of M whose points will be called integral

points or 0-dim integral elements of 7. Restricted to a point m £ M, I defines a

system of exterior equations 7(772) in the vector space T (M). At an integral point

m, a &-dim subspace E of T (M) is called a /«-dim integral element of 7 if E

annihilates 7(772). A submanifold N of M is called an integral manifold of 7 if

for every m £ N, T (N) is an integral element of 7, that is, the restriction (i.e.

pull back) of / to M vanishes identically. For definiteness, let M now be a

manifold of dimension n + p (later on, we will also consider differential systems

on manifolds of other dimensions) and 8,, • ■ • , 8 be 72 independent Pfaffian

forms defined on M. An important problem in the application of Cartan-Kähler

theory is to see if a differential system / has an integral manifold on which

(1.2) ö,A...A^0.

To study this problem we adjoin to 6y, ■ • ■ , 6 , p Pfaffian forms 8 +1, • • ■ , 8 +p

such that 6. A ... A 8 +fi / 0. Then the forms in / can be expressed in terms

of the d's. We put

,1 « 8 = Y"1 b .8.(1.3) a L-J ai 1

i

and denote by b- the vector whose components are b +. .,-•-, b +fi .. Sub-

stitute (1.3) into the forms of 7. and let I (m, b. , • • ■ , b.) denote the set of co-

efficients of 8. A . . . A 8. in these forms. Clearly every equation of the set

(1.4) 7.U, bv ..., b.) = 0 (; = 1, ... , 72)

is linear in each of the variables b (v = 1, • ■ ■ , j). At an integral point m, a

system of solutions (m°, b°, • • • , b°) of (1.1) and (1.4) defines an n-dim integral

element of I by (1.3) with ba- = b°a.. We will call an integral point m of I

simple if there exists a neighborhood U of ttz in M such that fe (/) O 11 is a

submanifold of U of dimension rQ with equations (1.1). We can now give

sufficient conditions for such an integral manifold of 7 to exist.

Definition 1.1. (a) 7 is said to be involutive with independent variables

\8,, • • • 8 \ (an ordered set) at a point 777o if there exists a system of solutions


336 D. S. P. LEUNG [August

(222° b?, ■ • • , b°) of (1.1) and (1.4) such that (i) 222o is a simple integral point of

/; (ii) in a neighborhood W of (222°, b\, ■ ■ ■ b°) in M x Rnp, the equation (1.4)

reduces to (n + p) — r. — j independent linear equations with respect to ba.

after taking account of the equations

(1.5) t._1(m,bl,...,b._1) = 0.

(b) Let E be the 72-dim integral element defined by a system of solutions

(722° b°, • • • b°) of (1.1) and (1.4) as in (a) and E . be the subspace of E on

which d +1 = • • • = 6 = 0. Then we have a chain of integral elements of /

(1.6) Í772°S= E.CE.C...CE .CE.U 1 72—1 TZ

Any chain of integral elements of / which can be obtained in this way by a

suitable choice of the independent Pfaffian forms jö., . . • , Q \ in (a) is called

an n-dim regular integral chain of /, E (p = 0, 1, ...,«) is called its pth

component. A iv — \)-dim integral element E of I is called regular if it

is the iv - l)th component of an 72-dim (72 > v) regular integral chain of /. In

this case, the nonnegative integer rv = r iE ., /) as defined in (a) is called

the character of Ev,.

Remark 1.1. It is clear that, in Definition 1.1 (a), / is also involutive at

any integral point sufficiently close to 222° and also that if E is another

integral element of / defined by (ttz, b., • • • , b ) sufficiently close to

(722° b°, • • ■ , b°), then the integral chain constructed on E as in (b) is also

regular.

Remark 1.2. The definitions of regular integral elements and regular inte-

gral chains given above are equivalent to other existing definitions of such,

but they are more convenient for our present purpose.

Remark 1.3. For a regular iv - l)-dim integral element E , at a point 272,

the set of all vectors Y £ T (zM) such that Y spans with Ey_ x an integral

element of / is called the polar space HiEv_x, I) of E .; it is a subspace of

T (zM) of dimension r ,,(£,, ,, ¡) + v. In other words, the set of all iv-dim inte-7Z7 V V— 1

gral elements which extend E . locally depends on r (E ., ¡) parameters.

Definition 1.2. A p-dim integral manifold N of I is called regular if, for

every 222 £ N, T (N) is a regular integral element of / and it is called a general

solution if, for every m £ N, T (N) contains a (p - l)-dim regular integral

element of /.

The fundamental theorem of Cartan-Kähler theory can be stated as follows:

Theorem 1.1. Let I be an exterior differential system on a manifold M of

dimension n + p. Let N . be a (v — l)-dim regular integral manifold of I



and rv the character of T 0iN .) at a point m° £ TV„_,• Let F be a

(n + p - r )-dim submanifold of M such that it contains N , and T 0(F) con-

tains a unique v-dim integral element E of I which extends E ., that is,

T 0(F) D H(Ev_., I) = E . Then in a sufficiently small neighborhood U of m°,

there exists a unique v-dim integral manifold Nv of I such that F O il ^Nv 3

Ti n/V„ , and T o(/V) = £,,

By applying Theorem 1.1 several times, we can prove the following theorem:

Theorem 1.2. Let I be a differential system on a manifold M of dimension

n + p which is involutive with independent variables \8,, • ■ ■ , 8 \ at m°. Let

E be the n-dim integral element defined by a system of solutions (m°, b°., • ■ • ,

b°) of (1.1) and (1.4) which satisfies the conditions in Definition 1.1 (a).

Then in a sufficiently small neighborhood of m°, there exist general solutions

of I on which 9, A ... A 9 / 0. In particular, there exists a general solution

N of I through m° such that T o(/V) = E .

Remark 1.4. The general solution TV in Theorem 1.2 is in general not unique-

ly determined. Classically this is described as depending on certain arbi-

trary choices of functions [3, p. 75]. For a modern and more precise description

of such, see [9].

2. The Main Theorem. Let 7 be an exterior differential system on a mani-

fold M of dimension n + p. Since all the results in this paper are local, we

will assume for simplicity that L = 0, i.e. the ideal 7 contains no scalar func-

tion. In order to motivate the following discussions, we will sketch a proof of

the following facts (see also [5, Theorem 3.1]).

For a one-parameter family of 72-dim submanifolds of M defined by f: N x

(- 1, 1) —*M (i.e. for every / e (— 1, l), f(N x t) is an 72-dim submanifold of

Al), the vector field v on f (N x 0) defined by

(2.1) v(/(y, 0)) = f^(d/dt(y, 0))

(where d/dt is the standard vector field on the interval (- 1, l)) is called the

deformation vector field of the one-parameter family on f(N x O).

Proposition 2.1. Suppose f: TV x (- 1, l) —► M defines a one-parameter

family of integral manifolds of I. Then its deformation vector field v 072

f(N x 0) satisfies the following system of differential equations:

(2.2) 0 = df*(8)/dt\t=0 = ¿/*(v J 8) + /*(v J d$) for all 8 £ I,

where J denotes the interior product of a tangent vector and a covector and

fQ: N —* M is the map defined by f0(y) = f(y, 0) for y £ N.



Proof. We will prove that for any form 6 defined on /M, we have

(2.3) df*(d)/dt\t=Q = df*iy J $)+ /*(v J dd).

It is obvious that (2.3) is true if 6 is a zero-form. By taking exterior derivative

on both sides of (2.3), we can see that if it is true for 6 it is also true for dd.

