Selected Papers of Chuan-Chih Hsiung

(3.6)

<5
ip + i

Si0(p + i}p + i, 7\ ipP ++ Ii); ij) dx ) ; »;/ 1

j dx 7o(P-D

— 7 —

The skew-symmetric tensor gI(p) as well as t h e form | ( p ) is said to be harmonic if df(p) = 0 a n d <5| (p) =0. P r o m equations (3.5) and (3.6) it t h u s follows t h a t the skew-symmetric tensor fZ(p) as well as t h e form f(p) is harmonic if p

(3.7)

(P + l)f[I(p);fl ^ f l f p ) ; 7 — -2 f/(p;£]«; i s = ° > s= l

»*yfi/(p-i);* = 0 .

(3.8)

I n particular, the vector ^ as well as the form | ( 1 ) is harmonic if (3.9)

&.,-£., = 0 ,

f*;i = 0 .

On t h e other hand, a vector ^ is called a (3.10)

KILLING

vector if

f.;. + | . ; . = 0 ,

which implies (3.11)

1^ = 0 .

I t is known (see, for instance, YANO and BOCHNER [21]) t h a t if there exists a nonzero KILLING vector field ^ on a E i e m a n n i a n manifold Mn, then t h e manifold Mn admits a one-parameter group of motions generated b y t h e vector field f\ Similarly, a skew-symmetric tensor f/(p) of order p is called a KILLING tensor if (3-12)

^il(p-l); j + £jHp-l);i = ° >

from which it follows t h a t (3-13)

£[/(p);fl

(3-14)

=

£l(p);j J

f/(p-l);a=0 •

Now we consider an a r b i t r a r y vector field f6 on the manifold Mn and the BJCCI identity for this field f„: (3.15)

b; i; c

" ^b; c; i

$a

l

bic >

120 _

8 —

where

(3.16) ^ H = a j ; j / ^ - a j j J / ^ H j,ij j i j -\fl\

i

l

j

sic \

j ^ ( denoting the CHRISTOFFEL symbols formed with respect to gif and the
1

qft

(89M • 8 &*

8

ft*\

Equation (3.15) can be written as (3.17)

Si;b;c

\±i;b

^b;i);c

H;c;i=

$a**

bic '

By multiplying equation (3.17) by gbc and summing for b and c, we can easily see that if the vector field f{ satisfies (3.18)

ff*8(fi;6-f6;i)!e

+r.aii = 0 ,

then it also satisfies 9bchb-,c=Kr ,

(3.19)

where Rjk is the RICCI tensor defined by Kik = Rsm-

(3-20)

From equations (3.9) it follows immediately that a harmonic vector field £f satisfies equation (3.18) and therefore equation (3.19). Equation (3.15) can also be written as (3.21)

— f i ; 6 ; c + (Si;b + h;i);c

— %b;c;i = ~ ^a^bic

•

By multiplying equation (3.21) by gbc and summing for b and c, we can easily see that if the vector field f< satisfies (3.22)

ff6e(fi;6+f6;*);c-^;O;i=0

then it also satisfies (3-23)

<76c
,

— 9 —

Prom equations (3.10) and (3.11) it follows immediately that a KILLING vector field ^ satisfies equation (3.22) and therefore equation (3.23). Similarly, if | / ( p ) is an arbitrary skew-symmetric tensor field of order p ( ^ 1) on the manifold Mn, then by using the EICCI identity p

f J(p); k; I ~ */(p); I; k ~

2

?l(v{7\ a) ^ iskl

s=l

we obtain (3.24)

S g s=l

(f/( P ; s N | ,); i s ; fe £l(p;^\1);

k;ij

p

1, — ,p

s—1

s< t

where E*f = gikEkj

(3.25)

.

On the other hand, by means of the BIANCHI identity (3.26) '

Bim + Bm+Riljk=0

,

a simple calculation suffices to demonstrate that (3.27)

2fz {p. -^i aj "7| b) g R\kit

= fi( P ; 7\ a, T\ b) R" isit >

where Ra% = gbkRakij

(3.28)

.

Prom equations (3.24) and (3.27) it follows immediately that (3.29)

<jf%(p).,. k -9ik(SI(py, 2 s=l

j-

1 l J ( p ; - , J); J: *

( _ l ) - l | a l(p;

a; i g 1, ... ,P

V

- ^ ?l(p; s\a)u s=1

S);

i +

^ s
?

I ( p ; s | a, t | 6) ^

i„i,

— 10 —

Thus, if the skew-symmetric tensor fJ(p) (3.30)

satisfies

v 2 l J ( ^ , ) ; i s ) ; , + I ( - l r-1 'toI ^ ^ ^ . ^ O ;

g>*(£I(p);i-

S= 1

.9=1

then it satisfies also V

(3.31)

1

g fj(P);,-;Jr= 2 fx (p. £| a) Bais +

V

2

s=l

£I(p.t^\ a^\

w

Ra

ti

,

s<(

and therefore (3.32)

£IlP>i9,k£i
,

which can be obtained by multiplying equation (3.31) by | / ( p > and summing for I(p), where (3.33)

J P ^ ^ R ^ I P - V ^ ^

+ PzJii2w^(p-2)f«j(j)_g) .

It is obvious that when p = l, F\£i\ becomes the EICCI curvature Ri^i1 with respect to the vector f\ Prom equations (3.7) and (3.8) it thus follows that a skew-symmetric harmonic tensor £I(py satisfies equation (3.30) and therefore equation (3.32). From equations (3.24) and (3.27) we obtain readily the following equation dual to equation (3.2.v): (3.34)

- Pfl% ( p ) ; , ; k+ 9jk (pSI(p)i i + 2 £l(pi?li);

«,). k

p •^ \ 1

x

)

?

V ^ Sl(p; sla)11 s=l

I(p; s); a; i

is +

1, ... . P ^ ^I(p; s\a,t\b)

a

•yah

isit

Thus, if the skew-symmetric tensor | J ( p ) satisfies (3.35)

g*(p£I
If/^i^i.):*-^^1)'"1^;*;-;*^0;

s=l

s=l

— 11 then it satisfies also (3.36)

0 % ( p ) . , , * = - - i £1{p:?\a) P s=l

%\ ]

1

V

P ~ ^

8
E . l l ''Up; (p; «s\a, | a, *t\b) | i) •"• isit

'

and therefore £7(p)(fl^/(p>i,;*)=--F|fi(P>j >

(3.37)

which can easily be obtained by comparing equations (3.3]), (3.32) a n d (3.36). F r o m equations (3.13) and (3.14) it follows immediately t h a t a skew-symmetric KILLING tensor | / ( p ) of order p satisfies equation (3.35) and therefore (3.37). Now for an arbitrary vector field | a on the manifold Mn we consider t h e Eicci identity ti $ ; j;k

ti

?

; /;; j" ~~

?

ta r>t • n ' ajk »

from which we obtain, b y contraction with respect to i and 1c, (3.38)

** (f* ,; i - £ * ; < ; , ) = - V * ^ •

Similarly, for an arbitrary skew-symmetric tensor field £ 7 ( p ) of order p ( 2 i l ) on t h e manifold Mn we consider tbe EICCI identity 2>

fil(P)

tUP)

—

V

,~.

t J ( p ; s | a) r> l s

«=1

from which we have, by contraction with respect to i1 and it,

ti7(p-l)

til(p-l)

_

__

J»-l

ys

y tU(P-1; s=l

s | a) p ' s

a n d therefore, by means of the relation (3.39)

Rim=Rlkji,

? / ( p-l) ( f " * - " ; j ; i - ^ < P _ 1 ) ; i; i) = B , , f " C - " ^ ( p _ „ + (p - 1 ) i2 WB ^ H <"- 8 > ^ / ( p _ 2 ) .

—

12 —

A simple calculation with the use of the BIANCHI identity (3.26) yields (•!401

J?

tiki (p-2)

til

_

_

7?

ttfUv-V) tkl

Substituting equations (3.33) and (3.40) in equation (3.39) we thus obtain (3.41)

^ Z(p _ 1) ( ^ e - "

4.

j : i

- ^*-]>.

i;

,) = F \iIip) j .

Integral formulas.

Let us now consider a compact orientable Eiemannian manifold Mn with boundary S" - 1 and a C3 positive metric (actually all manifolds in this paper need only a C2 positive metric). By applying the result of NASH [15] on C3 isometric imbeddings, without loss of generality we can assume that the manifold Mn has a C3 isometric imbedding in a Euclidean space En + m oi dimension n + m Si 3 with a sufficient large m. Therefore § 2 can be applied to this manifold Mn, and for non-empty connected boundary B n _ 1 we shall choose the local coordinates x1, ...,xn to be boundary coordinates so that on the boundary Bn~l the a?"-curve is orthogonal to the ^-curves, i = 1,2, ...,n—1. Thus, from equation (2.4) it follows that on the boundary Bn~l (4.1)

gu = gin = o

(» = 1, ..., n - 1 ) .

The tangential and normal components of the skew-symmetric tensor fJ(J)) and the form f(p) defined by equation (3.1) are respectively given by (4.2)

[t£I(p)] = iI(p) ,

n$I(p),

(4.3)

t£(p) = £I(p) dx1™ ,

n$I(p),

(4-4)

r«f/(P)] = f/(p) »

nel(p),

(4.5)

n

%(P) — f(p) ~ ^(p) •

— 13 —

By using equations (2.5), (2.6), (2.7), (2.13), applying the ordinary rules for differentiation of determinants, and noticing that in covariant differentiation gtj can be treated as constants, from a skew-symmetric tensor field f/(p) on the Biemannian manifold Mn we can obtain the differential form (4.6)

fZ(9

tt

j^>f/(p);,.r;„

dY, ..., dY, e„ + 1 , ..., e n + n | ) TO — 1

= (n - 1)! e (fJ(p); kSnvY. i + ZHV) £nv); u

u)dAn.

Integrating equation (4.6) over the manifold Mn, applying STOKES' theorem to the left side of the equation, and making use of equations (2.4), (2.8), (2.9), (4.1) and the boundary coordinates x1, ...,xn on the boundary Bn~x, we have f {g'k^Hnp),

,, k+ fJ(">: 'f J ( p ) : ,) dAn

Mn

(TO — 1 ) !

/ Bn x

~

I

] \n+m-l

(n ~ 1) ! - I)"" 1 |

^

I

^/(P);?

.fc' "" 1 •"» *

» en + U'--1

e

7i + m |

n-1 n—1 ginZI(p)tIip).,iY;n.(dY®...®dY®eH+1®...®en+J (/-f"1" SIlp). , T. n . Y. ^

^

,

where JV is the unit vector tangent to the manifold Mn and normal to the boundary B*"1. Thus we arrive at the integral formula (3) (4.7)

f (jf*f <">f/(W!,; & + p™ >f/(J>)! ,) dA„ Mn

o«-l

(3) After this paper had been written, in the July, 1957 issue of the Journal of the Mathematical Society of Japan T. NAKAE published a paper [14], in which he obtained this formula (4.7) and some special cases of Theorem 5.5A by using a different method.

—

14 —

In case p = l, equation (4.7) becomes

f (gbc?%;b;c+ f*'£,.,) dA^l-l)"-1

(4.8)

J p^.^dA^

.

By the same arguments as above, from a skew-symmetric tensor field f/(jo) on the manifold Mn we can also obtain the differential form (4.9)

a | f j(P_i) ?

;

j * ; i> «-*> ••• ) « - £ , e n + l )

••• J en + m \

n—1

and therefore arrive at the integral formula (4.10)

J (?Iip_1);aiI(p-1);i+

^(p-D^^iJi) ^ -

Mn

Similarly, we obtain (4-11)

/ (£'/(„-,);, £" (p - 1> : i + ^ / ( p - l ) ^ * " " ; <;*) <^« / — \

l \n-l >

l

I J Bn-l

;M <-U(.p-l) XT j A Sj(p-l)? ;iJynaJln~l

•

Subtraction of equation (4.11) from equation (4.10) gives immediately /4.1<>\

/

V*-- 1 ^

[>£T(P-D;;

J L?

t

>Al(p-\)

??7(p-l);i

?

£j

;i?Hp-l);i

Mn

, +

— \

1

)

si /i-iZ(p-l) s / t o - l ) I? ;j;i

J Bn-\

VCJ(p-l)?

J-IKP-D n /7/1 ? ; i; jM a ^ «

;j

? J(p-l) ?

;il-DlnUJ±n-l}

—

15 —

where gUie-i); i = £iT(P-»..ac,ai

(4.13)

.

In case p = l, equation (4.12) becomes (4-14)

\ it'Uui

-^iZ'-.i+SH^-.i-?;«,)}

dAn

Mn

= ( - 1)" ^ /

(? I " , - 11* i) K ^An-l

rj»-l

Substituting equation (3.38) in equation (4.14) we thus arrive at the integral formula for an arbitrary vector field | 4 on the manifold Mn: (4.15)

j (By??

+ f-*£ui-

f,11'.,)

dAn

Jf»

= (--l)n_1 /

^?;j-Zn?;i)Nn
•

B

For a skew-symmetric tensor field | I ( p ) of order p ( ^ l ) on the manifold J P , using the definition of €ll(p)] in equation (3.7) and the similar one of £ [r(p)] we can easily show that ^

„

s

?[7(P);j] — - ?

P

?/(p);j

?

?il(p-l);i

'

P

Substituting equations (3.41) and (4.16) in equation (4.12) we thus arrive at the integral formula f(F\SIip)\+lt1(py>UIM;i

(4-17) Mn

P+

— \—x)

tll(p);i] t

J B»-l

l?/(p-l)?

£i/(p-D

;i

?I(p-l)S

ti

\ dA

;i) i, n a Vl-

— 16 —

Subtracting equation (4.15) from equation (4.8) we readily obtain (4.18)

\

[{9ike;r,k-Rii?)$i

Mn 1

+ - if-1 - sr-l) (Su i - if. i) + s\ i ?., ,i dA n

= (-i) n _ 1 /

^^^-^(r'f;,-^^)]^-,

Similarly, addition of equations (4.8) and (4.15) gives (4.19)

/[(fl^f*;,;* + # , £ ' ; £ <

+ 1^^+

f' ') (Si; i + Si; i)-S\i

= (-I)"' 1 f

f'; ,1 AAn

V^neSi;n + #»(£'£";, ~ T^i)] dAn_t .

jjra-1

For a skew-symmetric tensor field fJ(p) of order p ( ^ 1 ) on the manifold Mn we introduce (4.20)

ASI(P) =- fhw.i:k

+ % SI{P;7\a) R\ s=l ],.... p ^

(4.21)

T£np)

= 91kSnP);

-^ $I{p; s\a, tlb)11 s< I

2 n,c+~ £l
J +

B

isit >

\

\,...,p ^

?i (p; s | a, 11 6) a

isit >

where /I is actually the LAPLACE-BELTRAMI operator dd + 3d. From equations (3.31), (3.32), (3.36), (3.37) then follow (4.22)

£™> z)f/(p) = - SI{P) 9ik Siw.,; k+pF\f

J(p)

I ,

— 17 —

(4.23)

^ni(p)=f^gikh(p);.;k+F^i(p)l

.