Using the facts that both the interior multiplication and exterior differentiation

are antiderivations in the ring of differential forms on M, it follows by a straight-

forward computation that if (2.3) is true for forms 6, and 62 it is also true for

the form 6. A 0 Now, since the ring of differential forms can be locally built

up from 0-forms using d and A, the proof of (2.3) is completed. ¡*i0) is identi-

cally zero for all del; therefore (2.2) also follows. Q. E. D.

Definition 2.1. Let g: N —> M define an integral manifold of /. A vector

field v: giN) —» TiM) defined on giN) is called an /-field if it satisfies the

following system of differential equations, called the equations of variation of

/ on giN):

(2.4) dg*i\ J (?) + g* (v J dd) = 0 for all del.

Remark 2.1. In the computation of the equations of variation it will be

sufficient to consider the forms in any set of generators of / as an ideal.

The main result of this paper can be stated as follows.

Main Theorem. Let N be an n-dim general solution of an exterior differen-

tial system I on a manifold M of dimension n + p. Let v: N —► T(M) be an

¡-field on N such that for m e N, \im) is either the zero vector or it is not in

T (zM). Then for every m° e N, there exists in a sufficiently small neighborhood

A of 272° ira M a one-parameter family of n-dim integral manifolds lN of I, where

t e (— e, c) and e > 0, such that N = N O A and the deformation vector field

of 'N on N is equal to the restriction of v to N.

3. Normal exterior differential systems. Under the assumptions of the Main

Theorem it is always possible to choose a system of local coordinates {x., zj

in a sufficiently small neighborhood U of 222o in M such that

(a) N fill can be defined nonparametrically as

(3-D Za. = Saixv---,Xn)

and on /V O U can be represented as

(3.2) vim) = £ Ta(xA\m), ■•■, x.U))!^-

for some analytic functions g and r defined on a suitable open set in R";



(b) restricted to il, 7 is involutive with independent variables {dx. , • • • ,

dx } at every point of ll.

Keeping the notations in § 1 except with 0 and 8. replaced by dza and

dx . (respectively), if we put

(3.3) b°a. = (dga/dx.)(Xl(m°), ■.., xn(m°)),

then (777°, ¿7°, • • • , b°) which defines the integral element T 0(N) of 7 satisfies

the conditions of Definition 1.1 (a). Let 7(1) be the exterior differential system

defined on the neighborhood W of (772° b., • • • , b°) which is generated by the

left-hand side of the following equations:

(3-4) I .(m, bv...,b) = 0,

(3-5) dJ.im, bv ..., 7O = 0,

(3-6) dz- Y\bn.dx. = Q,a. i^ a.1 1 '

i

0-7) Ydb Adx. = 0.i

One can easily recognize that 7(1) is the total prolongation of 7 (restricted to

U) as defined in [ll]. It follows from Theorem 2 in [ll] that 7(1) is also in-

volutive with independent variables {dx. , • • ■ , dx ! at every point of W. By

a straightforward computation, we can see that, upon setting

(3-8) b =dz /dx.,

(3-9) rn. = dr/dx.,ai a z'

(3.1) and (3.8) define a general solution of 7(l) and

(3.10) HTaT- + 2-ra«äT"a "Za a, 1 uvai

defines a iP(7)-field on it. Let 77 : W —» M be the map defined by

(3.H) 77(772, fej, .--, TO = 772.

Similarly we can also verify that, if w is a j(7)-field on an integral manifold

B of 7(1) on which dx, A . . . A dx / 0, then 77 (w) is an 7-field on the inte-

gral manifold 7r(B) of 7. It is also well known that (3.4) defines a submani-

fold ë"(7) of W. In fact, we can select q = S"=1 (r. + j - n) of the b's such

that they form together with {x., zj a system of coordinates of &"(!) [7, p. 42].

The restriction of j(7) to ë"(7) is also involutive with independent variables

{dx. , ■ ■ ■ , dx I. Note also that a ?(7)-field must be tangent to &"(/). The

above considerations lead us to consider a class of Pfaffian systems which

will be called normal systems.



Definition 3.1. An exterior differential system / on an open set ffl C R"+i,+9

is called normal with n independent variables, p primary dependent variables and

q secondary dependent variables, or simply a normal system of type (22, p, q) if it

satisfies the following conditions:

(i) /„ = 0.(ii) There exists a system of coordinates \x., z , z \ of (0, called normal

j % tv s

coordinates of /, and functions Ba- on il) so that / is generated as an ideal by

the following differential forms

(3.12) dza+ Y,ña.idxV T,dBaiAdxi>i i

and / is also involutive with independent variables \dx., ■ ■ • , dx | at every

point of u). \za\ and {z \ ate called respectively primary and secondary de-

pendent variables.

Definition 3.2. A normal solution of a normal system / of type (72, p, q) with

a fixed system of normal coordinates \x., z , z ! is an 72-dim integral manifold

zV of / on which dx, A . . . A dx / 0 and such that for m e N, the chain of inte-1 zz

gral elements on T (zV) obtained by setting successively dx, = dx,+. = • • • =

dx = 0 is an 22-dim regular integral chain /.

We can see now that to prove the Main Theorem it is sufficient to prove it

for normal solution N of a normal system / of type (72, p, q) with an /-field v

on it such that

(i) with respect to a system of normal coordinates \x ., za, z \ of / for

which N is a normal solution, N can be represented nonparametrically in an

open subset of iß as

(3.13) zrr=gaixl,.-.,xn),

(ii) v can be represented as

(3.14) v(m)= £rffUi(»n), ••■,*„(*))dz„

fot some analytic functions ga and t defined on a suitable open set of R".

Definition 3.3. A normal solution N of a normal system / of type (72, p, q)

together with an /-field v on it which have properties as in (i) and (ii) above

is called a set of normal data of /.

4. Two associated differential systems. In this section / denotes a normal

system of type (72, p, q) on an open set iß in Rn+p+q with a fixed choice of

normal coordinates \x ., z . z \ and it is generated as an ideal by (3-12). Let1 a. s °

£a be the coordinates of R^ and t the coordinate of the open interval (- 1, l).

Denote by / , the differential system on W x Rp x (- 1, l) generated by the

following differential forms,


3411972] DEFORMATION OF INTEGRALS

(4.1) dza+YéBaidxi-Çjt' I>Ba¡ A^.-Â^.■ i

Let

(4.2) irx: BxR" x(- 1, 1) — S

be the natural projection map. We will denote by ?72 (= (m, ¿; , t)) a point of (0 x

R" x (- 1, D and that of ffl by 772.

Proposition 4.1. Let

(4.3) 77,(772°) = En C E, C .-. C E ,C£

èe an n-dim regular integral chain of I such that dx.A> -A ¿x./O on E .. Then

there exists a chain of integral elements of I ,

r\j '>_'

(4.4) ^=E0CE1C...CEnCEn + 1,

such that 77, AE .., ) = E ., dt / 0 072 E, and dt A dx, A . . . A dx . 4 0 072 E ..,.

Furthermore any such integral chain is a regular (n + l)-dim integral chain of I

and

(4.5) rx(EQ, l) = n + P + q, r. + 1(E.,l) = riE._yl).

*\.7?. follows that I is involutive with independent variables dt, dx , ■ ■ ■ , dx at

any point of ffl x Rp x (- 1, l).

Proof. Any point m £ ll) x R^ x (- 1, l) is a simple integral point of 7 . We

put

(4.6) dzo-=bo-0dt + T, haidxi> dta=aaOdt + 2>'aid*V1 i

Denote by b (v = 0, 1, • • • , 72) the vector whose components are b , aay.

Substituting (4.6) into 7 , we can see, by a straightforward computation, that

7.(772, bA) = 0 is generated by the following equations

(4-7) Ko ~ €a = 0

and in general, T + Am, bQ, by, ••• , b.) = 0 after taking account of

7.(772, bQ, b., • • • , b._.) - 0 is generated by the following equations (7 is fixed),

b . + B . = 0,a.) a.j >

(4.8)

Z (Bau; *K, - Baj; «rV + * aß;i - Baj; p = 0 (M = 1, •••,;- D

(4.9) ZBa,;^ô + fla; = °-


342 D. S. P. LEUNG LAugust

where we have set B„ .„= dB„ Adz„ and B„. ■ = dB„ ./dx.. If we set dz„ =a¡;cr dj cr clj ;i a.j i cr

Z.b^dx. in / and let b be the vector whose components are è ., then it

follows by a similar computation as above that I .in Am), bx, ■ ■ • , b.)= 0 is also

generated by (4.8) after taking account of /._ ,(27^772), by, ■■■ b._x) = 0. There-

fore, if (4.3) is defined by (27,(722°), b°., ■ ■ ■ , b°), we can extend it to an (72 + 1)-' I /\,l 72

dim regular integral chain (4.4) of / by choosing b 0, aaQ arbitrarily, and define

ba0 , aa. by (4.7) and (4.9) respectively. It follows from our constructions that

27. (E) = 77,(222) and 27. (E.+.) = E .. (4.5) now follows readily from the defini-

tion of the character of a regular integral element and the fact that (4.7) as a

linear system in è as well as (4.9) as a linear system in aa. ate of rank p.

Q. E. D.Remark 4.1. An examination of the above proof shows that for any (p + 1)-

dim'integral element E,,.. of / which extends E,, , i.e. 27, (E ,. ) = E,, , there¿2."1"! r^j /-- r-^j 1^ /-i 2 r^ fJ.

exists a unique (p + 2)-dim integral element E +2 of / such that E +2 D

EM+1 and 771*(Em+2)^£m+1' P =!»•■•, »-1.

Remark 4.2. If /V is an 22-dim integral manifold of / on which dt A dx. A

• • ■ A dx / 0, then n,(N) is a one-parameter family of integral manifolds of /

on which dx, A . . . A dx 4 0.7^ 1 zz

/ will be used to construct the one-parameter family of integral manifolds

we need. But to prove the deformation vector field of the one-parameter family

to be constructed is the given /-field, we need to consider another associated

differential system.

Let f be coordinates of Rp*q. Denote by m' (= (m, ¿¡¿)) a point of iß x

R^+9 and also by 77,: iß x Rt>+q —* iß the natural projection map. Define a

vector field <f on iß x Rp ¥q by

(4.10) £(«'>= 2x(¿r) •er \OZcr I m

Then let / be the exterior differential system on UJ x Rp+q which is generated

as an ideal by the following differential forms,

(4.11) dz + Y B .dx., *YdBn.r\dx.,v a Á—7 o-i 1 ¿—1 ai 1'

(4.12)d\$J[dz + Vs„.A.| + |<f J YdB.Adx.i,

1= \ a. i^t CLl ill 1' i—d CLl z(' \ i I ) ( i )

^J^dBa.r\dxl.



Remark 4.3. With functions gCT and ra as in (3.13) and (3.14), the submanifold

N' of Ù5 x Rp+<? defined by

(4.13) za = Scr(xl, •••,*„),

(4.14) C=rtTî» •■•>*»)

in an 72-dim integral manifold of 7 . This follows readily from the fact that if we

substitute (4.13) into (4.12), we have the equations of variation on the integral

manifold N of 7 defined by (4.13) for the special type of vector field which can

be represented as the right-hand side of (4.10). Conversely, for any 72-dim integral

manifold TV of 7 defined (4.13) and (4.14) for some analytic functions ga and

ra, I.Q.Tç.d/dz^ is an /-field on the integral manifold nAN ) of 7.

We will prove later on that at any point 772', 7 is also involutive with inde-

pendent variables {dx., • • • , dx \. However, we will conclude this section by

making some important observations related to 7 . We put

(4.15) dz = Y b dx.,CT Á—I a I l'

i

(4.16) <,= ¿X .dx..i

Denote by b. the vector whose components are b ., a^ and as before, b. the

vector whose components are b .. By a straightforward computation, we can

see that lAm', b^, • • • , b'A = 0 after taking into account that /, _ ^m', bx, • • ■ , bk_ ¡)

= 0 is generated by the equations (k being fixed):

' z. + ß„, = 0,ak °-k

(4.17,)Z(Baß;*b«k-Bak;o-Kj + Baß;k-Bak;u=0 (p. = 1, • • • , k - l)

a

aak+ T,Bak;Jo- = °,

(4.18k)Z (Baa; o-aak ~ Bak;craat) + Z (ßaM; a; pb pk ~ B ak; a; pb p^cra er, p

+ Z^Baa;a;k-Bak;a;p.K = 0 (p. = 1, ■ ■ ■ , k - l)

where we have set B^.a.p=d Bjdzadzp, Bafl.a.. = d2 B^/dzJx..

Remark 4.4. It is easy to see that /, (772(772'), b. , ■ ■ ■ , bA = 0 is generated

by (4.17,) after taking account of 7, _ . (77,(772'), b. , ■ ■ • , b,_ ) = 0. If we con-

sider (4.17.) and (4.18.) as linear equations in ba , and aa , respectively, then

the coefficient matrix of b , in (4.17.) is exactly the same as the coefficient

matrix of a , in (4.18,).



5. Proof of the Main Theorem. In this section, / will denote the same

normal system of type (72, p, q) as in the previous one. We will now prove the

Main Theorem for a normal solution (Definition 3.2) N of / with an /-field v on

it such that they can be defined by a set of normal data (Definition 3.3) of /. As

noted earlier, this will prove the Main Theorem in general.

Let \x., za, z \ be the fixed normal coordinates of /. At any point 222 e il),

let ií¡uim) iv = 1, • • • , 72 - l) be the submanifold of iß defined by

(5.1) xv+l =xt,+ iWz •••>*„ = xnim).

We will simply write UJ when the point m is clear in the context. Denote by

J\ '(772) the differential system on (v (m) obtained by restricting / to (l) (222).

One can easily verify that Ap¡(m) is a normal system of type (v, p, q) on UJ^.

In fact, \x,, ... , x . z ,,..., z „, z . ,,..., z . S is a system of'1 ' V 72+1 n+p n+p + i n+p+q '

normal coordinates of Jw(?72) such that Áv¡(m) is involutive with independent

variables \dx., • ■ • , dxy\ at every point of (l) ; za and z ate still the primary

and secondary independent variables respectively; the restriction of (3.12) to

iß (722) also generates J\ l(m) as an ideal. This is the only set of normal coor-

dinates of Â ¡im) we will use. Let Av¡'im) and. Â l im) be the restrictions of

/' and T to ißÛ) x np+q and ¡JÛ) xRfx(- 1, l) respectively. They are

actually also the differential systems associated with jijim) (with respect to the

normal coordinates of 9\ 2(722) fixed above) as defined in the previous section.

Therefore, by Proposition 4.1, 3v^/ (272) is also involutive at every point of

ißv x R* x (- 1, l). For uniformity, we will also denote by WQim) the submani-

fold of (0 defined by

(5.2) *1 -*iW.x« = *>»)•

We will call any point of Wn(222) a normal solution or 0-dim integral manifold of

í\0/(?7z) and any tangent vector at a point 222° e fflgW of the form ô-^d/dz^)^

an 5l0/(272)-field. A set of normal data of J\0/(222) consists of a point of Í0o(t22)

and an i<L/(7?z)-field at that point. We also put 91 ¡im) = /, 5\ ¡'im) = /', J\ ¡im),_ U A 72 22 22

= / and ffi = iß.7Z

It follows directly from our definitions that a normal solution or a set of

normal data of 31,2(222) restricts to a normal solution or a set of normal data ofk

j\ .¡im) respectively.