Multiplying equation (4.17) by p and subtracting the resulting equation from equation (4.7) we obtain, in consequence of equation (4.22),

(4.24) f [ - f ™ A£I(P) + (p + 1) ! [J(P); ' ] ?U(P). n + ptI(p-'\

i f V i ) : ,] dA„

Mn

= (-l)"_1

[NniI(p)Zi(Py,n

/ Bn-1

Similarly, addition of equations (4.7) and (4.17) and use of equation (4.23) yield (4.25)

f

[ ^ T ^ - r m

V

1 1

; , ^ . ) ; ,

itI(p);j \?

e[/(p);?']\

it > U/(p);j

?

t \l HA ?lI(p);jVi aJin

( - 1 ) - 1 J [^|/(P)fJ(p);m Bn-1

-J- AT f£?

£nl(p-l)

j.n

tH(p-l)

X] A A

since from the definitions of S[I(p);i] and ft™-'?1 we can easily show that

For the case of empty boundary B"-'1, equations (4.15), (4.17), (4.18), (4.19), (4.24), (4.25) were obtained by YANO [19, 20]. Finally by the same method as that used in deriving equations (4.7), (4.10) and (4.11), from a vector field | { on the manifold Mn we can obtain the differential form (4.27)

&\?Y.a,

dY, ...,dY,en+1, n— 1

..., e „ + m | = ( n - 1 ) ! | % dAn ,

—

18

—

and therefore, by means of the boundary local coordinates x1, ..., xn on the boundary B11'1, (4.28)

j^adAn

= (-l)^

un

5.

f

?NndAn_x.

Bn-1

Harmonic and Killing vector fields.

In this section we shall discuss some applications of integral formulas (4.8), (4.15), (4.18), (4.19), (4.28) to harmonic and KILLING vector fields on a compact orientable Eiemannian manifold Mn with boundary Bn~x. By noticing that Nn= fcj™ and that »i; j t — nab.icd s Si; j — (J 'J

t £ ta; c Sb; d

is a positive definite form in | a ; c , from equations (4.1) and (4,8) we can easily obtain THEOREM 5.1. If a vector field f4 on a compact orientable Eiemannian manifold Mn with boundary Bn~1 has zero tangential component on the boundary Bn~1 and satisfies

abc t. ,

= T--P

T--PP > 0

with the condition £'.$= 0 on the boundary Bn~x; then it must satisfy fi;, = 0 and Tii^i^ = 0. In particular, on the manifold Mn if the form T-.^P is positive definite everywhere, then there exists no such vector field ff other than the zero vector field. From equations (3.18), (3.19), (3.22), (3.23) and Theorem 5.1 we further have THEOREM 5.2 If a vector field | f on a compact orientable Eiemannian manifold Mn with boundary JS™"1 has zero tangential component on the boundary Bn~l and satisfies equation (3.18) and Bypg'^.Q with the condition k~i.ti = Q on the boundary Bn~l, then it must satisfy
— 19 — THEOREM 5.3. If a vector field gi on a compact orientable Biemannian manifold Mn with boundary Bn~x has zero tangential component on the boundary B m_1 and satisfies equation (3.22) and Rij&'S1 5g0 with the condition

For the case of empty boundary Bn~x, Theorem 5.1, 5.2 and 5.3 were obtained by BOCHNER and YANO [3, 21]. Similarly, from equations (4.1) and (4.8) we can also obtain THEOREM 5.4. Theorems 5.1, 5.2 and 5.3 are still true; if on the boundary Bn~l the condition £l;i=0 is replaced by gi;}- — £j. t = 0, and the vector field f4 has zero normal component, instead of zero tangential component.

If a vector field £t on the manifold Mn is harmonic and has zero tangential (normal) component on the boundary Bn~x; then from equations (3.9), (4.2) and (4.4) we have l n . n = 0 ( f . ; „ = f n ; i = 0 for i ^ n) on the boundary Bn~x, and therefore equations (4.8) and (4.15) both become, in consequence of equations (3.19), (4.1) and the relation Nn=]/gnnt (5.1)

f {Rii$i?+£i,'ii;i)dAn

= 0,

Mn

which was obtained by YANO [19] for the case of empty boundary J3 n_1 . By applying the theorem of DUFF and SPENCER mentioned in §1, from equation (5.1) we thus have THEOREM 5.5A. If a harmonic vector field gi on a compact orientable Biemannian manifold Mn with boundary Bn~x has zero tangential or normal component on the boundary Bn~1 and satisfies B^g1^'^. 0, then it must satisfy l i;3 -= 0 and Bijglgi= 0. In particular, on the manifold Mn if the BICCI curvature is positive definite everywhere; then there exists no such harmonic vector field f4 other than the zero vector field, and the absolute first BETTI number of the manifold Mn and the relative first BETTI number of the manifold Mn modulo the boundary Bn~l are zero.

— 20 —

Theorem 5.5A can also be obtained as a special case of Theorems 5.2 and 5.4. As mentioned in § 1, for the case of empty boundary Bn~l Theorem 5.5A was proved first by MYERS [13] and then by BOCHNER [1]. On the other hand, if the vector field £t is a KILLING vector field and has zero tangential (normal) component on the boundary B n _ 1 ; then from equations (3.10), (4.2) and (4.4) we have £n.n= = ~%n;n=Q (h;n= — ^n;i = 0) on the boundary B"" 1 , and therefore equations (4.8) and (4.15) both become, in consequence of equations (3.23) and (4.1),

(5-2)

f

(RiiSU'-fhJdA^O,

Mn

from which we thus obtain THEOREM 5.5B. If a Killing vector field f4 on a compact orientable Biemannian manifold Mn with boundary Bn_1 has zero tangential or normal component on the boundary Bn_1 and satisfies it^.^f'^gO, then it must satisfy £ i;3 —0 and jRy
Theorem 5.5B can also be deduced as a special case from Theorems 5.3 and 5.4. For the case of empty boundary JBm_1, equation (5.2) was obtained by YANO [19] and Theorem 5.5B by BOCHNER [1].

Now suppose that a nonzero vector field f* on the Biemannian manifold Mn has zero normal component on the boundary Bn~l and satisfies (5-3)

l < ;/ i * + ^ « f , = 0 •

It is known (see, for instance, YANO and BOCHNER [21]) that the manifold Mn admits a one-parameter group of affine collineations which leave the boundary B™"1 invariant and are generated by the vector £\ Multiplying equation (5.3) by gjk and summing for j

— 21 —

and k, we obtain (5-4)

9bCii;b;e=-Raiia

,

from which and Theorem 5.4 it follows immediately 5.6. If on a compact orientable Riemannian manifold M with boundary Bn~x there exists a one-parameter group of collincations which leave the boundary Bn~l invariant and whose generating vector |* satisfies R^S1? ^ 0, then the vector | l must satisfy £i;j=0 and RtjflP = 0. In particular, if the manifold Mn has negative definite RICCI curvature everywhere, then it admits no one-parameter group of affine collineations leaving the boundary Bn~l invariant. For the case of empty boundary Bn~1, Theorem 5.6 was obtained by BOCHNER [1]. It is known (see, for instance, YANO and BOCHNER [21]) that if a nonzero vector field f* on the manifold Mn defines an infinitesimal eonformal transformation; then we have THEOREM

n

and therefore equation (4.15) becomes / [Ra t? - f'j Si- , - n (n - 2) 2] dAn Mn

= (~ D""1

/

(l* ?; i ~ t i*; <) *n <^«-l

Bn-1

from which and equations (4.1) we can easily obtain 5.7. On a compact orientable Riemannian m,anifold Mn with boundary Bn~x if a vector field |* with zero normal component on the boundary Bn~x defines an infinitesimal eonformal transformation and satisfies i ? i ? | l | ? ^ 0 , then it must satisfy ^.^-=0 and RijSlii= 0. In particular, on the manifold Mn if the Bicci curvature is negative definite everywhere; then there exists no such vector field f* other than the zero vector field, and the manifold Mn admits no oneparameter group of eonformal transformations leaving the boundary Bn~x invariant. THEOREM

,

134

For the case of empty boundary Bn~x, Theorem 5.7 was obtained by BOCHNER [1] for n = 2 and by TANO [19] for n > 2. It has been shown that if a vector field ^ on a Biemannian manifold Mn is harmonic, then it satisfies equations (3.9) and therefore equation (3.19), which can also be written as (5.5)

0>* f *.,. fc _ 22*^ = 0 .

Now suppose that a vector field ^ on a compact orientable Eiemannian manifold Mn with boundary B71"1 has zero tangential component on the boundary Bn~1 and satisfies equation (5 5) with the condition f*. { = 0 on the boundary Bn~x. By means of equations (-1.1) it is easily seen that the integrands on the right and left sides of equation (4.18) are zero and non-negative respectively. Thus the vector | { also satisfies equations (3.9), and hence THEOREM 5.8A. On a compact orientable Biemannian manifold Mn with boundary Bn~x, a necessary and sufficient condition that a vector field | f with zero tangential component on the boundary B"'1 be harmonic is that it satisfy equation (5.5) with the condition f1 i=0 on the boundary Bn'x. Theorem 5.8A was obtained by CONNER [5]. Similarly, from equations (3.10), (3.11), (3.23), (4.1) and (4.19) we obtain THEOREM 5.8B. On a compact orientable Biemannian manifold Mn with boundary Bn~x, a necessary and sufficient condition that a vector field £t with zero tangential component on the boundary Bn~l be a KILLING vector field is that it satisfy

(5.6)

gi*?;.;k

+

B)? = 0 ,

f';i = 0 .

For the case of empty boundary Bn~x, Theorems 5.8A and 5.8B were obtained by de EHAM [16] and YANO [19] respectively. Similarly, from equations (4,18) and (4.19) we can also obtain THEOREM 5.9. Theorems 5.8A and 5.8B are still true ; if on the boundary Bn'x the vector field | 4 has zero normal component, instead of zero tangential component, and in Theorem 5.8A the condition i\i = 0 is replaced by ^ . y — £,. 4 = 0. The first part of Theorem 5.9 corresponding to Theorem 5.8A was obtained by CONNER [5].

— 23 —

Prom the definitions, it is obvious that an infinitesimal motion on a Kiemannian manifold Mn is necessarily an infinitesimal affine collineation on the manifold Mn. For the converse, suppose that a nonzero vector field f* on the manifold Mn has zero normal component on the boundary Bn~x and generates an infinitesimal affine collineation on the manifold Mn leaving the boundary Bn~x invariant. Then the vector field f* satisfies equations (5.3) and (5.4). By putting i=j = a in equation (5.3), summing for a and making use of the identity Baakl = 0 we obtain ! a ; o ; f t = 0 and therefore £a.a = const., which and equation (4.28) imply that 1% = 0 because of the vanishing of the integrand on the right side of equation (4.28). Prom Theorem 5.8B it follows that f* is a KILLING vector on the manifold Mn. Hence THEOREM 5.10. On a compact orientable Riemannian manifold Mn with boundary I?™"1, an infinitesimal affine collineation leaving the boundary JB""1 invariant is necessarily an infinitesimal motion leaving the boundary Bn~l invariant. Por the case of empty boundary B""1, Theorem 5.10 was obtained by YANO [19].

6.

Harmonic and Killing tensor fields.

Similar to §5, we shall in this section discuss some applications of integral formulas (4.7), (4.17), (4.24), (4.25) to harmonic and KILLING tensor fields on a compact orientable Kiemannian manifold Mn with boundary B"" 1 . By noticing that ti"(p); it

is a positive definite form in lj( P ) ; ^, from equations (4.1) and (4.7) we can easily obtain THEOREM 6.1. If a skew-symmetric tensor field £ /( } of order V{=^) on a compact orientable Riemannian manifold Mn with boundary Ba~l has zero tangential component on the boundary B71'1 and satisfies it

Slip); j; k — -LI(p)J(p) I(

T i f) -1I(p) J(p) S

t

tJ{p) > 0 ? = U

1

— 24 —

with the condition ilI(^p^;i = 0 on the boundary satisfy f J ( p ) ; ? -= 0 and m

tI(v)

tJ(.p)

_

J

?

s

— u

-I{V)J(.P)

In particular,

on the manifold -LKP)J(P)?

Bn

l

; then it must

o .

Mn if the form £

is positive definite everywhere, then there exists no such tensor field fJ(p) other than the zero tensor field. F r o m equations (3.30), (3.32), (3.35), (3.37) and Theorem 6.1, we further have THEOREM 6.2. If a skew-symmetric tensor field fJ(p) of order p (2i 1) on a compact orientable Biemannian manifold Mn with boundary B"" 1 has zero tangential component on the boundary Bn~x and satisfies equation (3.30) and F\$IiP)\^.0 with the condition £ii(,p-i)-,i= 0 on the boundary J5 m_1 , then it must satisfy l/( P ) ; , = 0 and F\£I{p)\=0. In particular, on the manifold Mn if the form •F{f /(p) j is positive definite everywhere, then there exists no such tensor field f/(p) other than the zero tensor field.

6.3. If a skew-symmetric tensor field | J ( p ) of order p (2il) on a compact orientable Biemannian manifold Mn with boundary Bn~x has zero tangential component on the boundary B"-'1 and satisfies equation (3.35) and F |fj(P)j ^ 0 with the condition | l J ( p _ 1 ) ; i = 0 on the boundary Bn_1, then it must satisfy | / ( p ) . ; - = 0 and _F J£J(p)j = 0 . In particular, on the manifold Mn if the form F\£Iip)\ is negative definite everywhere, then there exists no such tensor field fJ(p) other than the zero tensor field. Similarly, from equations (4.1) and (4.7) we can also obtain THEOREM

THEOREM 6.4. Theorems 6.1, 6.2 and 6.3 are still true; if on the boundary Bn~x the condition f^p.])^ = 0 is replaced by S[I(p);i] = 0 , and the tensor field | / ( 2 J ) has zero normal component, instead of zero tangential component. If a skew-symmetric tensor field f7(p) of order p ( > 1) on t h e manifold Mn is harmonic and has zero tangential (normal) component on t h e b o u n d a r y Bn~l; t h e n from equations (3.7), (3.8), (4.2), (4.4) we have tnI(p.1);n = 0 (f7(W.B = 0 for n$I(p)) on the b o u n d a r y Bn~x, a n d therefore equations (4.7) a n d (4.17) both

- - 25 —

become, in consequence of equations (3.32) and (3.33), (6.1)

f

(pF\tIlp)\

+ f^>

' £I(P); ; ) dAn = 0 ,

•which was obtained by T A N O [20] for the case of e m p t y boundary Bn'x. Applying t h e theorem of D U F F and SPENCER, from equation (6.1) we t h u s have THEOREM 6.5A. If a skew-symmetric harmonic tensor field | 7 ( p ) of order p ( > 1 ) on a compact orienlable Riemannian manifold Mn 1 with boundary 5 " " has zero tangential or normal component on the boundary Bn~x and satisfies F j | j ( P ) j > 0 , then it must satisfy fr< P );7=0 and F\£I(P)\ = 0. In particular, on the manifold Mn if the form F\£np)\ is positive definite everywhere ; then there exists no such harmonic tensor field £J(p) other than the zero tensor field, and the absolute p-th BETTI number of the manifold Mn and the relative p-th BETTI number of the manifold Mn modulo the boundary B*1'1 are zero for p = l, 2 , . . . , n—1. Theorem 6.5A can also be deduced as a special case from Theorems 6.2 and 6.4. F o r the case of e m p t y b o u n d a r y Bn~l, Theorem 6.5A was obtained independently by LICHNEROWICZ [11], MOGI [12]

and

TOMONAGA

[18].