The Main Theorem will follow from this proposition:

Proposition 5.1. (1) Let m'° e iß x Rp+q and m° e iß be such that n2im'°)

= 222° 2.e. 222'°= (772°, £°). Tie72, for /= 1 ,•••,«, we have P.: //

(5-3) Í722°i = E0 C Ej C ••• C E ,_j C E.


1972] DEFORMATION OF INTEGRALS ,-,

is a j-dim regular integral chain of 7 l(m°) such that dx, A . . . A dx /= 0 on7 r, ^

E (l < p < j), then there exists a j-dim integral chain of 7l'(m°),

(5.4) {m'°\=E'Q CEj' C--.CE/_! CE/,

which projects onto (5-3), that is, 77, (E) = E . Furthermore, any such integral

chain of 7.1 (m°) is regular and

(5-5) ry(E,' -1 • V(w0)) = 2ri{Ei-1 ' V*")).

(2) At any point m° £ (Ö, iê have, for v = 0, 1, ■ • • , n, P2: Let Ny be a

normal solution of 7 l(m°) which contains m° and v be an 7 I(m°)-field on N

such that they can be defined by a set of normal data of 7 l(m°). Then in a

sufficiently small neighborhood uv of m° in ll)Am°), there exists a one-parameter

family of integral manifolds *Nv of 7vl(m°), where t £ (- e , ej) and ev > 0, such

that Nv = Nv Cl U and the deformation vector field of lN on TV coincides

with the restriction of v to N .

We will prove the proposition in the following order:

(a) Pj and P. are both true,

(b) P1 and P2 , imply P2,

(c) P1 and P2 imply P1 ,.; 7 r ' i +1

We put, for j = 1, • • • , 72,

(5.6) dza = Z VV dô = Y,aorudxM=l M=l

7

a dx

in 91.7(772) and J\.7'(?72) (btt and h' will have the same meanings as in the pre-; 1 M ß °

vious section); we will find, tot p = 1, • • • , j, (7 .l(m)) Am, by, ■ ■ • , bA and

(J\.7'(t72)) ,(772, b, , ■ • • , b'A are the same as I,,(m, b, , • • • , bn) andj p. 1 p. p. 1 ¡j.

l'(m, b'. , • • • , b' ) respectively. For these reasons we will write 7 (. • .) and

/,',(•••) instead of (3v ./(m))(. • •) and (7 ./'(to)) (• • •) respectively.p. j p. j p.

Proof of (a). Let (4.17,)*and (4.18,)* be respectively the linear equations

in b'. obtained by setting m = m° and rf_ = tf" in (4.17,) and (4.18j). Then

the equations /j (to'0. b'x ) = 0 are generated by (4.17j)*and (4.18,)*. By Remark

4.4 the equations /x(m°, b.) = 0 are also generated by (4.17j)*. Clearly (4.17j)*

and (4.18,)* as linear equations in 7>CTl and aQ.l (resp.) are both compatible and

of rank p. They are compatible and their ranks are independent of the points

to' or 772. Therefore to'°ís a regular integral point of 7^1 (m°) and

(5.7) 7j(to'°, 7yl'(m°)) = 27j(to°, ÍRj/(to°)) = 2q.

This proves P}. As for P2, let v = Sa¿^(d/dzj^. The '7Vn can be defined

by any analytic mapping a: (- eQ, eQ) —» (fî0 such that a(0) = m°, a'(0) =

âeß/dza)mo and e0>0.



/t2 the remainder of this section j will be fixed.

Proof of (b). We can assume that x, (222°) = ■ • • = x (222°) = 0. In a neighborhood

of 222° in iß (222o), N. and v are defined by7 ; '

(5.8) za=gaixl, ...,x),

(5.9) \{m)=YlT_ix,(m),...,x.(m))(—) fotmeN..' \dza/m '

We put

(5.10) C=rJ°> ---.O)

and denote by 222'° the point (722°, f°) e il). x R^**. Then when we set

(5.11) Ço-^'o-^V '"'*?'

(5.8) and (5.11) define in a neighborhood of 222'0 in iß. x Rp+<? a /'-dim integral

manifold zV'. of %.¡'(m°).J t

The restriction of N. to iß. ,(722°) defines a normal solution N . , ofj ;- 1 ;-1

3\._j/(722°) and the restrictions of (5.8) and (5.9) to lÙ^îm0) define a set of

normal data of Jv._ , ¡im°). Therefore by P ._,, there exists in a neighborhood

of 222° in UJ._,(722°) a one-parameter family of integral manifolds zV._j of

Â . . ¡im°) it e i- e . ,, e . A, e. , > 0) which can be defined as;- 1 7-I ;-l ;.- 1

(5.12) z(r=faix1,...,xi_l;t)

such that

(5>13) f¿xv ••;. Vi; 0) = g¿xv • • •' Vi' 0)-

idfa/dt)ix1, ■ ■ . , x._x; 0) = 2-a(x,, • • • , x._v 0)

in a suitable neighborhood of the origin of R,_ . For every fixed t

2a idfjdùid/dz^ is an §.._ x¡im°)-íield on 'zV._,. It follows by an easy com-

putation that if we put

(5.14) ta=idfa/dt)ix1,..-,x._l;t)

then (5.12) and (5.14) define a /-dim integral manifold <V_j of Jv._ j / (722°) in a

neighborhood of the point m° = (222°, f°, 0) e il). , x Rp x (- 1, l). N. , is

also an integral manifold of J\.I(m°). Put E. = T^0{N. .). Then it is clear

that

(5.15) 27^°) =222°, 27u(i.)=rm0(N/_1).

Since T 0(zV._.) is a regular integral element of Jx./(z22°), it follows readily

from Proposition 4.1 and Remark 4.1 that E. is also a regular integral element

of J\ .¡im°). If we put



(5.16) '° = ? ¿T moiN, _ t), 7 l(m°))

then the character of E . is given by

(5.17) ',■*_<*/• 7Ï(m°)) = r°.

If we set

(5.18) b°Cru=tt8a/dxf)(0,---,0) (77=1,...,;)

and denote by b° the vector whose components are b° , then T 0(/V._.) and

T 0(7V.) as integral elements of 7._yl(m°) and 7.1(m°) (resp.) are defined by

(to°, ¿7°, ... , b°_y) and (m°, b°, ■ • • , b°) (resp.). ¿>° . constitute a solution

of the following linear equations in b .,

(5.19) T(m°,bl,...,b°_yb) = 0

whose rank is p + q — r°. Therefore it is always possible to choose a subset

]C{n+l,...,n + p + q] consisting of r° distinct members such that (5-19)

together with

(5.20) \/-K, = °> A£''

form a compatible linear system of rank p + q. In other words, if F is the

(p + q + j — 7°)-dim submanifold defined in a suitable neighborhood of m° in

IÖ.(to°) byz. — e (x, , • ■ ■ , x .) = 0, X £ ],

(5.21) A * l >

then we have

(5.22) FD7V D./V ,

and since T o(F) as a subspace of T o(u).) is defined by the equations dz, —m r m j J ^ A

S , 7>? dx = 0, we have also

(5.23) Tm0(F) n H(TJNH1), 7^)) = Tm0(N;)-

Associated with F we will construct a submanifold F oí W. x R x (— 1, l).7

Since all the functions under consideration are real analytic, by writing down

appropriate convergent power series explicitly or otherwise, we can always find

analytic functions G^ defined on a neighborhood of (0, • • • , 0; O) in R; x

(-1, l) such that

Gx(x1, ■ ■ ■ ,x j, 0; t) = fx(xr, ■ •• ,*._,; t),

(5.24) Ae/.(dG,/dt)(x, , ■ ■ ■ , x .; 0) = rAx. , ■ ■ ■ , x .),

7 A 1 ,


348 D. s. P. LEUNG [August

Then F is the (2p + q + j + 1 - r°)-dim submanifold in a neighborhood of 222° in (l. x

Rp x (- 1, l) which is defined by

(5.25) zx- Gxixx,--- ,x.; t) = 0, \ej.