On the other hand, if a skew-symmetric tensor field fi(p) of order p(>:l) on the manifold Mn is a KILLING tensor field and has zero tangential (normal) component on the b o u n d a r y Bn~x ; then from equations (3.12), (4.2) a n d (4.4) we have ? n / ( p-i ) ; n = = — fnj
(6-2)

f (F | | J ( r t J - £*<»* > S l ( p ) .,) dAn = 0,

from which we thus obtain THEOREM 6.5B. / / a skew-symmetric KILLING tensor field f/(p) of order i ? ( > l ) on a compact orientable Riemannian manifold Mn n l with boundary B ~ has zero tangential or normal component on the boundary J3™"1 and satisfies 2? j £ J(p) j :g 0, then it must satisfy

— 26 —

£i(P);j = () and F\£I(p)\=0. In particular, on the manifold Mn if the form F\iI(p)\ is negative definite everywhere, then there exists no such KILLING tensor field f/(p) other than the zero tensor field. Theorem 6.5B can also be deduced as a special case from Theorems 6.3 and 6.4. F o r t h e case of e m p t y b o u n d a r y JB™_1, equation (6.2) was obtained by YANO [20] and Theorem 6.5B by MOGI

[12].

A skew-symmetric tensor field | 7 ( p ) of order p^l on the manifold Mn is called a conformal KILLING tensor field if (^•3)

l i T ( p - l ) ; j + SjHp-1);

i = *®I(p-l)

9if ,

where 1 ®I(p-l)

= ~

.. 3%1 f / ( p - l ) ; j •

I t is obvious t h a t a KILLING tensor is a conformal KILLING tensor. Using equations (3.41) a n d (6.3) we can reduce equation (4.12) to

(6.4)

J (F j t7(p)j - £"<*-»•• 'Znw. i-Mn-2)

— I— 1 \ » _ 1 — \

1

)

I j Bn-1

I ti

tn

£»/(p-l)

\5l(p-l)S

;j

SI(p-l)S

riKp-l)

0 ^ - » 0np_1}) dAn \ iy- J A ; i)

ly aJL

n

n-l

>

a n d therefore obtain THEOREM 6.6 If a skew-symmetric conformal KILLING tensor field fI(p) of order p ( > l ) on a compact orientable Riemannian manifold n m_1 M with boundary _B has zero tangential or normal camponent on the boundary ~Bn~x and satisfies _FJfJ(p) J :g 0, then it must satisfy fj(„) ; ; = 0 and F\$Iip)\ = 0. In particular, on the manifold Mn if the form F\£I(p)\ is negative definite everywhere, then there exists no such tensor field f/(p) other than the zero tensor field. For t h e case of e m p t y b o u n d a r y JB"" 1 , Theorem 6.6 was

obtained b y YANO [20].

I t has been shown t h a t if a skew-symmetric tensor field | J ( p ) of order p ( > l ) on a Ttiemannian manifold Mn is harmonic, then it satisfies equations (3.7), (3.8), (3.31), t h e last of which can be written as, b y means of equation (4.20), (6.5)

Miw

= 0 .

— 27 —

Now suppose that a skew-symmetric tensor field fJ(p) of order p ( > l ) on a compact orientable Eiemannian manifold Mn with boundary Bn~l has zero tangential component on the boundary Bn~l and satisfies equation (6.5) with the condition (3.8) on the boundary B™"1. By means of equations (4.1) it is easily seen that the integrands on the right and left sides of equation (4.24) are zero and non-negative respectively. Thus the tensor | J ( p ) also satisfies equations (3.7) and (3.8). Hence THEOREM 6.7A. On a compact orientable Riemannian manifold Mn with boundary Bn~x, a necessary and sufficient condition that a skew-symmetric tensor field | J ( p ) of order p ( > l ) with zero tangential component on the boundary Bn~x be harmonic is that it satisfy (6.5) with the condition (3.8) on the boundary Bn~x. Theorem 6.7A w-as obtained by CONNER [5]. Similarly, from equations (3.12), (3.13), (3.14), (3.36), (4.1), (4.21), (4.23) and (4.25) we obtain THEOREM 6.7B. On a compact orientable Riemannian manifold Mn with boundary Bnl, a necessary and sufficient condition that a skew-symmetric tensor field £IiP) of order p ( > l ) with zero tangential component on the boundary Bn~l be a KILLING tensor field is that it satisfy

(6.6)

T£Ilp) = 0 ,

f*J(p-i); i = 0 •

For the case of empty boundary Bn~x, Theorems 6.7A and 6.7B were obtained by de EHAM [16] and YANO [20] respectively. Similarly, from equations (4.24) and (4.25) we can also obtain THEOREM 6.8. Theorems 6.7A and 6.7B are still true; if on the boundary Bn~x the tensor field £J(p) has zero normal component, instead of zero tangential component, and in Theorem 6.7A the condition (3.8) is replaced by the condition (3.7).

7. Flatness and deviation from flatness. In this section we shall apply Theorems 6.5A and 6.6 to a compact orientable Eiemannian manifold Mn with boundary Bn~* and the property that certain inequalities are satisfied by one of the

— 28 —

four curvature tensors: EIEMANN-CHRISTOFFEL, WEYL'S projective, W E Y L ' S conformal, concircular. I n each case t h e determination of a positive or negative definite form F\£I(p)\ on t h e manifold if" for certain values of p is the same as t h a t for t h e case of e m p t y b o u n d a r y Bn'x (for the details see, for instance, YANO and BOCHNER [21], Chapters I V and

V).

At first, suppose t h a t t h e sectional curvature of a Eiemannian manifold if" is equal to a constant J?/[r*(«—!)]. Then it is well known t h a t R

R from which we have Rjk = — gjk.

Substitution of these two

equations in equation (3.33) gives immediately

B y applying Theorem 6.6 we t h u s obtain THEOREM 7.1 On a compact orientable Eiemannian manifold if" with boundary Bn~l and negative constant sectional curvature, there exists no skew-symmetric conformal KILLING tensor field | J ( p ) of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~1 for p = l, 2 , . . . , n—1. Now consider a conformal transformation

~9jk=e29jk

(7.1)

of the metric of a E i e m a n n i a n manifold if", Q being a scalar function. Under this transformation t h e EIEMANN-CHRISTOFFEL curvature tensor R{ikl of the manifold if" will be changed, while unchanged the following one, called Weyl's conformal curvature tensor of the manifold if" : (7.2)

C\kl = R)kl

^

(Rlkd\ - Rfldi + gibR\ -

(n —1) (n — 2)

g^)

(9iA — 9n*\) ,

— 29 —

where R=gi1Rij. Obviously, if the manifold Mn can be reduced to a Euclidean space by a suitable conformal transformation; then &m = 0, which has been proved also to be sufficient by J. A. SCHOUTEN for n > 3. Accordingly, a Eiemannian manifold Mn(n>3) with C)kl = 0 is said to be conformally flat. If C)kl = Q, from equations (3.33) and (7.2) it follows that F

\tnv)\ = n~~J~- * ^ i I , H ) 4 - i ) +

(7-3)

+

(»-l)(»-2)

On the other hand, if the EICCI form R^t? then (7.4)

BylV^^l'fi,

22 7(P)

'

^"> •

i s positive definite, R>nL>0,

where L denotes the smallest (positive) eigenvalue of the matrix I By J. Substituting equations (7.4) in equation (7.3) and at a fixed point on the manifold Mn choosing a local coordinate system with respect to which gxi = 6tj, for n > 2p we obtain, at the fixed point, n—v

From equations (7.3) and (7.5) it follows immediately that for n>J2p the quadratic form -Fj£J(p)j is positive definite. Applying Theorem 6.5A and the duality theorem of LEFSCHETZ mentioned in §1 we thus have 7.2A. On a conformally flat compact orientable Biemannian manifold Mn(n>3) with boundary I?™-1, if the EICCI curvature is positive definite everywhere ; then for p = l, 2,... , [w/2] there exists no skew-symmetric harmonic tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary B n _ 1 , and for p = l, 2 , . . . , n—1 the absolute p-th n BETTI number of the manifold, M is zero. For the case of empty boundary Bn~x, Theorem 7.2A was obtained by LICHNEROWICZ [11] and MOGI [12] independently. Similarly, if we assume that the EICCI form Rti
— 30 —

definite, then (7.6)

Riifs'^—MiUi,

R^-nM<0

,

where — M denotes the largest (negative) eigenvalue of the matrix || 22i?-1[. For gij = b'ij and n>=2p, substituting equations (7.6) in equation (7.3) we obtain

A direct application of Theorem 6.6 thus gives THEOREH 7.2B. On a conformably flat compact orientable Riemannian manifold Mn(n>3) with boundary Bn~l, if the EICCI curvature is negative definite everywhere, then there exists no shew-symmetric conformal KILLING tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~l for p = l, 2 , . . . , [nj2~\. IsTow suppose that for any skew-symmetric tensor | t J on a Eiemannian manifold Mn

(7.7)

0
m

r ' <,B Pit'I?

A and B being constants. Then it is easy to show that (7.8)

Rii^^a>.\

(»-l)^f% •

From equations (3.33), (7.7) and (7.8) we have

^k/(P,i =4 [(n-l)A-(p-l)Btf™£I(p)

,

1 . which is positive for AjB > (p—l)/(w—1). Since — > (p—l)/(n—1) for p = 1,2, ..., [n/2], we may assume that A = —B, so that the form F\i-I(p)\ is positive definite for p = 1, 2, ...,[»/2]. By applying Theorem 6.5A and the duality theorem of LEFSCHETZ we thus obtain

— 31 —

7.3A. On a corn/pact orientable Riemannian manifold M with boundary Bn~l, if the RIEMANN-CHRISTOFFEL curvature tensor Rijkl satisfies, for any skew-symmetric tensor £w of the second order THEOREM

n

imS

0 < - B<2 ~

S


?hi

B being a constant; then for p = l,2, ...,[n/2] there exists no skew-symmetric harmonic tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~l, and for p = 1, 2, ...,n — 1 the absolute p-th BETTI number of the manifold Mn is zero. For the case of empty boundary Bn~l, Theorem 7.3A was obtained jointly by BOCHNER and YANO [4]. Similarly, if we assume that J?..,, tij tkl Hj p . . s

then F\fi„J \^(P)\^1 <

F

2

Si]

[(p-l)A-(n-l)B]£«p)£I(p)

which is negative for (p— l)/(n—1) < BjA. we thus have

, Applying Theorem 6.6

7.3B. On a compact orientable Riemannian manifold M with boundary Bn~l, if the RIEMANN-CHRISTOFFEL curvature tensor RiW satisfies, for any skew-symmetric tensor £1' of the second order, THEOREM

n

J?..,, tij tkl

1

A being a constant; then there exists no skew-symmetric conformal KILLING tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~x for p — 1 , 2,...,[n/2]. A Riemannian manifold Mn is said to be locally projectively flat, if for any coordinate neighborhood of the manifold Mn there exists a one-to-one correspondence between this neighborhood and a domain D i n a Euclidean space such that any geodesic of the

— 32 —

manifold Mn corresponds to a straight line in the domain D. For w > 3 , a necessary and sufficient condition that the manifold M* be locally projectively flat is that WEYL'S projective curvature tensor (7.9)

Wmi = Rm

-

- i — [Rjkgu it

R,l9ik)

J.

vanish. It is easy to show that a locally projectively flat Eiemannian manifold is of constant sectional curvature, and vice versa. Now for any skew-symmetric tensor f*' of the second order on a Eiemannian manifold Mn we introduce IW

(7.10)

w = sup,

l

y

S& £kl I

LA .

If the RICCI form R^gg is positive definite; then by substituting equation (7.9) in equation (3.33) for Bijkl and making use of equations (7.4), (7.10) we can obtain, for g^ = <5i?.,

and therefore, by applying Theorem 6.5A, THEOREM 7.4A. On a compact orientable Riemannian manifold Mn with boundary Bn~l and positive definite EICCI curvature everywhere, if

where W is given by equation (7.10), and L is the smallest (positive) eigenvalue of the matrix |-R^|; then for p = l, 2, ..., n — 1 there exists no skew-symmetric harmonic tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary B71'1, and the absolute p — th BETTI number of the manifold Mn is zero. For the case of empty boundary B""1, Theorem 7.4A was obtained by BOCHNER [2].