It follows from (5.24) that 77, (TÂ,F)) = T o(E). If E. , is any (2 + l)-dim inte-n ~ '* m ~ m ~ ^7 + 1 ' J

gral element of J\.I (m°) such that T~°(F) I) E . , DE., then we havee 7 zzz 7 + 1 7'

(5.26) Tm0(E) 3 271+(E; + 1) Dw^&j) = rÂ,^).

But nx¿Ei + x)C H(TmdN._x), %.I(m°)), therefore by (5.23)we have nxßi + 1) =

T o(zV.). However, we know by Remark 4.1 that such an E always exists222 7 ^j *v *\, 7 +1

and is unique. That is, T~o(E) O //(T~o(zV ■_ j), 31 ■/ (2220)) = E . t. By Theorem

1.2, there exists in a neighborhood U . C il. x R? x (l, l) of 222° a (7 + l)-dimr\j f\j J J r\¿ r\j f\j ñj

integral manifold N . of 3t./ (m°) such that F n ll. D N . D N . , O il.. We can6 1 j 7 ; j- 1 7

assume that N . n II. is defined by7 7 r

(5.27) z = 77 (x,, • • ■ , x ; r),cr ct 1 2

(5-28) ^^■"•-V')'

Then, in a suitable neighborhood of (0, ... , 0; 0) x R;_ x (- 1, 1), we have

baixx, ■ ■ ■ ,x._v 0; t) = fcrixl, ■ ■ ■ , x._x; t),

(5.29) À e /.

¿>x(*,, • • • , x.; t) = Gx(xx, ■ ■ ■ , x.; t).

Let N . be the one-parameter family of integral manifolds of 3v .2(222°) defined by

(5.27).7

If we put

(5.30) za= ha(xx, ■■■ ,x.;0), Ça= (dha/dt)(xx, ••• , x.; 0),

then, in a neighborhood of 222'° in iß. x Rp+q, (5.30) defines a /'-dim integral mani-

fold %'. of 3\.í(722°). The restriction of ft', to H>._x x Rp+q defines a (/'- l)-

dim integral manifold W of 31._ /'(222°), consequently also of j{.l'im°). We

put E. , = T zc(A/'. ). Since n-,AE'. ) = T d(/V. ,) is a tegular integral* 7—1 zzz 7—1 ¿* 1 — I 7ZZ 7 — 1 ° °

element of J\./(?72°), it follows readily from P. that E'. , is also a regular inte-

gral element of J\ ./'(722o) and

(5.31) r.(E^r S./'(272o)) = 2r°.

Let E be the (2(p + q) + j - 2r°)-dim submanifold in a neighborhood of 722'0 in

iß. x R which is defined by the following equations:

z.- G Ax,, ••-,*,.; 0) = 0,(5.32) X X e J.

ix-idG/dt)ixx, ... ,x.;0) = 0,

Then in a suitable neighborhood il'. C iß. x Rp+q of 222'° we have (by (5.13) and

(5.29))



F' nil' dJI.' nil' D/v.' , nil',(5.33) ? 7 7 7-1 7'

f' nii; d/v; nii; dtv;_, nW.

Since 77,(E ) = F and 77,(/V ._ A = /V._., (5.23) and the uniqueness assertion of

Theorem 1.2 imply that 77 (?l'.) = ttAN'). That is,

<5-34> ha(xl,...,x.;0) = goixl,..-,x-0)

in a suitable neighborhood of (O, • • < , O) x R;.

Now if we put

(5.35) a°erK = (d2b(r/dtdxK)(0,...;0) (1<k<;),

then T ,AF') as a subspace of T ,o((l'. x R^+l?) is defined by the equationsm r m j

7

^X-ZèA>K = 0>

(5.36) Ae/.

K = l

Let ¿7° be the vector whose components are bc\K,a°crK- Then E- , and Tm,0(J[')

as integral elements of J\._ ./'(to°) and 5\./'(ra°) (resp.) are defined by (m'°,

b'°, • . . , b'.°_ y) and (to'°, ¿j0, . - . , b'°) (resp.). i>^.° are solutions of the follow-

ing system of equations in b.\

(5.37) /;(to'°, 7»;°,.■ .,b'«_x,b¡) = o,

(5.20) ^-«¿ = 0, A e/,

(5.38) «.-<. = 0. A e /.A7 A7

Since E' is a regular integral element of 7.l'(m°), it follows from (5.31) that

(5-37) is a compatible system of rank 2(p + q) - 2r°. Denote by (4.17)* and

(4.18.)* the linear equations in b. obtained by setting b = b° , a = a° ,

rfCT= cf °a and ra = to° in (4.17 .) and (4.18.) (resp.). Then (5.37) is generated by

(4.17.)* and (4.18)*. (4.17)* also generates (5.19). By Remark 4.4 we know

that the coefficient matrix of b .in (4.17.)* is the same as that of a .in

(4.18)*. Now, by our choice of the subset /, (5.19) and (5.20) have rank p + q

as linear equations in b .. Therefore (4.18.)* and (5.38) have rank p + q as

linear equations in a .. In other words, (5.37), (5.20) and (5.38) have rank

2(p + q) as linear equations in b'. (maximal rank). That is,

(5.39) Tmlo(F') n H(E;_V 7f(m°)) = T^ÖU).



Since rm,o(/Vp C T m,o(F') n H(E'._V 9,.¡'im0)), therefore Tm,0(N') = Tm,o0l'.).

By Theorem 1.1 and (5.33) we can conclude that N'. O U. =71. fï U .. In particu-

lar, this implies that the restriction of the J\ ./(772°)-field v to zV.O 27,(U'.) coin-

cides with the restriction of the deformation vector of the one-parameter family of

integral manifolds *N . on °N. to °N . n »7,(lL').B 7 ; ; 2 j

Proof of (c). Let

(5.40) [m0! = E0 C Ex C ■ • • C Ej C E ,+1

be a regular integral chain of 31. jKttz0) such that dx A ... A dx 4 0 on E

( 1 < p < /' + 1) and 272'°= (222° rfp e ffi. + 1(222°) x Rp +q. Let

(5.41) Í772'°S = E¿ CE; C".CBÍ.,CBJ

be an integral chain of A ./ (222°) such that

(5.42) n2AE'v) = Ev (0<V</).

By P. such integral chain always exists and furthermore it is also regular.