— 33 —

Similarly, if the EICCI form R^f? is negative definite, then making use of equations (3.33), (7.6), (7.9), (7.10) we obtain, for 9n = dn >

and therefore, from Theorem 6.6, THEOREM 7.4B. On a compact orientable Biemannian manifold Mn with boundary Bn~x and negative definite EICCI curvature everywhere, if

n~p n—1

mr

p — 1 ._, 2

where W is given by equation (7.10), and —M is the largest (negative) eigenvalue of the matrix JjRi7.||; then there exists no skewsymmetric conformal KILLING tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bm_1 for p = 1, 2,..., n — 1. On a Eiemannian manifold a geodesic circle is defined to be a curve whose first curvature is constant and all other curvatures are zero. It is known that an arbitrary conformal transformation (7.1) carries any geodesic circle on a Eiemannian manifold into a geodesic circle on another Eiemannian manifold if and only if the function Q satisfies Qj;k — QjQk = 0gfk, where Q^dlogQJdaP'. Such a conformal transformation is called a concircular transformation. The tensor (7.11)

Z% = R)kl - - J ^ -

(gjkb\ ~ gfl6i)

is invariant under any concircular transformation, and Ziikl=0 is a necessary and sufficient condition that a Eiemannian manifold be reducible to a Euclidean space by a suitable concircular transformation. Accordingly, a Eiemannian manifold with Zijkl=ii is said to be concircularly flat. It is easily seen that a concircularly flat Eiemannian manifold is of constant sectional curvature, and vice versa. Now for any skew-symmetric tensor £1' of the second order

-

34: —

on a Biemannian manifold Mn we introduce (7.12)

Z = sup^

If the EICCI form B^gg is positive definite, then making use of equations (3.33), (7.4), (7.11), (7.12) we obtain, for <7i? = <^,

and therefore, by applying Theorem 6.5A, THEOREM 7.5A. On a compact orientable Biemannian manifold Mn with boundary Bn'x and positive definite EICCI curvature everywhere, if

r

P-l n (n — 1)

B>P=lZf

where Z is given by equation (7.12), and L is the smallest (positive) eingenvalue of the matrix ||jRi?-1|; then for p = 1, 2, ..., n — 1 there exists no skew-symmetric harmonic tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~1, and the absolute p-th BETTI number of the manifold Mn is zero. Similarly, if the EICCI form B^fg is negative definite, we have, in consequence of equation (7.6) for <7# = <5i?-,

and therefore, by applying Theorem 6.6, 7.5B. On a compact orientable Biemannian manifold M with boundary B11"1 and negative definite EICCI curvature everywhere, if THEOREM n

n(n — 1)

2

where Z is given by equation (7.12), and —M

is the largest

— 35 —

(negative) eigenvalue of the matrix | Rtj | ; then there exists no skew-symmetric conformal KILLING tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~l for p = l, 2, ..., n—1. For the case of empty boundary B"" 1 , Theorems 7.5A and 7.5B were obtained by YANO [20]. Pinaly, by means of equation (7.2) we introduce G = supf ' Jim If the Eicci form RM f* | 7 is positive definite; then making use of equations (3.33), (7.2), (7.4), (7.13) we obtain, for n>2p,

and therefore, by applying Theorem 6.5 A and the duality theorem of LEFSCHETZ, THEOREM 7.6A. On a compact orientable Riemannian manifold Mn with boundary B™-1 and positive definite EICCI curvature everywhere, if n 2

~P n —2

L+

P - l (n — 1) (n—2)

B>Pj=±0i

where C is given by equation (7.13), and L is the smallest (positive) eigenvalue of the matrix \R^\; then for p = l, 2 , . . . , [w/2] there exists no skew-symmetric harmonic tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary _Bn_1, and for p=l, 2,..., n—1 the absolute p-th n BETTI number of the manifold M is zero. It is easily seen that Theorem 7.2 A is a special case of Theorem 7.6A. Similarly, if the EICCI form R^PP is negative definite, for n >^2p we have, in consequence of equation (7.6),

» l f t j s ( - ^ M+^J-^i and therefore, by applying Theorem 6.6,

+ Vzl c)f»bm ,

148 — 36 —

THEOREM 7.6B. On a compact orientable Riemannian manifold Mn with boundary Bn~l and negative definite EICCI curvature, everywhere, if

»-2

M

(n-l)(n-2)

2

°'

where C is given by equation (7.13), and —M is the largest (negative) eigenvalue of the matrix ||-Ky|; then there exists no shew-symmetric conformal KILLING tensor field of order p, other than the zero tensor field, with zero tangential or normal component on the boundary Bn~l for p=l, 2, ...,[n/2]. For the case of empty boundary B"1'1, Theorems 7.6A and 7.6B were obtained by BOCHNER [2]. LEHIGH UNIVERSITY BETHLEHEM, U. S. A.

PENNSYLVANIA

BIBLIOGRAPHY

[1] S. BOCHNER, Vector fields and Ricci curvature, Bull. Amer. Math. Soc, vol. 52 (1946), pp. 776-797. [2] S. BOCHNER, Curvature and Betti numbers, Ann. of Math., vol. 49 (1948), pp. 379-390. [3] S. BOCHNER, A new viewpoint in differential geometry, Canadian J. Math., vol. 3 (1951), pp. 460-470. [4] S. BOCHNER and K. YANO, Tensor-fields in non-symmetric connections, Ann. of Math., vol. 56 (1952), pp. 504-519. [5] P. E. CONNER, The Neumann's problem for differential forms on Riemannian manifolds, Mem. Amer. Math. Soc., No. 20, 1956. [6] G. F. D. DUFF and D. C. SPENCER, Harmonic tensors on Riemannian manifolds with boundary, Ann. of Math., vol. 56 (1952), pp. 128-156. [7] L. P. EISENHART, Riemannian Geometry, Princeton, 1926. [8] W. V. D. HODGE, The Theory and Applications of Harmonic Integrals, 2nd ed., Cambridge, 1952. [9] E. HOPF, Elementare Bemerkungen iiber die Losiingen partieller Differ entialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungsber. Preuss. Akad. Wiss., vol. 19 (1927), pp. 147-152. [10] C. C. HSIUNG, Some global theorems on hypersurfaces, Canadian J. Math., vol. 9 (1957), pp. 5-14. [11] A. LICHNEROWICZ, Courbure et nombress de Betti d'une variete riemannienne compacte, C. R. Acad. Sci. Paris, vol. 226 (1948), pp. 1678-1680. [12] I. MOGI, On harmonic field in Riemannian manifold, Kodai Math. Seminar Reports, vol. 2 (1950), pp. 61-66. [13] S. B. MYERS, Riemannian manifolds with positive mean curvature, Duke Math. J., vol. 8 (1941), pp. 401-404. [14] T. NAKAE, Curvature and relative Betti numbers, J. Math. Soc. Japan, vol. 9 (1957), pp. 367-373. [15] J. NASH, The imbedding problem for Riemannian manifolds, Ann. of Math, vol.63 (1956), pp. 20-63. [16] G. DE RHAM, Remarque au sujet de la theorie des formes differentielles harmoniques, Annales de l'Universite de Grenoble, vol. 23 (1947-48), pp. 55-56. [17] G. DE RHAM and K. KODAIRA, Harmonic Integrals, Mimeographed notes, Institute for Advanced Study, Princeton, 1950. [18] Y. TOMONAGA, On Betti numbers of Riemannian spaces, J. Math. Soc. Japan, vol. 2 (1950), pp. 93-104. [19] K. YANO, On harmonic and Killing vector fields, Ann. of Math., vol. 55 (1952), pp. 38-45. [20] K. YANO, Some remarks on tensor fields and curvature, Ann. of Math., vol. 55 (1952), pp. 328-347. [21] K. YANO and S. BOCHNER, Curvature and Betti Numbers, Ann. of Math. Studies, No. 32, Princeton, 1953.

A UNIQUENESS THEOREM ON TWO-DIMENSIONAL RIEMANNIAN MANIFOLDS WITH BOUNDARY Chuan-Chih Hsiung 1. INTRODUCTION The purpose of this paper is to establish the following THEOREM. Let Mz and Mf be two oriented two-dimensional Riemannian manifolds of class C2 imbedded in a Euclidean space E N + 2 of dimension N+ 2 (N > 0), with boundaries C and C*, respectively, and with positive Gaussian curvatures in every normal direction. Suppose that there exists an orientation-preserving differentiable homeomorphism H of the manifold M2 onto the manifold M£ such that at corresponding points the manifolds M2 and MJ have parallel tangent planes and equal sums of the principal radii of curvature associated with every common normal direction. If the homeomorphism H restricted to the boundary C is a translation (strictly speaking: is induced by a translation in the space E j ^ ^ carrying the boundary C onto the boundary C*, then the homeomorphism H is a translation carrying the whole manifold M2 onto the whole manifold MJ. For the case where N = 1 and the boundaries C and C* of the two manifolds are empty, this theorem was proved by Christoffel [3]; for the general case where N = 1, it is due to the author [4]. The method used in this paper is an extension of that used by Chern [2] in proving the uniqueness theorem for Minkowski's problem for closed convex surfaces imbedded in a three-dimensional Euclidean space. (We recall that, by a result of Nash [6] on C3 isometric imbeddings, every two-dimensional Riemannian manifold with a C3 positive metric can be imbedded in some Euclidean space.) 2. PRELIMINARIES Let M2 be an orientable two-dimensional Riemannian manifold of class C3 imbedded in a Euclidean space E^+2 of dimension N + 2 (N > 0). To avoid confusion, we shall use the following ranges of indices throughout this paper: (2.1)

a, 0 = 1 , 2;

3 < r < N + 2;

1 < i, j , k < N + 2 .

Associated with a point P in the space Ej\j+2 w e introduce a right-handed rectangular frame P e j * " e N + 2 such that e j , •••, ejsj+2 form an ordered set of mutually perpendicular unit vectors with the determinant ( e j , •••, ejs[+2) equal to +1. Let X denote the position vector of the point P with respect to a fixed point O in the space E N + 2 ; then we can write dX = 2 i

oJjej,

(2.2) dej =

Z, u}{j e ; , J

R e c e i v e d D e c e m b e r 7, 1956. 25

151 26

CHUAN-CHIH HSIUNG

where W£ and w^ are Pfaffian forms in the manifold of frames and satisfy (2.3)

Wij + ujji = 0.

Since the exterior derivative of an exact differential is zero, d(dX) = 0 and d(dej) = 0, and therefore we have, from equations (2.2),

J (2.4) dw

ij = ^

w

ikA%,

where A denotes the exterior product. To study the manifold M2, we consider the submanifold of the frames Pei ••• e N + 2 such that P eM 2 and e 1; e2 are two tangent vectors of the manifold M2 at the point P . Denoting by the same symbols the forms on this submanifold of frames induced by the identity mapping, we have (2.5)

wr = 0,

and therefore, from the first of equations (2.4), (2.6)

du)r = £ va/\uar a

= °-

By a Lemma of E. Cartan (see, for instance, [l], p. 117) on exterior algebra, equation (2.6) implies that for each value of r (2.7)

war = £

Kae'^B

with the conditions

Let Oil ••• iN+2 be a frame in the space Ejsj+2 such that i j , •••, i^+2 f ° r m an ordered set of mutually perpendicular unit vectors at a point O. With respect to this frame we define the vector product of N + 1 vectors A 1? ••-, A N f i through the point O in the space E N + 2 to be the vector A N + 2 (denoted by A 1 x - - x A N + 1 ) satisfying the following conditions: (a) the vector A N + 2 is perpendicular to the (N + 1)-dimensional subspace of the space E N + 2 spanned by the vectors A x , •••, A N + 1 , (b) the magnitude of the vector A]sj+2 is equal to the volume of the parallelepiped whose edges are the vectors A j , •••, A N + 1 , (c) the two frames OAx •••A N+1 A N + 2 and Oi x " - i N + 2 have the same orientation. Let a be a permutation on the N + 1 numbers 1, •••, N + 1; then A

a(l) >< "" >
152 RIEMANNIAN MANIFOLDS WITH BOUNDARY

27

where sgn a is +1 or -1 according as the permutation a is even or odd. Furthermore, the scalar product of the vector A i X " - x A N + 1 and a vector I through the point O in the space E N + 2 * s g i v e n by (2-9)

I - ( A i X . . . x A N + i ) = (-1) N + 1 (I, Ai, •••, A N + 1 ) .

From equation (2.9) it follows that (2.10)

eiX-.Xer.1xer+1x--xeN+2 = (-l)N+rer.

Let dA be the area element of the manifold M2 at a point P, and let K r and H r ) respectively, be the Gaussian curvature and the mean curvature of the manifold M2 associated with the normal vector e r at the point P. Then, by means of the combined operation ® of the vector product x and the exterior product A (for this operation (x) see, for instance, [5]), we obtain (2.11)

d X ® dX® e 3 ® - ® e r _ ! ® e r + 1 ® -

x eN+2 = (-l)N+r2erdA,

(2.12)

de r ® d e r ® e 3 ® •••® e r _ ! ® e r + 1 ® — ® e N + 2 = (-l) N + r 2K r e r d A ,

(2.13)

d X ® d e r ® e 3 ® •••® e r _ i ® e r + 1 ® •••® e N + 2 = ( - l ) N + r + 1 2 H r e r d A .

From equations (2.2, (2.5), (2.10), (2.11), (2.12), (2.13) it follows that (2.14)

dA =

ulAu2,

(2.15)

K r dA = w r l A^r2>

(2.16)

2H r dA = u) r 2 A'^i - w r i A w 2 .

It is known that the vector £> = Z H r e r is independent of the choice of the mutually r

perpendicular unit vectors e 3 , •••, e N + 2 in the normal space of the manifold M2 at the point P; the vector £> and its magnitude are respectively called the mean curvature vector and the mean curvature of the manifold M2 at the point P . 3. AN INTEGRAL FORMULA Suppose that M2 and M* are two orientable two-dimensional Riemannian manifolds of class C3 imbedded in a Euclidean space E N + 2 of dimension N + 2 (N > 0) with boundaries C and C*, respectively, and with positive Gaussian curvatures in every normal direction. Furthermore, suppose that there exists an orientationpreserving differentiable homeomorphism H of the manifold M2 onto the manifold M* such that at corresponding points the manifolds M2 and M* have parallel tangent planes. Then the definitions in Section 2 can be applied to the manifold M2; and for the corresponding quantities and equations for the manifold M* we shall use the same symbols and numbers with a star, respectively. By using equations (2.2), (2.3), (2.5), (2.9), (2.10), (2.16), (2.5)* and the first of equations (2.2)*, and applying the ordinary rules for differentiation of determinants, we can obtain the differential form

28

CHUAN-CfflH HSIUNG d(X*, dX, e 3 , - , e N + 2 ) = (-1)N+1 e 3 • (dX*® d X ® e 4 ® - ® e N + 2 ) + (-1) N + 1 X*- E d X ® e 3 ® - " ® e r _! ® d e r ® e r + 1 ® ••• ® e N + 2

(3.1)

r

= ajAMj - W * A ^ ! - 2 Ep*HrdA, r

where (3.2)

p* = X * - e r .

Integrating both sides of equation (3.1) over the manifold M2 and applying Stokes' Theorem to the left side, we then arrive at the integral formula (3.3)

j(X*, dX, e 3 , • " , e N + 2 ) = J"J(u>*AW 2 - w$Ao^) - 2 j"J £ p * H r d A . M;,

M^

C

r

4. PROOF OF THE THEOREM First we should notice that the given differentiable homeomorphism H between the manifolds M2 and M* induces a homeomorphism between the two frames P e i ••• e n + 2 and P*ei ••• e^+2 at two corresponding points P and P* of the manifolds M2 and M*, whence (4.1)

^*a=«

r a

.

From equations (2.7), (2.14), and (2.15) it follows that <4-2)

K

r = ArllAr22-Arl2Ar21;

this and the assumption that K r > 0 for each value of r imply that the matrix (ArQ,o) is nonsingular for each value of r. Therefore by equations (2.7) and (2.8) we can write, for any value of r, (4.3)

u>a = S X r a „ u p r ,

where (A.rap) is the inverse matrix of (ArQ,o) and where < 4 - 4)

X

r ^ =

X

re-

using equations (2.3), (2.15), (2.16), (4.1), (4.3), (4.4), (4.3)*, (4.4)*, we find immediately that 2H r = (A. r ll + X r 2 2 ) K r , (4.5) a>*Aw2 - co*A^, = (Xr*u X r 2 2 - 2 X * 1 2 X r l 2 +

x y ^ l . ^ A , ^ .

RIEMANNIAN MANIFOLDS WITH BOUNDARY

29

The first of equations (4.5) implies that the sum of the principal radii of curvature at the point P of the manifold M 2 , which are associated with the normal vector e r , is equal to (4.6)

2H r /K r = X r l l + X r 2 2 .