Applying Theorem 1.2, we can find a general solution N'. of 3x7'(772°) through 722'°

such that T ,o(/V'.) = E.'. Restricted to a subset of zV'., if necessary, this willzzz' ; ; Í7J

give us a normal solution nAN') = N . of 3\ ./(222o) with an J\ ./(?720)-field v on it0 2777 7

which can be defined in a neighborhood 272° by

za =£>!' Y •»*/>•

We may assume that x (722°) = . .. = x .(722°) = 0. (5-43) is a set of normal data of

Jv./(772°). Applying P. we can construct a one-parameter family of integral mani-

folds *N . it e i- e., e) and e . > O) of 3v .¡im°) in a neighborhood of 722° in

il .(222°) which can be defined as

(5.44) *<r-/oi*i» •••.*•; t)

and such that

faixx, ... , x.; 0) = gaixx, ..., x.),

°-45) (¿>/CT/ai)(x1,...,X7;0) = r(T(x1,...,x;).

We put

*„W = idfo/dxK){0, - - -, 0; <)

(5-46) (1< #c < /).*«<'> = (d2fo/dtdxK){0, ..., 0; i)

Denote by & W the vector whose components are & (r), au) and also by

&,Xz) the vector whose components are ¿ ,X¿). Let c: (- e., e.) —» (1.(722°) be theK * CT K 11 J

(5.43)



analytic mapping such that for every fixed t £ (- e-, e) the coordinates of c(t)

are given by

E'. and E. as integral elements of 7 I '(ra°) and 7.l(m°) (resp.) are defined by

(ra'°, & \ (0), • i - , l\ (0)) and (m°, 1^0), ■■• , &Í0)) (resp.). T^^N) as an

integral element of 7.l(m°) is defined by (c(t), £'«), • •• , £.0)). Let (4.17. j)'

be the system of linear equations in b . , obtained by setting b = &a¡{t)

and ra = c(t) in (4.17. ,). Also let (4.18. ,) be the linear system in b'. ,v ; + l v 7 + 1 ' 7 + 1

obtained by setting ba.pi=&cr/J0), aa/_L= ao-^' £r = €°a and m = c^ in

(4.18 +1). Then it is clear that (4.17- ,)' generates I. ¿At), £,(<•), ■•■ , ¿.(r), 7>;+1),

(4.17;+1)° and (4.18.+ 1)° generate 7;' + 1(ra'°, i[(o), ••- , V (O), b'.+y). Since

E . is a regular integral element of 7. j/(to°), (4.17 . ,) is a compatible linear

system of rank p + q - r° ,, where r°. , = r. ,(E., ft. ,/(772°)). Since T ,,.( 7V.)

= E ., for 2" sufficiently close to 0, say |i| < e, T ( X'TV .) is also a regular

integral element (see Remark 1.1) and (4.17. .)' is also of rank p + q - r°.

For every fixed |/| < e, consider the following linear system in the variables

^_ • i = h„ . , + ta . ,:CT.7+ 1 ^7.7+1 C7.7 + 1

Aá,, + l+S*,7+l(c(í)) = 0'

(5.48)' Z^Bap.,M^K,, + i -Batj+liMt))bati(t)\a

+ "«■«,>!«'» - öa,; + l;M(c(/)) = ° tl<|l</)

(Ba, and Ba, a ate the same functions as in previous section.) Note that

(5.48)' is obtained by setting fc^ = 4*. . x in (4.17)'. Therefore (5.48)'

is a compatible linear system of rank p + q - r°+1. It is important to observe

that if we differentiate each equation of (5.48)' with respect to 7 at 7 = 0 and

making use of the facts that de. /dt = a and B = B , we will° dp: ap. au;a;p ap.;p;cr'

have precisely (4.18., .) . For t of sufficiently small absolute value, there is

an obvious one-to-one correspondence between (4.17. ,) and (5.48)' which

preserves generators as linear equations in b . , and A . , respectively.

Denote such a correspondence by K : (4.17. ,)° —» (5.48)'. For L £(4.17. y)°,

Ü (L) depends analytically on t. Define a one-to-one correspondence 5:

(4.17, + 1)0^(4.18; + 1)°by

(5.49) 8(L) = (5Ht(L)/<5<|t=0 for L £ (A.M. +,)c


352 D. s. p, LEUNG [Auguugust

Let S = {Lj, .. . , L I be a set of linearly independent generators of (4.17 , ) .

We claim that S U diS) is a set of generators for the linear system (4.17 . , )° U

(4.18 . x)°. Indeed, let 8(L) e (4.18 • i)°, then there exist functions cfexit), ■ . ■ ,

<fe it) defined for t sufficiently close to 0 such that

(5.50) K,(L) = cfex(t)Kt(Lx) + ■■■+ cfea(Mt(La).

Using elementary linear algebra and the fact that ft (L ), . . . , ft (L ) as well as

ft (L) depend analytically on t, one can easily show that cfex(t), • • • , cfe (t) ate

actually analytic functions of t. Therefore we have

8(L) = dKt(L)/dt\t_0(5.51)

= cfe[ (0)LX + ...+fp'a (0)La + cfex(0)8(Lx) +■■■ + cfe>a(0)8(La).

Therefore there are at most 2(p + q — r° , ) linear independent equations in the

linear system (4.17 . ,) u (4.18 . ,) . However, it follows from Remark 4.4 that

the coefficient matrix of aa .. in (4.18. ,) is exactly the same as that of

baj+2 in (4-17/+1)°. Therefore (4.17.+ 1)° u(4.18.+1)° is a compatible linear

system in the variables a„ . , , b„ . , and it is of rank 2(p + q - r° .). Thus we

can always find a (/' + l)-dim integral element E. x of Jv. ^1 (m°) such that

E'. , 'D E' and 77,.(£'. , ) = E. We have also proved that r. , (E* 3\. ,l'(m°)) =

2r. ,(E ., Ä. ,l(m°)). Now if E. is a /-dim integral element of Jt 7 (272°) defined

by (272 , bx , • • • , b. ) in a sufficiently small neighborhood of (222 °, ê.(0), • • • ,

¿'(0)), we can construct an integral chain of 3v./ (722°)) using subspaces E of E.

on which dx , = ■ . . = dx ■ = 0 (l < p < j). Under n2^ this will be projected

onto an integral chain of &.¡(m°) which is also regular, since it is 'close' to the

regular integral chain (5.40). Repeating the above argument, we can prove that

/ . , (zzz *, b .*,..., b. , b. ,) is a compatible linear system in b. ,, and7+1 7 7 7+1 r 7 1 +1

77+1(e;*, 3l+1/'(t22°)) = 27;+1(t72,(e;*), SR.+1/0*°))

(5'52) = 2r.jB., %j + xl(m°)) = rj+x(E¡ , K;>1/'(m°)).

The second equality follows since 272 (E, ) is 'close' to E .. We have thus

proved that we can extend (5.40) to an integral chain of Jv. yl (222°) and any

such extension is also regular. This completes the proof of Proposition 5.1.

6. Applications to Jacobi fields on minimal submanifolds. Let M be a

Riemannian manifold of dimension n + p and F(zM) be the bundle of orthonormal

frames of M with bundle projection 77: F(M) —> M. Let a>A be the solder 1-forms

and cúab be the 1-forms which define the Riemannian connection of F(M) (see

for example [l]). The local geometry of M is completely determined by the

following structure equations,



d(0A = ¿«BA wRd, coA D + co„A = 0,'A - *-* "'S "*"BA> UJAB+U)BA

B

(6.1)

dco Z "4C AaCS + °Aß' °AS =-T Z RABCDWC A WD "Aß ^ AC"TBT"/1B' "'Aß ~ 2

C CD

Let / be the ideal in the ring of analytic differential forms on F(M) which is

generated by the following forms,

"a- Z w¿ A ̂ a-

(6.2)

&a~ Zwi A ••• A toi_l A <u¿a Aw.+1 r\.../\ù)n.i

It is easy to see, using (6.1), that i/7 C /. Therefore, / defines an exterior dif-

ferential system on F(M).

Proposition 6.1. / 7s involutive with independent variables icu., ••-,&> .,

co ,.,■■-, co , , co , ,,•••, o> „ , „, co ! at every point e £ F(M)12 n-\n rc + 1,72+2 n+P-l,n+p n ' r

({co,^, ■ • ■ , co , \ and {co , ,,■••, co „ , .1 are sowe orderings of12 fz—l,n n + l,ra+2 n+P— 1,72+p ° ■

¡"Ae set's [tu..; z < /'! awe/ {co^; a< /3! respectively).

Proof. We put

a ^^ a; z x—/ az; i; ¿-^ a/3 y py'i z<7 /8<r

(6.3)

w,a = T.hi^j+^hiaik(ú,k+ T/biaßy<»ßyi j<k ß<y

Let h. be the vector whose components are a , h, .. Also let h.., i< j

(h^, a< ß) be the vector whose components are a , bky.. iayaß, hkya$-

Also put

(6.4) A = 72(72 - l)/2+p(p - l)/2.

We will find, for 1 < p < ?2 - 1, that the equations 7 (e, h , ■ ■ ■ , h ) = 0, after

taking into account that / Ae, h , ■ ■ ■ , h A - 0 is generated by

(6.5) <V=°. ¿W-V-=0 Â<v<p-\).

It is also not difficult to see that there is only one way to extend any system

of solutions of the equations 7 ,(e, h,,■■■, h ,) = 0 to a system of' n- 1 1 7Z- 1 '

solutions of the equations

'n + X-lie- hV ' •••*—.!• ¿12' ••■'è„,„-lV

(6.6)

¿77 + l,« + 2' " "' bn+p-l,n+p' = °'


354 D.S. P. LEUNG [August

namely, by setting

(6.7) b.. = haß = 0.

After taking account of the equations

(6-8) Wi^' bV ■■■>hn-V ¿12' ■■■'hn+p-l,n+P) = °>

the equations lê, h%, ... , hn_y byv ■■■ , b„^_UnH>, bj = 0 are generar-

ed by

zz =0,azz '

(6.9)

h -h =0 (1<2Í<72-1)Uan ñau v¿ \ zx v « *■/,Aazz naß

77-1

¿ + y h =o.Z2az2 Z_z fJ-aß

It follows that any system of solutions (e°, h°, ■•■ , h°_x> h°2, ••• , b° x ,

h°) of (6.5), (6.7) and (6.9) defines a regular integral flag \e°\ = EQ C Ey C . ■ • C

E . . C E . of /. It is also easy to see thatn+X-l n+\ '

,,- ion \ (£. ., /) = 0. Q.E.D.(6.10) A+ZZ A+72-1' ^

Remark 6.1. (6.10) implies that through any (X + n - l)-dim integral element

E j of / on which

tu, A • • • A tu , A ca. , A . . •' zz— 1 12

(6.11)Aoj . A cl> , . A •■■ Aw , 2^0

72-1, ZZ ZZ+1.Z2+2 /V 7Z+/>-l,72+p

there exists a unique (À + 72)-dim integral element E of / on which

cox A • • . A a> . /\ cu A ■ ■ ■1 72— 1 1 I

(6.12)i\ cu , A cu , , A • • ■ A <y , A a) 4 0

72-1.7Z 71 + 1,72+2 7Z+/J-l,Z2+¿2 A 72 ^

and E , DE, ..\+7! X+Z2-1

Remark 6.2. We have proved above that any integral manifold of / on which

(6.11) holds is a general solution of /.

If /: Fa —► F(M) defines an integral manifold of / on which (6.12) holds,

then dim Fa = 72(72 - l)/ 2 + p(p - 1)/ 2 + 72 (= X + n). fiFa) is the bundle space

of the bundle of adapted frames (see for example [l]) of the submanifold 77 ° fiFa)

of M. In fact, 77 ° fiFa) is an 22-dim minimal submanifold of M since the pull-

backs of ®a through any section of the adapted frame bundle are the mean cur-

vature forms of 77 ° f(Fa) (see for example [4]). Conversely, given any 72-dim

minimal submanifold N of M, the bundle of adapted frames over N considered

as a submanifold of F(M) is a general solution of /. The involutiveness of /

implies the local existence of minimal submanifolds of dimension 77. More pre-

cisely, we have the following theorem (see also [lO] and [12]).



Theorem 6.1. Let G . be an imbedded (n — l)-dim submanifold of M and

P be an n-dim distribution along G , such that T (G ,) C P(m) for all to e° n— 1 m n— 1 '

G _.. Then assuming the data are real analytic, for each m° £ G _, there

exists in every sufficiently small neighborhood U of m° a unique imbedded

analytic minimal submanifold N of dimension n such that

1. llDAOG . n H,

2. T (TV) = P(m) for all m £ G , nUm ; n— 1

Proof. Let F'(G ,) be the subset of F(M) consisting of all frames (ra, ey,

• • • , e ) (i.e. e. , • • • , e ^ is an orthonormal basis of T (M)) such thatn+p 1 n+p m

m £ G , and e, , • • • , e span P(m). It can be easily seen that F'(G ,)n— 1 1 ?2 r 7 rz— 1

is a submanifold of F(M) of dimension A + 72 - 1. Indeed, if G , is defined

by the imbedding g: G _, —► M, then F'(G _,) realizes a reduction of the group

0(72 + p) of the induced bundle g_1(E(M)) over G _, to 0(72) x 0(p) (for defini-

tions and facts related to induced bundle and reduction of structure group see for

example [8]). Denote also by a>A , co. „ their pullbacks to F'(G _,). Then, at

every point of F'(G _,), co.., co ̂ and co . span the cotangent space of F'(G _,),

but only 72 - 1 of the co . ate linearly independent while the co .., co^, ate

independent among themselves and independent of the co .. On F(G _x) we have

(6.13) wa = °-

Taking the exterior derivative on both sides of (6.13), we have

(6.14) Zwz Aû>.a =0.z

This implies that, on F'(G _.), co . ate independent of co .., co^ and they are

only linear combinations of the co ■ Since on F'(G A only 72 - 1 of the co ■' 1 n—1 ' z

are linear independent, the restriction of ©a (in (6.2)) to F'(G _,) vanishes

identically. F'(G _,) is therefore a (A + 72 — l)-dim integral manifold of 7 on

which (6.11) holds. By Remark 6.1, we have that at every point e° £ F'(G A

there exists a unique (A + 72)-dim integral element E of / on which (6.12)

holds and E. 3T Af\g A). Since r, (T ÄG ,), /) = 0, we have byX+n e° n—l \+n e° n— 1 ^ '

Theorem 1.1 that there exists, in a sufficiently small neighborhood ll of e°

in F(M), a unique integral manifold Fa of / such that T 0(F") = E and U

3Fa DF!(Gn_j) n í. Then, ll = tt(ÎÎ) is a neighborhood of m° = 7z(e0) and 7V =

ir(Fa) is a submanifold of M which have the required properties. Q.E.D.

Let /: Fa —> F(M) define a (A + 72)-dim integral manifold of 7 on which

(6.12) holds. We will now compute the equations of variation of 7 on f(F ). Let

7: f(Fa) —► T(F(M)) be an /-field defined on f(Fa). For simplicity, we will

assume that 0 = r. = (r, co .) . In fact, only the components t a= (r, coa), T a =

(7, <u-a) are of geometric significance. The equations of variation are



(6.15) df*(r J cja) + f*(r J dcoa) = 0,

(6.16) rf/*fjÇ«iAa)^ + /*(rJrf£<BiA»i^-Of

(6.17) df*(r J@a) + f*(T J d®a) = 0.