From the assumption and from equations (4.6) and (4.6)* we thus obtain (4-7)

*2ll + *?22 = * r l l

+

*r22-

Substituting equations (2.15), (4.4), (4.5) in equation (3.3), we have J~(X*, dX, e 3 , •••, e N + 2 ) C

(4.8) v '

= f f (A* , ,X , , - 2X* X . , + X* X . . ) « , A « -, r r J J r*l r " ^ r ^ ^ 2 r l l r l r2 " j"J 2?pJ(A- r il + X r 2 2 ) U r l A ^ r 2 • M2

The replacement of the manifold Al, by the manifold M* in equation (4.8), together with the use of equation (4.1), gives | ( X * , dX*, e 3 , - , e N + 2 ) c (4.9) =

2

Jf(X?HX?22-ni22'UrlA^r2 M2

- JJ*i;P^ll M2

+

^2>rlA*> r 2 -

Subtracting equation (4.8) from equation (4.9), and using equation (4.7) and the a s sumption that dX* = dX along the boundary C, we obtain (4.10)

JJ[2(A* r l l A* 2 2 - X*rl22) - (X* r l l X r 2 2 - 2X*12 A r l 2 + X*22 X r U ) ] e o r l A w r 2 = 0. M

2

Subtracting equation (4.10) from the one obtained by interchanging the roles of the two manifolds Mj and M* in equation (4.10), we get (4.11) jj{\*11X*Z2-X*lzZi^1AUxZ= M2

J](>rllXr22-Xrl22)wrlAU>r2M2

Thus the substitution of equation (4.11) in equation (4.10) yields (4.12)

jjlX*n

- X r l l )(X* 2 2 - X r 2 2 ) - (X*12 - X r l 2 ) 2 ] W r l A o > r 2 = 0 .

30

CHUAN-CHIH HSIUNG

But (*r*ll-*rll>(X?22-Xr22>

= ^(Kll

- X r l l ) + ( V 2 2 " \22>]2 - ^ ( ^ l l - X r l l ) 2 + ( ^ 2 2 - X r 2 2 ) 2 ] -

which is reduced, by means of equation (4.7), to (4.13)

(X* u - Xrll)(X*22

- \T22)

= - i [ ( A * H - X r l l ) 2 + (X*22 - X r 2 2 ) 2 ] .

By the assumption and equation (2.15), w r l Ac»)r2 > 0. Thus the integrand in equation (4.12) is nonpositive, and therefore equation (4.12) holds when and only when (4.14)

X* a/? = A rff p

(a,/3 = 1,2).

Using equations (4.14), (4.3), (4.3)*, we obtain (4.15)

oo* = ua

( a = 1, 2).

From equations (4.15), (2.2), (2.5), (2.2)*, (2.5)* it follows that dX* = dX over the whole manifold M2, and hence the proof of the theorem is complete.

REFERENCES 1. E. Cartan, Lecons sur la theorie des espaces a connexion projective, 1937. 2. S. S. Chern, A proof of the uniqueness of Minkowski's faces, to appear in Amer. J. Math.

Paris,

problem for convex

sur-

3. E. B. Christoffel, Ueber die Bestimmung der Gestalt einer krummen Oberfldche durch lokale Messungen auf derselben, J. Reine Angew. Math. 64 (1865), 193-209. 4. C. C. Hsiung, A theorem on surfaces with a closed boundary, Math. Z. 64 (1956), 41-46. 5.

, Some global theorems

on hypersurfaces,

Canadian J. Math. 9 (1957), 5-

14. 6. J. Nash, The imbedding problem for Riemannian manifolds, (1956), 20-63. Lehigh University

Ann. of Math. (2) 63

156

A Uniqueness Theorem on Closed Convex Hypersurfaces in Euclidean Space S. S. CHERN, J. HANO & C. C. HSIUNG 1 The object of this note is to prove the following theorem: Let 2, 2 ' be two closed, strictly convex, C -differentiable hypersurfaces in a Euclidean space of dimension n + 1 (2; 3). Let f: 2 —> 2 ' be a diffeomorphism such that 2 and 2 ' have parallel outward normals CM! p e 2 and p' = f(p) respectively. Denote by Pi(p) (respectively Pi(p')) the lih elementary symmetric function of the principal radii of curvature of 2 {resp. 2') at p (resp. p'). If, for a fixed l,2^l^n, we have (!) Pi-i(p) g P,-x(pO, for all points p t 2, then / is a translation.

Piiv) S£ Pi(p')

The interest in this theorem lies in the fact that the conditions (1) involve only inequalities. The proof of the theorem depends on some integral formulas established in a previous paper of one of us 2 and on an algebraic inequality. Let (X,i), (Mi), 1 ^ i, k ^ n, be positive definite symmetric matrices. We define the polynomials P r .(X, X') by means of the equation (2)

det (8„ + y\ih + y'XQ =

w!

£

r y Pr8(X,

\')yry".

P r ,(X, X') are thus homogeneous polynomials of degrees r, s in \ik , \/h , respectively. I t is easy to derive from the definition that (3)

rPr_Xil(X, X') =

Z ^ ^

5

-

Since Pr0(X, X') is independent of X't , we shall write Pr0(X, X') = P r (X); then P„r(X, X') = P r (X'). Lemma. Let a = (ait), X = (Xik) be positive definite symmetric matrices such that, for a fixed I, 2 ^ I ^ n, (4)

P,_,(a) ^ P,-i(X),

Pt(a) ^ P,(X).

x

The first two authors are partially supported by the National Science Foundation, and the third author is supported by the Air Force Office of Scientific Research. 2 S. S. CHERN, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, this Journal, 8 (1959), pp. 947-955. This paper will be quoted as IFH. 85 Journal of Mathematics and Mechanics, Vol. 9, No. 1 (1960).

86

S. S. CHERN, J. HANO & C. C. HSITJNG

Then (5)

Q,(a, X) s P,(«) + P,(X) - 2P,- I . 1 (a, X) g 0.

Proof. Let Z> be the convex domain of all positive definite symmetric matrices of order n. Let 5 = (5,t) e D be the unit matrix. We have

det (8ik + X.-#) = £ ("W)»', so that 'n JP r (X + f 8)yr

det (5 U + \iky + t 8iky) = £ OSrSn

But the left-hand side can also be written

det (8„(1 + ty) + \thy) = £ (jpr(X)y'(l + ty)-\ OSrSn

\''

l

Comparing the coefficients of y , we get P,(X + i 5) = P,(X) + lP,_x(\)* + £

c«P,_,(X)*',

c< > 0.

2£iS!

Similarly, comparing the coefficients of y'~1y' on both sides of the equation det («„ + yaik + y' 8ik) = ^ . H . K n - r - , ) ! ^ " ' = det ((1 + y') 8ik + yaik) = £

n!

W .,P r (a)» r (l +

yT',

we get (6)

Pi-iM,

5) = Pi-i(«).

It follows that (7)

Pi-i.i(«, X + * 5) = P,_ ltl («, X) + tfV.fe),

and therefore that g«(a, X + t 8) = Q,(a, X) + {ZP.-xQO - 2P,_ 1 (a)}< +

£

^ . . . ( X ) * * ' , C, > 0 .

2£>SI

Hence if * > 0 and P,-i(X) ^ P.-i(«), then Q,(a, X + tS) > Qj(a, X). The tangent hyperplane at a of the hypersurface {X | P,(X) = P,(a)} has the equation £ 0«« - «.*) 1 T ^ = *(iV...(«, /*) - P.(«)) = 0. Since the set {X | Pj(X) ^ P«(«)} is strictly convex, we have Pi_,,,(a, X) —

CLOSED CONVEX H Y P E R S U R F A C E S

87

Pj(a) > 0 if P»(X) > Pi(a). (This also follows directly from GARDING'S inequality.) We now prove that Pi-^a) ^ Pi-i(X), Qi{a, X) ^ 0 imply P,_i,,(a, X) ^ P i (a). Suppose the contrary be true. Then there exists X 0 1D such that Pl-i(a) ^ P/-i(X 0 ), Qi{a, Xo) ^ 0, Pi_i.i(a, Xo) < P,(a). The ray X0 + t8, t > 0, does not belong to the hyperplane P,_i.i(a, X) = P^1A{a, X0), and hence meets the hyperplane P,_ l f l (a, X) = Pt{a) in a point X0 + t0S, t0 > 0. Then we have P;-i,i(a, Xo + t08) = Pt(a). From the result of the last paragraph this implies that P,(X0 + t0S) ^ Pi (a). It follows that (2,(0, Xo + t0 8) g 2{P ; (a) - P i _ 1 , 1 (a, A „ + U ) l = 0. On the other hand, we have, since P,_!(«) ^ P,_i(X0), Qi(a, Xo + t0 6) > QM, Xo) ^ 0. But this is a contradiction, and our statement is proved. Suppose therefore that P ( _i(a) g P,_i(X), Q;(Q;, X) > 0. Then P,(X) - ?,_,.,(«, X) > - P , ( « ) + P.-x.^a, X) ^ 0 or P,(«)

^P.-LI^X)


This obviously implies the statement of the lemma. To prove the theorem we utilize the integral formulas established in IFH. Suppose the first hypersurface to be denned by the mapping x: S0 —» S, where So is the unit hypersphere in the Euclidean space and x(g), £ e S0 , is the point of 2 whose outward unit normal vector is £. Similarly, the second hypersurface is defined by a mapping x'\ S0 —* 2'. From their second fundamental forms we construct the mixed scalar invariants P „ , 0 ^ r, s ^ n (cf. IFH). Then we have the following integral formula: (8)

f

{P„ - h'Pr,.-d

dV = 0,

l ^ s ^ ,

J So

where dV is the volume element of S0 and &'(?) is the support function of 2 ' . By a proper choice of the origin we can always suppose h'(£) > 0. From (8) we derive the formula (9) f

{(P,0 + Po, - 2P,_,.0 + (Poi ~ Pio) + 2A'(P,_ ll0 - Po.,-0} ^

= 0.

The hypotheses of the theorem give P,_i, 0 ^ Po,*-i , Poi ^ PJO • From these and the lemma it follows that P 0 i + Pio — %Pi-i.i ^ 0. Hence the integrand in (9) is ^ 0, and must vanish identically since it is continuous. I n particular, we have P 0 i = Pio for all p t 2. Our theorem then follows from the theorem of ALEXANDROFF-FENCHEL-JESSEN.

88

S. S. CHERN, J. HANO & C. C. HSIUNG

Corollary. Let 2 be a closed strictly convex hypersurface in Euclidean space. If there is a constant c such that (10)

PY-T\p)

^ c ^ PY\p),

peS,

then 2 is a hypersphere. Remark. The theorem can obviously be extended to a pair of convex hypersurfaces with boundaries. University of Chicago Chicago, Illinois and Lehigh University Bethlehem, Pennsylvania

R e p r i n t e d from I L L I N O I S J O T J B N A L O F M A T H E M A T I C S

Vol. 4, N o . 4, D e c e m b e r 1960

SOME UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY1 BY CHUAN-CHIH HSITJNG

1. Introduction Let Mn be a differentiable manifold of dimension n, and X: Mn —> En+m a mapping of M„ into a Euclidean space En+m of dimension n + m for any m > 0. Mn , or rather Mn together with the mapping X, is called an immersed submanifold of En+m if the functional matrix of X is of rank n everywhere. The submanifold Mn is said to be imbedded, if X is one-one, that is, if X(P) = X(Q), P, Q e Mn , implies that P = Q. In particular, when m = 1, an immersed (imbedded) submanifold Mn of the space En+m is called an immersed (imbedded) hypersurface. Throughout this paper all manifolds are supposed to be of class C3, and the dimension of a manifold Mn is understood to be n. Now let us consider an oriented immersed manifold Mn • Then to each point P e Mn there is a unique linear space N of dimension m normal to X(Mn) at the point X(P). For any unit normal vector er(P) at the point X(P) in the space N, we put (1.1)

I = dX-dX,

IIr = der-dX,

IIIr = der-der,

where dX and der are vector-valued linear differential forms on Mn , and the dot denotes the scalar product of two vectors in the space En+m . The eigenvalues kri, • • • , km of IIr relative to / are called the principal curvatures of the manifold Mn associated with the unit normal vector er(P). If the GaussKronecker curvature K, = kri • • • krn associated with the vector er(P) is nonzero, the reciprocals l/krl, • • • , l/krn , called the radii of principal curvatures associated with the vector er(P), are the eigenvalues of IIr relative to Illr, which is also positive definite due to the assumption Kr ^ 0. In this case we introduce the ath elementary symmetric function (1.2)

(I)

Pra =

E

V*rt ••• I / * "

(1 ^

« g

n).

If M„ is a hypersurface, then at each point X{P) of Mn there is only one unit normal vector eT, and for Pra associated with it we shall simply write Pa . Let Mn be a closed oriented Riemannian manifold immersed in a Euclidean space En+m . By a normal frame Xen+i • • • en+m on the manifold Mn we mean a point X of the manifold Mn and an ordered set of mutually perpendicular unit vectors en+i, • • • , e„ +m normal to the manifold Mn at the point Received August 13, 1959. 1 This research was partially supported by the United States Air Force Office of Scientific Research of the Air Research and Development Command. 526

UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS

527

X. Mn is called a star manifold,2 if there exist a point 0, called a pole, in the manifold M„ and a class C2 field of normal frames Xen+i • • • en+m over the manifold Mn such that the Gauss-Kronecker curvature Kr of the manifold Mn and the support function X • er with respect to the pole 0 are positive for every vector er, n + 1 g r ^ n + m, at every point of the manifold M„ . This normal frame Xen+i • • • en+m is called a fundamental normal frame of the star manifold Mn at the point X. An n-dimensional star manifold with boundary is an n-dimensional compact subset of an n-dimensional star manifold. An n-dimensional convex hypersurface with boundary is an n-dimensional compact subset of the boundary of a convex region in an (n + 1)dimensional Euclidean space En+1, or is equivalently an n-dimensional compact subset of an n-dimensional imbedded hypersurface with positive GaussKronecker curvature everywhere. An n-dimensional convex hypercap is an n-dimensional convex hypersurface with boundary such that in the space En+i there is at least one fixed direction, along which every line either is a tangent to the hypersurface or intersects the hypersurface at most at one point. It is obvious that a convex hypercap can never be closed. Since Christoffel [5] established in 1865 his well-known uniqueness or rigidity theorem on closed convex surfaces in a space Ez, various uniqueness theorems of the same type on closed convex hypersurfaces have been obtained by different authors with different methods. It is natural to ask whether we can extend some of these uniqueness theorems on closed convex hypersurfaces to general immersed manifolds with boundary. In recent years the present and other authors have succeeded in deriving some new integral formulas, by means of which most classical uniqueness theorems can easily be extended to convex hypersurfaces with boundary satisfying a natural boundary condition. For uniqueness theorems on general immersed manifolds with boundary, due to the complication arising from the immersion, the only result we have so far is the generalization [9] of Christoffel's uniqueness theorem to two-dimensional immersed manifolds with boundary. The main purpose of the present paper is to further extend this uniqueness theorem to immersed manifolds of a general dimension n > 2 with boundary, and to establish a uniqueness theorem on convex hypercaps by proving the following theorems. THEOREM I. Let Mn and Mn be two star manifolds, with boundaries 5„_i and Bn_i respectively, in a Euclidean space E„+m for any m > 0. Suppose that there exists an orientation-preserving diffeomorphism f of the manifold Mn onto the manifold Mn such that, at each pair of corresponding points, the manifolds Mn and Mn have a common fundamental normal frame en+\ • • • en+m and equal PT,n-i defined by equation (1.2) and associated with each common unit normal vector er, r = n + 1, • • • , n + m. If the diffeomorphism f restricted to the boundary Bn_1 is a translation {strictly speaking, is induced by a transla2

The author is indebted to the referee for his comment which leads to the definition of a star manifold in the present form.