Set dA = f*ia>A), dAB = f*icoAB). Considering ra and t¿a as functions defined

on ,Fa, (6.15) can be written as

(6.18) dTa + T<rß®ß*-Y< Tiad1 - °-/S

One will find (6.16) is merely the exterior derivative of (6.18), that is,

(6.19) Ç idTß A dßa + zyö^) - X (*ia A di + r.a¿0.) = 0.P z

As for (6.17), we have, by straightforward computations,

(6.20) dfHr J. 0a) = Ç (. Di-1 L a + £ r.ftö J a öl A ... A f5. A ... a ön,

rf0a= £ Z-rß<"i A ••• Aû)._! A w.o A û2. .A í ;'¿1

(6.21)

A ••■ A coi_x f\ù>iaAù)ul A-A«,

^^<x/3r/3wl A ••■ A wn/3

- ¿^(- 1)!r,/3wâ A«] A • • ■ A co• A • • • A conß, i

, I.+ terms at least linear in co ,

where we have set R^ = 2. R-a-a- Since /(Fa) is an integral manifold of /, we

have

(6.22) /*K) = 0, f*ia>ia)= IX,.*.,7

where h .„ . = h .„. and 2 . h . . = 0. Using these facts, we havezCX; ¡ai i jttz o

/*(r J d8a) = ¿(- O'-1^^ AÔ, A • • • A fj. A • • ■ A ̂

(6.23) Äi

- E(Ra/8+%sV, A-AvB,/3

where we have set ^=2.^. b.a.b.ß..

Therefore, (6.17) can be written as



Z(-i)'-I(â+Zv?7z+ ZÔaâ.Z I P, '

he

(6.24)

A ■ • • A 8n - Z iRaß + oaß)rß8x A • • • A 0„ = 0.

(6.18) and (6.24) give the equations of variation of 7. Observe that (6.18) implies

the r.a components of an /-field for which r. = 0 are completely determined once

the 7 components are given.

Recall that f(Fa) is the bundle space of the bundle of adapted frames over the

minimal submanifold f(Fa) of M. Let e = (e., ea): W —> f(Fa) be an arbitrary

section of the adapted frame bundle over an open set W of N. Let r be an /-field

as above. If we put r = 77^(7), then we can write

(6.25) T*(x) = £ ra(e(x))ea(x) d=f £ r*(*)e>).

a. a

Pulling (6.18) back to W,

(6.26) Z 'ia(*<*))e*(0.) = dr*a + Z r|**( V = Z 'S. z e*^>/3

where a.; z here denotes the covariant derivative with respect to the induced

normal connection on the submanifold 77 ° f(Fa) in the direction of e.. If we write

R aa= e*iaaß), then the pullback of (6.24) after taking (6.18) into account is

(6-27) J2- <% , i + Z (Raß + °Ar*Ae*(8, A • ■ A 8 ) = 0•27) jZ^:z;z+ Zâ/J + ̂ Ûiö,

where we have put

(6.28) K; z + Z '% ̂CV + Z ^ ; ,*( V = Z ^ !; ,• ̂(6).7 P 7

Therefore, the normal vector field r* on 77 ° f(Fa) satisfies the following system of

of partial differential equations,

(6.29) Z 'a; i;,-+ Z(«a/8+ffa/3)^=0,i /3

which is the so-called Jacobi equation defined on the minimal submanifold 77 ° f(Fa)

(see, tot example, [4] and [12]). A?2y normal vector field which satisfies (6.29)

z's called a Jacobi field. The Jacobi equations were obtained previously in,

for example, [4] or [6] by considering the second variations of the 72-dim volume

integral. It is given in the present form in [4].

Proposition 6.2. Let r be an ¡-field on an integral manifold Fa of I such

that (t, co ) = 0, then r*= 77^(7) is a Jacobi field on n(Fa). Conversely, for any

Jacobi field 7* 072 rr(Fa), there exists an l-field r on Fa such that n\\r) = 7*.

Proof. We have already proved the first half of the proposition. If 7*=

Sar*ea is a Jacobi field on n(Fa), by taking different sections of the adapted


358 D. S. P. LEUNG

frame bundle Fa over 27(Fa) we can define the ta components of an /-field r

on Fa by (6.25) and hence also its r. components by (6.26). If we put 0 =

(r, o>.) and choose the components r -= (r, cy o), r . = (r, oj .^ arbitrarily,

then we have an /-field r on Fa such that z7#(r) = r*. Q.E.D.

Theorem 6.2. Let M be a Riemannian manifold of dimension n + p and N

be an n-dim minimal submanifold of M. Let v: N —► TÍM) be a Jacobi field de-

fined on N. Under the assumption of real analyticity, for any point m° e N there

exists a one-parameter family of minimal submanifolds lH it £ i— c, c) and e > 0)

z« a neighborhood u) of m° in M, such that N = iß C\ N and the deformation

vector of N on N coincides with the restriction of v to N.

Proof. The bundle space of the bundle of adapted frames FaiN) of N is an

integral manifold of /. In fact, as noted earlier, it is a general solution of /. By

Proposition 6.2 there exists an /-field r on Fa(N) such that nAr) = v and 0 =

( r, coAi = (r, Cu^ß) = ( r, a>..). Then r is an /-field which satisfies the conditions

of the Main Theorem. Since a one-par am et er family of integral manifolds of / on

which (6.12) holds is mapped under 27 onto a one-parameter family of minimal

submanifolds, Theorem 6.2 follows now directly from the Main Theorem Q.E.D.

BIBLIOGRAPHY

1. R. L. Bishop and R. J. Crittenden, Geometry of manifolds, Pure and Appl. Math.,

vol. 15, Academic Press, New York, 1964. MR 29 #6401.

2. G. A. Bliss, Lectures on the calculus of variations, Phoenix Science Series, 1963.

3. E. Cartan, Les systèmes différentiels extérieurs et leurs applications géométrique,

Actualite's Sei. Indust., no. 994, Hermann, Paris, 1945. MR 7, 520.

4. S. S. Chern, Minimal submanifolds in a Riemannian manifold, University of Kansas,

Department of Mathematics Technical Report 19 (New Series), Univ. of Kansas, Lawrence,

Kan., 1968. MR 40 #1899.5. R. Hermann, E. Cartan's geometric theory of partial differential equations, Ad-

vances in Math. 1 (1965), fase. 3, 265-317. MR 35 #520.6. -, The second variation for minimal submanifolds, J. Math. Mech. 16 (1966),

473-491. MR 34 #8348.7. E. Kahler, Einführung in die Theorie der System von Differentialgleichungen,

Chelsea, New York, 1949.

8. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. 1,

Interscience, New York, 1963. MR 27 #2945.

9. M. Kuranishi, Lectures -on exterior differential systems, Tata Institute of Funda-

mental Research, Bombay, 1962.

10. A. B. Poritz, A generalization of parallelism in Riemannian geometry, the C

case, Trans. Amer. Math. Soc. 152 (1970), 461-494. MR 42 #3710.

11. Y. Matsushima, O21 a theorem concerning the prolongation of a differential system,

Nagoya Math. J. 6 (1953), 1-16. MR 15, 428.12. J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88

(1968), 62-105. MR 38 #1617.

DEPARTMENT OF MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RHODE ISLAND

02912

Current address: Department of Mathematics, Purdue University, Lafayette, Indiana

47907


Documents

DEFORMATIONS OF INTEGRALS OF EXTERIOR DIFFERENTIAL SYSTEMS