528

CHUAN-CHIH HSIUNG

lotion in the space En+m) carrying the boundary Bn_\ onto the boundary B*_i, then the diffeomorphism f is a translation carrying the whole manifold Mn onto the whole manifold M„ . THEOREM II. Let M„ be a star manifold with a spherical boundary Bn-i such that at every point Pr,re_2 P',n-i is constant for n -\- v > 0, M = 0, v > 0 and for each vector er of a fundamental normal frame of the manifold Mn . Then the manifold Mn is a compact subset of an n-sphere. THEOREM III. Let Mn and Mn be two oriented convex hypercaps with boundaries Bn-i and i3„_i respectively. Suppose that there exists an orientationpreserving diffeomorphism f of the hypercap Mn onto the hypercap M* such that at each pair of corresponding points the hypercaps Mn and M* have the same outer normal vector and satisfy either

(1.3)

Pj ^ Pt,

ft

k

Pt,

P2 g

Pt,

or (1.4)

Pi 5; Pt,

where Pa and Pa are defined by equation (1.2) for the hypercaps Mn and Mt respectively. If the diffeomorphism f restricted to the boundary B„_i is a translation carrying the boundary 5„_i onto the boundary Bn-i, then the diffeomorphism f is a translation carrying the whole hypercap Mn onto the whole hypercap Mt • COROLLARY. Let Mn be a convex hypercap with a spherical boundary Bn_i . If there is a constant c such that, at each point of the hypercap Mn , either

(1.5)

Pi ^ c g Pi12

or (1.6)

Pi 2: c ^ Pi12,

then the hypercap Mn is a compact subset of an n-dimensional hypersphere. It should be noted that when n = 2, the conditions (1.3) and (1.4) together are obviously weaker than the condition of Alexandroff [2], which can be stated as follows: At each pair of corresponding points the hypercaps M2 and Mt satisfy the condition P(2P X , P 2 ) = F(2P?, P * ) , where F(U, V), for U > 0, U2 ^ 4 F > 0, is a continuous function monotonely increasing in both variables U and V. Furthermore, Grotemeyer [8] obtained Theorem III for n = 2 in terms of the condition of Alexandroff, but actually only used the conditions (1.3) and (1.4) together in his proof. 2. Immersed submanifolds in Euclidean space Suppose a Euclidean space En+m is oriented. By a frame Xei • • • en+m in the space En+m we mean a point X and an ordered set of mutually perpendicular unit vectors ei, • • • , en+m with an orientation coherent with that of

UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS

529

the space En+m so that the determinant | ei, • • • , en+m | is equal to + 1 . To avoid confusion we shall use the following ranges of indices throughout this paper: 1 ^ a, |3, 7 ^ n,

(2.1)

n + 1 ^ r, s, t ^ n + m,

1 ?g i,j, k ^ n + m.

Then we have (2.2)

ei-ej

= 5*y ,

where S,y are the Kronecker deltas. Let F(n, m) be the space of all frames in the space En+m, so that dim F(n, m) = \(n + m)(n + m + 1). In F(n, m) we introduce the linear differential forms co* , w,-,- by the equations (2.3)

dX = X); "i ei i

^e; = X)y «»> ei

J

where (2.4)

co,y + «,-,• = 0.

Since d(dX)

= 0 and d(dei) — 0, from equations (2.3) we have

(2.5)

don = zlj W; A can ,

dom = ilk u>ik A to*,-,

where A denotes the exterior product. As explained in §1, by an immersed submanifold in the space En+m we mean an abstract manifold Mn and a mapping X: M„ —> En+m such that the induced mapping X* on the tangent space is univalent everywhere. Analytically, the mapping can be defined by a vector-valued function X(P), PeM„. Our assumption implies that the differential dX(P) of X(P), which is a linear differential form on Mn with value in En+m , has as values a linear combination of n, but not less than n, vectors U , • • • , tn . Since X* is univalent, we can identify the tangent space of M„ at the point P with the vector space spanned by t\, • • • , tn . A linear combination of the vectors i i , • • • , t„ is called a tangent vector, and a vector perpendicular to them is called a normal vector. The immersion of Mn in En+m gives rise to a bundle B, whose bundle space is the subset of Mn X F(n, m) consisting of (P, X(P)ei

• • • en+i • • • en+m) e Mn X F(n, m)

such that ei, • • • , e„ are tangent vectors and tors at the point X(P). Consider the inclusion mapping 0 and the projection p: (2-6)

B —£-> Mn X F(n, m) - ^ U F{n, m).

By putting (2.7)

CO; =

(p4>)*w'i ,

Olij =

(p0)*co'-y ,

normal vec-

530

CHUAN-CHIH HSIUNG

from equations (2.4) and (2.5) we have (2.8) (2-9)

con + wji = 0, dwi = 2-,j uj A (Jin ,

don, = z^k uik A ukj •

From the definition of the bundle B it follows that cor = 0 and that coa are linearly independent. Thus the first equation of (2.9) gives /,<*. W» A Olar =

0,

from which we have (2.10)

C0„r = 2^9 Arap <<>3/ ,

Arap = Arfla .

If det (Arafi) 7^ 0 for some r, by introducing the matrix (\rap) inverse to the matrix —(Ara$) we have (2.11)

C0„ = 2_,p Xra0 C0r|3 .

B y means of equations (2.2), (2.3), (2.7), (2.10), a n d (2.11), equations (1.1) can be written as L

=

2—ia 1 . ,

lllr

= 2-ii Uri t

(2-12) Suppose (2.13)

det (Sap + \rap y) = Zo S 7S» (")

Pn(X)y\

where y is a parameter. Then Pry{\r) is a polynomial of degree 7 in Xr„0 for a fixed r, and it is easily seen that P ra (X r ) is equal to the invariant Pra defined by equation (1.2). Through a point in a Euclidean S p a c e Min+m l e t A.\ , • * • , .An+m—i be n + m — 1 differentiable vector functions of n variables x1, • • • , a;71, and let J be any vector. Then the scalar product of the vector J and the vector product Ai X • • • X An+m-i of the vectors A\, • • • , -4„+m_i is given by (2.14)

J-(AXX

••• X i » + , - i ) = ( - l ) " + m - V » ^ i . •••

,An+m-i\,

from which it follows that (2.15)

ex X • • • X eT X • • • X en+m = ( - 1 )

n+m+r

er,

where the circumflex over er indicates that the vector er is to be deleted. In a previous paper of the author [10] we have combined the vector product of vectors and the exterior product of differentials to define the vector Ai ® ••• ® Ai-i ® dAi ® • • • ® (2.16)

dAn+m-!

= (At X • • • X Ai-i X Ai.at X • • • X -4»+—!.»,+„_!) dx" A • • • A ete"n+m-1,

UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS

531

where i = 1, • • • , n + m — 1 and A(,ai = dAi/dxai. It is obvious that the vector (2.16) is independent of the order of the vectors dAt, • • • , dAn+m-i. Let dA be the area element of an immersed submanifold Mn in the space En+m . Then by means of the combined operation ® we obtain dX ® • • • ® dX ® en+i ® • • • ® er ® • •• ® en+m (2.17)

'

n =

{-l)n+m+rn\erdA,

der ® • • • ® der ® en+i ® • • • ® er ® • • • ® en+m (2.18)

'

„

' =

(-iy+m+rn\erKrdA.

From equations (2.3), (2.7), (2.15), (2.17), and (2.18) it follows that (2.19)

dA = wi A • • • A wn ,

(2.20)

Kr dA = wrl A • • • A cora . 3. Integral formulas for a pair of immersed manifolds with boundary

Let M be a compact differentiable manifold of dimension n with boundary, and let Mn and M* be immersed manifolds with boundaries _B„_i and 5*_i given by X:M —> En+m and X*:M —> EnJrm , respectively. Then §2 can be applied to the manifolds Mn , and for the corresponding quantities and equations for the manifold Mn we shall use the same symbols and numbers with a star respectively. Suppose that there is a diffeomorphism / of the manifold Mn onto the manifold Mn such that at each pair of corresponding points the manifolds Mn and Mn have parallel tangent spaces. Without loss of generality we may assume that (3.1)

e* = d

(i = 1, • • • , n + m).

From equations (2.3), (2.7), (2.3)*, (2.7)*, and (3.1) it follows that (3.2)

0) r „ = COra •

Now for the pair of immersed manifolds Mn and Mn we introduce the following differential forms: R

v

_ "VVn+OT i

' -\x,x*

i v—1 der,dX

, ••• ,dX ,dX*, •••

,dX*\,

532

CHUAN-CHIH

Cr/3,n—1-0 =

(3.4)

HSIUNG

( — 1)

• I X, en+i, • • • , 4 , • • • , en+m , der, dX, • • • , dX, dX*, • • • , dX* I , P r0,»-l-0 =

(3.5)

n - 1 - /3

(. — 1 ;

• | X * , e n + 1 , ••• ,ir, ••• ,en+m,der,dX,---

,dX,dX*,--y

,dX*\, v

P

'

n- 1- p

-Dr/3,j>-l-/3 = ( — 1 ) ™

(3.6)

-1e r ,e n + i, • • • , er_i, e r + i, • • • ,en+m ,der,dX, • • • ,dX,dX*, • • • ,dX*\, -— ~ v v P n - 1- p where 0 ^ a ^ n — 2 and 0 | |3 g n - 1. By means of equation (2.14) and the operation ® we obtain

cr(J,n_1H) = i-iy+'-'x-E,

c%,n-t^ =

(3.7)

i-iy^-'x*^,

n+r _ x

•Dr/S,n-l-0 =

( —1)

r

er-2?,

where E = en+i ® • • • ® S, ® • • • ® en+m ® deT ® dX ® ••• ® dX ®
7 1 - 1 - / 3

From the definition of the operation ® and the last equation of (3.7) it follows immediately that (-l)n+r-^ = Z W ^ e r .

(3.8)

Thus the substitution of equation (3.8) in the first two equations of (3.7) gives (3.9)

CrjS.re—1-3 =

hrDrp,n—1-0

,

Cr0,»—1—0 =

/jr-Dr0,n—i_0 ,

where we have placed (3.10)

fcr

= X-er,

A? = X*-e r .

By using equations (3.3), (3.4), (3.5), and (3.9), applying the ordinary rules for differentiation of determinants, and noticing the pairwise cancellation of terms, we can easily obtain (^•t>a,n—2—a (o.ll)

=

^ ^ r = n + l \,(->ra,n—1— a

=

2—lT=n+l (fir JJroc,n—1— a

^ r,a+l,n—2— a) " r - L ' r . a + l ,n—2— a )

(0 ^ a ^ n - 2). Integrating both sides of equation (3.11) over the manifold Mn and applying Stokes's theorem to the left side, we can arrive at the integral formulas

UNIQUENESS THEOREMS ON EIEMANNIAN MANIFOLDS

/Q irt\

/

£>cL,n—2—a ~

/

/

'

\ " r ^ra,n—1—a

533

"T -LSr,a+l,n—2—a/

(0 ^ a ^ n - 2). To apply the formulas (3.12) we introduce the differential forms of order n Dra0=(-l)m+r

(3.13)

* \ er, en+i, * • • , e r _i, e r + i, * • •, e n+m , ae r , * • •, der, dX, • • •, aX, aX , • • • , dX \, '

s S^

- ^

.. V

n - (a + |8)

K.

V.

a

-*

*-

/3

'

( 0 ^ « , ^ n). In virtue of equations (2.3), (2.7), (2.8), (2.11), (2.14), (2.15), (2.20), (2.3)*, (2.7)*, (2.11)*, (3.2), and (3.13) we can easily obtain, for two parameters y and y*, y y* Drali

a^a+fiin al fil (n — a — j3)! '

= (-l)^1"-" (3.14)

X)

^...^(yco^ + y*u*ai + Wrai)

1S.1.-.-.S.

A • • • A (j/eoa„ + y*« a „ + « W ( +1>(ro_1)

= ( —l) "

w! (t/wi + y*w* + wri) A • • • A (yw„ + j/*w* + &>„>)

= (-l) ( " + 1 , C m - 1 ) n! det (yX^ + j/*Xr*a
(3.15)

Draii = {-lYn+mm-l)n\PTafKrdA

(3.16)

(0 ^ a, 0 £ n ) .

Substituting equation (3.16) in equation (3.12) we therefore obtain J

(3.17)

/ £>a,n—Z—a B„-1

= ( _ 1)(n+1)(m _ 1)n ,

J

" £ (hrPra,n+a

- h*r Pr,a+1,n+a)Kr

dA,

(0 S a g n - 2 ) . If the diffeomorphism / of the manifold Mn onto the manifold Mn restricted to the boundary Z?„_i is a translation carrying the boundary B„_i onto the boundary 5 „ _ i , then on the boundary £„_!, dX* = dX, and therefore

534

CHUAN-CHIH

HSIUNG

Bo,n-2 = 5 n _ 2 ,o. From the two equations of (3.17), for which a = 0, n — 2 respectively, we can easily obtain the integral formula £

*•

n+m

I

2—1 "rvPrO.n—1

PT,n—2,1

(3.18) /»

=

tt+TTC

/ 2—1 IAr (Prl.n—2 — P r,n—l,o) •
— " r ( P r ,n—2,1 — Pr0,n—l)

J-K^r

flA.

Addition of equation (3.18) to the one obtained by interchanging the roles of the two manifolds Mn and Mn in equation (3.18) thus gives *• (3.19)

n+m

/ E [hr(Pr0,n-l •"Afn r - n + 1

-

Pr,n-2,l)

+

tf(P,.*-1.0

~

Prt.«-«)]Xr dA

=

0.

Let Fr(ui, • • • , w„), r = n + 1, • • • , n + m, be m functions in n positive variables Wi, • • • , un . We shall say that each function Fr is of type n — 1, if the following two conditions for each r are satisfied: (i) Fr(Prl0, • • • , Prn0) = Fr(Pm , • • • , Pron) implies that Pr,n_2,i ^ Pro,n-i, and therefore, by interchanging the two manifolds Mn and Mn , that Pri,„_2 ^ P r?n _i,o, (ll)

-

Pr(PrlO ,

•• ,

Prno)

=

Fr(Pr01

, ••• ,

PrOn)

and

Pr,ji-2,1 =

Pr0,n-1

(or Pri,B_2 = P,,„_i,o) if and only if Xra^ = Xr„^ for a, /3 = 1, • • • , n. THEOREM 3.1. Le< Af„ and Mn be two star manifolds, with boundaries Bn-i and Bn_i respectively, in a Euclidean space En+m for any m > 0, and let Fr(ui, • • • , un), r = n + 1, • • • , n + m, be m functions of type n — 1 in n positive variables Suppose that there exists an orientationpreserving diffeomorphism f of the manifold Mn onto • the manifold Mn such that, at each pair of corresponding points, the manifolds Mn and Mn have a common fundamental normal frame en+i • • • en+m, and the functions P r (Prio, • • • , Prno) and Fr(PM , • • • , Pr0n) have the same value for each r. If the diffeomorphism f restricted to the boundary B n -i is a translation carrying the boundary P„_i onto the boundary B„_i, then the diffeomorphism f is a translation carrying the whole manifold Mn onto the whole manifold Mn .

Proof. Applying a translation in the space En+m if necessary, without loss of generality we may assume the poles in the star manifolds Mn and M„ to be coincident, so that hr > 0 and hr > 0 ior r = n + 1, • • • , n + m over the whole manifolds Mn and M„ . Thus, due to the first property of the functions Fr and the assumption that Kr > 0, each term of the integrand of equation (3.19) is nonpositive, and equation (3.19) holds when and only when PrO.n—1

=

Pr,n—2,1 ,

Pr,n—1,0

=

Prl.n—2

(3.20)

(r = n + 1, • • • , n + m).

By the second property of the functions Fr we therefore obtain (3.21)

A*ap = A™/?

(a, P = ! , - • • , n).

UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS

535

Substitution of equation (3.21) in equations (2.11) and (2.11)* gives immediately (3.22)

u* = ua

(a = 1, • • • , n).

From equations (3.22), (2.3), (2.7), (2.3)*, and (2.7)* it follows that dX* = dX over the whole manifold Mn , and hence the proof of the theorem is complete. In particular, if the second manifold Mn in Theorem 3.1 is a compact subset of an n-sphere, then Theorem 3.1 becomes THEOREM 3.2. Let Mn be a star manifold with a spherical boundary BM_i in a Euclidean space En+m for any m > 0. If there are m functions

Fr(ui,

• • • , un),

r = n + 1, • • • , n + m,

in n positive variables ux, • • • , un with the following two properties for each vector er of a fundamental normal frame at every point of the manifold Mn : (i) Fr(Pri, • • • , Pm) = Fr(a, • • • , a") = constant implies that Pr,n—i = a

,

Fr(Pri, • • • , Prn) = Fr(a, • • • , an) and Pr,n-2 = an~ \r«p = a$ap for a, /3 = 1, • • • , n, then the manifold Mn is a compact subset of an n-sphere of radius a. (ii)

imply that

For m = 1, the integral formulas (3.12), (3.17), (3.18) and Theorems I, II, 3.1, 3.2 were obtained by Chern [3]. 4 . Proofs of Theorems I and II Proof of Theorem I. Theorem I follows from Theorem 3.1 immediately if we can show that the m functions Fr = Pr,n-i, r = n -\- 1, • • • , n -\- m, are of type n — 1. To this end we need the following inequality of Carding [7]: Let Pr.n-iiX^, • • • , Xr" x>) be the completely polarized form of the polynomial P r , n _i(X r ) defined by equation (2.13), so that P r,n—l(,Xr, "

" ' ' , \r) v

ri — 1

= rr,n—\\Ar), '

rr%n—l\Kr, v

• • • , Xr , X r J = v

Pr,n—2,1-

'

n —2

Then for positive definite symmetric matrices (\l%), • • • , (\lrZj1)) the following inequality is valid: (4.1)

Pr^iQi?,

••• , X'"- 1 ') £ Pr„_J(A<1))1'<"~1) " • • P r ^ A i - 1 ' ) 1 " - " ,

where the equality holds when and only when the n — 1 matrices are pairwise proportional. Suppose now Pr,n-i,o = Pro,n-i, which can be written as Pr,„_i(Xr) =

536

CHUAN-CHIH HSIUNG

P r ,n-i(Xr). Since (\rafi) and (Xr«^) are positive definite, from the inequality (4.1) it follows that Pr,n—2,1

=

" r , « - l ( X r , ' ' ' , Xr , X r ) i

>

(4.2)

n- 2

a= P r .^. 1 (x r ) ( " H ' )/< - 1) P r> »_ 1 (xr) 1/( - 1> = P r ,„_i(x?), which is the first condition for the functions Pr,n-i to be of type n — 1. By interchanging the two manifolds M n and Mn we have (4.3)

P rl ,„_ 2 £: P r i 1 ^(X,).

The equality holds in (4.2) and (4.3) when and only when Xra0 = pXra0 for a, |8 = 1, • • • , n. On the other hand, as in the proof of Theorem 3.1, by using equations (4.2), (4.3), and (3.19) it is easily seen that the equality holds in (4.2) and (4.3). Since Pr,„_i(Xr) = P r ,„_i(X*), p = 1, and therefore the second condition for the functions Pr,n-i to be of type n — 1 is satisfied. Proof of Theorem II. By putting Ar<*0 =

• • • — Ara/3

=

Arcr/3 ,

Arap

=

CLOap

{a, p =

1, • • • , 71),

from inequality (4.1) we obtain (4.4)

p ^

2 )

^ pj^r",

where the equality holds when and only when Xra(j = 65„# for a, ft = 1, • • • , n. Let a > 0 be defined by IA r \ ^4.0;

pe p» *>,n-2 "r,n-l

_ _«(n—2)+»(n—1) — CL ,

X>,n-1 — O,

rr,n-:

so that (4.0)

From inequality (4.4) and equation (4.6) it follows that /,

m\

(,4./J

p

-»

p(n-2)/(n-l)

i > , n - 2 £S * r , i i - l

n (n-2KW.)(»-2)/(«-l)+l)n-(n/i)(>-2)/{»-l)

= «

/>,n-2

,

which implies Pr,n_2 2: a" -2 , where the equality holds when and only when it holds in (4.4). Thus, if Pr,n-2 = an~2, then Xr«^ = bda$ for a, /? = 1, • • • , n, and therefore 6 = a in consequence of equation (4.5). Hence Theorem I I follows from Theorem 3.2 by taking P r ( P r l , • • • , P r ») = P?,n-2 P',n-i •

5. Integral formulas for convex hypercaps Let I f be a compact differentiable manifold of dimension n with boundary; and let Mn be a convex hypercap with boundary 5„_i, so that M„ is an imbedded manifold given by X:M —» En+1 with positive Gauss-Kronecker curvature Kn+i everywhere. Then §2 with m = 1 can be applied. In the space En+i, let £ be a fixed direction along which every line either

UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS

537

is a tangent to the hypercap Mn or intersects the hypercap M„ at most at one point, so that by the definition of a convex hypercap we have (5.1)

r = £-e„+1 ^ 0.

For the hypercap M„ we introduce the following differential forms: Aa = I £, X, den+\, • • •, den+1, dX, • • •, dX I 71 — 1 — a

(5.2)

(0 ^ a ^ w — 1),

a ( 0 g ^

Dp = I £, de n + i, • • •, de„+i, dX, • • •, dX | n - /3

n).

j3

As in §3, by using exterior differentiation and Stokes's theorem we obtain (5.3)

f

Da+1 = f

Aa

(0 g a g n - 1).

Now let M n be another convex hypercap given by the imbedding X*: M —> 2? n + 1 , and suppose that there is a diffeomorphism / of the hypercap Mn onto the hypercap Mn such that at each pair of corresponding points the hypercaps Mn and Mn have the same unit normal vector cn+i • For this pair of hypercaps Mn and Mn we introduce the following differential forms: AaB = £, X, de n + i, • • •, de„+i, dX, • • • , dX, dX*, • • •, dX* |, v

^

v

^

n - 1 - (a + j9) A *

j

«

J

a

0

£, X*, de n + i, • • •, <2e„+1, dX, • • •, dX, dX*, •••, dX* L N^

"

« — 1 — (a + /3)

Y

a

,3

Bap = X, X*, c?en+i, • • •, den+i, dX, • • •, dX, dX*, • • •, dX* I, n — 1 — (a + j8)

(5.4) ^ pa

—

p

o-

X*, cfen+1, • • •, den+1, dX, • • • , dX, dX*, • • •, dX* I, V

v

n — (p + a-)

z>„

y

p


£, de n + i, • • • , den+i, dX, • • • , dX, dX*, • • •, dX* I, rc. — (p +


p

where 0 ^ a, 0 ^ n — 1 and 0 i£ p, o- ^ n. (5.5)

TC, = W ) „ ,


As in §3, it is easily seen that

T ( £ = h*D„.

where we have placed (5.6)

^

X, c?en+i, • • •, rfe„+i, dX, • • •, dX, dX *,---,dX*\ n — (p + a-)

p<7

a

h = X-en+i,

h* =

X*-en+1.

(0 ^ p, o- ^ n ) ,

538

CHXJAN-CHIH

HSIUNG

Exterior differentiation gives (5-7)

dAap = Da+i,p, dAa& = Da#+\ , * = Cafi+i - Ca+ifi = (l/r)(hDa,$+1 - h*Da+lfi),

dBal>

if r * 0.

Integrating both sides of each of equations (5.7) over the hypercap Mn and applying Stokes's theorem to the left side we have /

J

Aap — I

Bn_i

(5.8)

f

Da+is,

Baf=

[

J

J

->Mn

Aaff — I

Bn_l

(l/T)(hDa,e+1

Da,p+i,

JMn

- h*Da+lif!),

if r * 0.

Essentially the same argument as that used in deriving equation (3.16) shows that Da,, = (-l)nnlrPa?Kn+1dA

(5.9)

(0 £ «, 0 £ n),

where Pap are defined, in terms of two parameters y and y*, by d e t (y\n+l,a$

+ y Xn+l.o/S +

5«jj)

(5.10)

yn! ,.«,.*0p a a? O ^ + J S . « ! / S ! ( n - a - P)lVy "

We shall also write Pao = P a (X B +i) and Poa — Pa(\n+i). Substituting equations (5.9) in equations (5.8) we thus arrive at the integral formulas [

J

A«3= (-1)"»! j

Bn_i

(5.11)

rPa+^Kn+1dA,

JMn

f

Aaf = ( - l ) " n ! f

[

Bap = ( - l ) " n ! f

TPa,?+1Kn+1dA, (l/r)(feP„. w .i -

h*Pa+1,p)Kn if

T

^ 0,

where 0 ^ a, /3 ^ ra — 1. 6. Proof of Theorem III From equations (5.11) we can easily deduce different integral formulas, but for proving Theorem III we need only the following one: / (6.1)

B

(A10 - An + A*i - A?0) -1

. = (-l)"n! /

r(P 2 0 + P02 - 2Pn)Kn+1

dA.

From the assumption of Theorem III that the given diffeomorphism / re-

UNIQUENESS THEOREMS ON RIEMANNIAN MANIFOLDS

539

stricted to the boundary Bn-i is a translation carrying the boundary Bn-i onto the boundary S „ _ i , it follows that over the boundary Bn-i, dX* = dX, and therefore Aw — Am. + A$i — Aio = 0. Thus the integral formula (6.1) is reduced to (6.2)

f

r(Pi0 + P02 - 2Pn)Kn+1

dA = 0.

On the other hand, in a recent paper [4] we have established the LEMMA. If p. = ( M ^ ) and n* = ( M ^ ) are two positive definite symmetric matrices of order n such that for a fixed y, 2 g y ^ n, (6.3)

Py^dx)

=g P 7 -l(M*),

Py(li) ^ P,(/**),

then (6.4)

Qy(», /**) = P 7 ( M ) +

w/iere
P7(M*)

- 2P,_ 1 , 1 ( M , „*) ^ 0,

= P7(ju*).

At each pair of corresponding points of the hypercaps Mn and M* under the diffeomorphism /, we take the common unit outer normal vector to be the vector e„+i ; so that the matrices (\n+i,ap) and (X n+ i, a|3 ) are positive definite everywhere, and the conditions (1.3) or (1.4) are equivalent to the conditions (6.3) with 7 = 2. Since Q2(M, M*) is symmetric with respect to the matrices yu and ju*> the above lemma gives (6.5)

P2(XK+1) + P 2 ( \ : + 1 ) - 2P U (X„ + 1 , X*+i) S 0,

where the equality implies that P2(X„+i) = P 2 (X„+i). From the inequalities (5.1), (6.5), and the assumption that Kn+i > 0, it follows immediately that the integrand of equation (6.2) is nonpositive, and equation (6.2) holds when and only when the equality holds in (6.5). Thus P2(Xre+l) = P 2 (X„ + l), and hence Theorem III follows from the uniqueness theorem of AlexandroffFenchel-Jessen for convex hypersurfaces with boundary. 3 REFEBBNCES

1. A. ALEXANDEOFF, Zur Theorie der gemischten Volumina von konvexen Kdrpern. II, M a t . Sbornik (N.S.), vol. 2 (1937), p p . 1205-1238 (Russian with German summary). 2. , Sur les Moremes d'uniciU pour les surfaces fermies, C. R. (Doklady) Acad. Sci. URSS, vol. 22 (1939), p p . 99-102. 3. S. S. C H E R N , Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, J. M a t h . Mech., vol. 8 (1959), p p . 947-955. 4. S. S. CHERN, J . H A N O AND C. C. H S I U N G , A uniqueness theorem on closed convex hypersurfaces in Euclidean space, J. M a t h . Mech., vol. 9 (1960), p p . 85-88. 3

For t h e uniqueness theorem of Alexandroff-Fenchel-Jessen, see [1], [6] for closed convex hypersurfaces, and [3] for convex hypersurfaces with boundary.

540

CHUAN-CHIH HSIUING

5. E . B . CHBISTOPFEL, Ueber die Bestimmung der Gestalt einer krummen Oberfldche durch lokale Messungen auf derselben, J . Reine Angew. M a t h . , vol. 64 (1865), pp. 193-209. 6. W. F E N C H E L AND B . J E S S E N , Mengenfunktionen und konvexe Korper, Danske Vid. Selsk. M a t h . - F y s . M e d d . , vol. 16, no. 3, 1938. 7. L. GABDING, An inequality for hyperbolic polynomials, J. M a t h . Mech., vol. 8 (1959), pp. 957-965. 8. K . P . GBOTEMEYEB, Zur Flachentheorie im Grossen: Tiber die Abbildung durch parotide Normalen, Arch. M a t h . , vol. 9 (1958), p p . 117-122. 9. C. C. H S I U N G , A uniqueness theorem on two-dimensional Riemannian manifolds with boundary, Michigan M a t h . J., vol. 5 (1958), p p . 25-30. 10. , Curvature and Betti numbers of compact Riemannian manifolds with boundary, Univ. e Politec. Torino, Rend. Sem. M a t . , vol. 17 (1957-1958), p p . 95-131. LEHIGH UNIVEBSITY BETHLEHEM, PENNSYLVANIA

Vlo. 73, No. 1, January, 1961

ISOPERIMETIC INEQUALITIES FOR TWO-DIMENSIONAL RIEMANNIAN MANIFOLDS WITH BOUNDARY BY CHUAN CHIH-HSIUNG

(Received February 8, 1960) (Revised May 31, 1960)

1. Introduction For a simple closed rectifiable plane curve C of length L and enclosed area A the isoperimetric inequality states that L2 - 4TTA ^ 0 ,

(1.1)

where the equality holds if and only if C is a circle. By using different methods many authors have given different proofs for this inequality (1.1), and extended the inequality to surfaces in ordinary euclidean space E3 under different conditions on the gaussian and mean curvatures of the surfaces (for this see [3], [1], [5], [10], [9], [11]). The purpose of this paper is to further extend the inequality to the most general two-dimensional riemannian manifolds with boundary, which can be imbedded in a euclidean space EN+2 of dimension N + 2 for any N > 0. The results are contained in Theorems 4.1 and 4.2. 2. Preliminaries Let M2 be an oriented two-dimensional riemannian manifold of class C with a boundary C imbedded in an oriented euclidean space EN+2 of dimension N + 2 for any N > 0. To avoid confusion, throughout this paper we shall use the following ranges of indices unless stated otherwise: 2

(2.1)

a, /9 = 1, 2 ;

3 ^ r g iV + 2 ;

l ^ i , j , k^N+2

.

By a frame Pet • • • eN+2 in the space EN+2 we mean a point P and an ordered set of mutually perpendicular unit vectors e^ • • •, e^+2 with an orientation coherent with that of the space EN+2 so that the determinant [ e,, • • •, eN+21 is equal to + 1. Then we have (2.2)

e4 • e} = St} ,

where the dot denotes the scalar product of two vectors in the space EN+2, and Sl} are the Kronecker deltas. Let X denote the position vector of the point P with respect to a fixed point 0 in the space EN+2, then we can write (2.3)

dX = I^co 4 e 4 ,

det = I ^ a ^ e . , ,

where o>4 and (ol} are pfaffian forms on the manifold of all the frames in 213

214

CHUAN CHIH-HSIUNG

the space EN+2, and satisfy (2.4)

a)i} + o)H = 0 .

To study the manifold M2, we consider the submanifold of the manifold of all the frames Pe^ • 'eN+2 such that P e M2 and eu e2 are two tangent vectors of the manifold M2 at the point P. Denoting by the same symbols the forms on this submanifold of frames induced by the identity mapping, we have (2.5) cor = 0 . Let A lt • • •, Ajy+j, I be AT+2 vectors in the space EN+2, and Ax x • • • x A^ +1 denote the vector product of the N + 1 vectors Alt • • •, A^+1. Then we have (2.6)

1 - ^ x . - . x AN+1) = ( - ir+111,

Alf • • -, A ™ | ,

from which it follows that (2.7)

e, x • • • x e r x • • • x e^+2 = ( - l)*+'e r ,

where the circumflex ~ over e,. indicates that the vector er is to be deleted. Let dA be the element of area of the manifold M2 at a point P, and Hr the mean curvature of the manifold M2 associated with the normal vector er at the point P. By means of the combined operation (g) of the vector product x and the exterior product A we then obtain (2.8) (2.9)

dX (g) dX (g) e3 • • •
Making use of equations (2.3), (2.5), (2.7), (2.8), (2.9) a simple calculation suffices to demonstrate that (2.10) dA = (o1A(o2, (2.11) 2HrdA = Q)r2 A (o1 — o)rl A o)2 . Let X0 be the position vector of an arbitrarily fixed point on the boundary C with respect to the origin 0 in the space EN+2, and suppose that (2.12)

Xx = X - X„ .

By using equations (2.6), (2.8), (2.9) and applying the ordinary rules for differentiation of determinants, we can obtain the differential form die,, • • • f e* + „X 1 ,dX 1 | = ( - 1)*+V(e,<8> • • • e*+IdX1) (2.13) - X 2 -E r e 3 • • • er_x de r • • • S= ^ iTrer is the mean curvature vector of the manifold M2 at

ISOPERIMETRIC INEQUALITIES

215

the point P. It is well known that this vector £> is independent of the choice of the normal frame Pe3- • -e^^. Intergrating both sides of equation (2.13) over the manifold M2 and applying Green's theorem to the left side, we then arrive at the integral formula (2.14)

( | e,, • • •, eN+2, X1dX1\ = 2A + 2\\ JO

X,-^dA

,

JJM2

which is essentially the same as a special case of an integral formula obtained by the author in a previous paper [8]. 3. Lemmas This section contains two lemmas, which will be needed for the proofs of our theorems. LEMMA 3.1. Let Alt • • • AN be N mutually perpendicular B any other vector in the space EN+2. Then (3-1)

vectors and

| Ax x • • • x A^ x B |2 = | A, |2 • • • | AN H B f - £f =1 (A 4 -B)' | A l | 2 ••• | As|2 ••• | AN\> ,

where the magnitude of any vector A in the space EN+2 is denoted by | A |. Lemma 3.1 is a generalization of the well-known Lagrange's identity for the case in which N=l; and can easily be proved by taking the vectors Aj, • • •, AN to be along N coordinate vectors such that all, except the ith, components of the vector A4 (i = 1, • • •, N) are zero, since both scalar and vector products are invariant under a proper motion in the space EN+i. LEMMA 3.2. If L > 0 and 7] (s) is a real-valued absolutely continuous function on the closed interval [0, L] satisfying 39(0) = 0 = f]{L), and with 7)'{s) of integrable square on the interval [0, L], then

(3.2)

\[(r - f>* ^ 0 ,

where the accent denotes the differentiation with respect to s, and the equality holds if and only if 7](s) is of the form c sin (TTS/L) with an arbitrary constant c. Lemma 3.2, which is obviously equivalent to Theorem 257 of [6, p. 185], is the extremal property of the smallest proper value of boundary problem 7]" + /JO] = 0 ,

39(0) = 0 ,

39(L) = 0 .

It is a direct consequence (for this see, for instance, [11, p. 898]) of the readily verified identity

CHUAN CHIH-HSIUNG

216

and the fact (see [6, Theorem 222, p. 164]) that under the hypotheses of the lemma we have rj(s) = o[s1/2] as s —* 0 + and rj(s) = o[(L — s)1/2] as s —> L~. The identity (3.3) is essentially the well-known Jacobi form of the Legendre transformation for the second variation of a nonparametric simple integral variational problem (see, for instance, [2, p. 64]). 4. Theorems THEOREM 4.1. Let M2 be an oriented two-dimensional riemannian manifold of class C2 with a non-empty connected boundary C imbedded in a euclidean space EN+2 of dimension N + 2 for any N > 0. Let L be the length of the boundary C, s (0 ^ s fS, L) the arc length measured along the boundary C in the positive sense induced by the orientation of the manifold M2, and X(s) the position vector of the point with arc length s on the boundary C with respect to a fixed point 0 in the space EN+2. Suppose that X0 = X(0) = X(L), and that other notations are the same as in § 2. Then

(4.1)

U - 4TTA - 4x[[

X^&dA^O.

Moreover, the equality holds in (4.1) if and only if (4.2) Ma)-er(8) = 0 (r = 3, • • -, N + 2) , (4.3)

Xt(s)

s f(s) sin (TZS/L) ,

where 0 ^ s :£ L, and f(s) is a solution of the vector differential (4.4)

f (») = j . [e, x • • • x eN+2 x f(s)]

equation

(0 ^ s <£ L) .

In particular, if the equality holds in (4.1) for a manifold M2 with a boundary C along which e3, • • •, e^+2 are constant, then C is a circle of radius

LI(2K).

For the case in which N = 1, Theorem 4.1 is due to Reid [11]. PROOF. Since by the assumptions of the theorem Xx(0) = 0 = XX(L), X[(s) == X'(s) and | X'(s) |2 = 1 almost everywhere on the closed interval [0, L], it follows from equations (2.14), (2.6) that the left side of the inequality (4.1) is equal to J = L\^\ X[{S) \*ds - ^

o

| e., • • •, e„+2, Xu dX11}

(4.5) = L J ' { | XI(s) I2 —|Lx;(s).[e 3 x . . . x eN+2 x X1(s)]jds .

179

ISOPERIMETRIC INEQUALITIES

217

Let f4(s) (i = 1, • • •, N + 2) be real solutions of the vector differential equation (4.4) such that the (N + 2) x (N + 2) matrix (4.6)

l|fi(»), • • • , W 8 ) I I

is orthogonal for s = 0. If f4(s) and f,(s) are two solutions of equation (4.4), then by using equations (4.4) and (2.6) it is easily seen that (f 4 'f,)' = fj-fj + f^f} = 0, and consequently the matrix (4.6) is orthogonal on the interval [0, L]. In particular, the equation (4.7)

Xx(s) = E 4 Ms)f*(s)

determines uniquely the scalar functions Xt such that \4(0) = 0 = >w(L) (i = 1, • • •, N + 2), since X^O) = 0 = X^L). From equations (4.4) and (4.7) it follows immediately that X{(«) = 72M

+ £,*•& = ^ [ e 3 x • • • x eN+2 x X^s)] + £ > & •

If X(s) denotes the vector function (X^s), • • •, X^+^s)), then we have, in consequence of the relation fl-f] = 8iJ(i,j = l, • • •, N + 2), I XKs) I2 = g I e3 x • • • x e„ +2 x X,(s) |2 + 2 ^ £4X{f.-[e, x • • -eN+2 x Xx(s)] + | X' |2 , Li

so that an application of Lemma 3.1 and the condition | X |2 = | Xj |2 yields I XI(s) |2 - ^ X J ( s ) - [ e 3 x . . . x e„ +2 x X^s)] = | V | 2 - ^ |2e 3 x . . . x e „ + 2 x X^s)! 2 L = | X' |» - — I X |* + — £ [e^-X^s)]2 . Thus the functional J of equation (4.5) satisfies the relation (4.8)

J :> Z, j ' ( | X' |2 - g | X |»)ds ,

where the equality holds only if er-X^s) == 0 for r = 3, • • •, N + 2 and 0 5S s ^ L. From Lemma 3.2, it follows that the right side of the inequality (4.8) is nonnegative, and is equal to zero if and only if \(s)=g sin {izsIL), where g=(<7i, • • •, gN+2) is a constant vector in the space EN+2. Hence we have J 2s 0, where the equality holds if and only if the relations (4.2) and

218

CHUAN CHIH-HSIUNG

(4.3) hold With f(s) = J^MBefore we prove the last part 1 of the theorem, it is desirable to give a geometrical interpretation of the conditions (4.2), (4.3), (4.4), under which the equality holds in (4.1). Let M* be the two-dimensional cone in the space EN+i with vertex at the point X0 and generated by the curve C, denned perimetrically by the vector X^s), and the limiting generator X'(0) =s X'(L). The cone M* is differentiable, if Xa(s) and X[(s) are linearly independent for 0 < s < L, or equivalently if (4.9)

| d \XMIds

| < | dX1(s)/ds | = 1

(0 < s< L) .

Now suppose that Xx(s) is a solution of equations (4.3) and (4,4). Then f(s) has a constant absolute value, say | f (s) | = c, and | dX,{s)lds | = | f'(s) sin (ns/L) + y f(s) cos (TZSJL) \ (4.10) = JL [ | e3 x • • • x eN+2 x f(s) |2 sin2 (jra/L) + c2 cos2 (TTS/L)]112 . Ld

From equation (4.10) it follows that | dXx{s)\ds | —» cz/L as s —<• 0 + , so that c = L\K. Moreover, | e3 x • • • x eff+2 x f(s) | g | f(s) | = L/7T, where the equality holds if and only if f(s)-er = 0 (r = 3, • • •, N + 2). On the other hand, equation (4.10) and | dX1(s)/ds | = 1 imply that | e3 x • • • x e^+2 x f(s) | = | f(s) | = Ljit. Thus equations (4.2) are a consequence of equations (4.3) and (4.4), and a use of equations (4.3) and (4.9) shows immediately that the cone M* is differentiable. Developing this cone M* into a plane, the curve C then goes into a differentiable plane curve parametrized by the vector Y(s) with s (0 < s < L) as its arc length and satisfying | Y(s) | = (L/K) sin (rus/L), which is true only for a circle of diameter L\TZ and passing through the origin O. Hence the proof of the theorem is complete. Using the same notation as in § 2, for the position vector X of a point P on a manifold M% with respect to a fixed point O in the space EN+2 we have (see, for instance, [4, Chapter IV]) AX =
= ESUe r = £> , where A is the Laplace-Beltrami operator, matrix \\gap\\ is inverse to the matrix || ga?, \\ of the induced riemannian metric of the manifold M2, the semicolon denotes the covariant derivative with respect to gafi, repeated 1 T h e author is indebted to the referee for his suggestions in regard to the present form of the prnnfo nf Thporem 4.2 and this part of Theorem 4.1. T h e original form of these proofs is an extension of the one given by Reid [11] for the special case in which N = 1.

ISOPERIMETRIC INEQUALITIES

219

indices imply the summation over their ranges, and D,riap (a, /3 = 1, 2) are the coefficients of the second fundamental form of the manifold M2 associated with the unit normal vector er at the point P. The manifold M2 is called a minimal manifold, if its mean curvature vector £> vanishes at every point. From equation (4.11) it follows immediately that the position vector X of every point on a minimal manifold M2 satisfies the equation AX = 0. Now suppose that a compact minimal manifold M2 has a boundary C lying in a linear subspace Ew+2-k of the space EN+2, so that 0 < k < N+ 2. Let x1, • • •, xN+2 be the components of the vector X with respect to a fixed orthogonal frame through the fixed point 0 in the space EN+2. Without loss of generality we may assume the subspace EN+2_k to be defined by the equations xl = 0, • • •, x" = 0. Thus x1, • • •, x" satisfy the equations Ax1 = 0, • • •, Ax" = 0, respectively, over the manifold M2, and vanish on the boundary C. By the maximum modulus principle [7], x1, • • •, x" must vanish identically over the whole manifold M2, and therefore the manifold M2 lies entirely in the subspace EN+2-k. By means of this remark and Theorem 4.1 we can easily reach 4.2. Let M2 be a two-dimensional riemannian manifold satisfying the same conditions as in Theorem 4.1. Then (4.12) U - ircA ^ 0 , where the equality holds only if the boundary Cisa circle and the manifold M2 is the circular disc bounded by the circle C. In fact, the inequality (4.12) follows immediately from the inequality (4.1) and the assumption that § = 0 at every point P of the manifold M2. Moreover, if the equality holds in (4.12), then it also holds in (4.1) for an arbitrary point X0 on the boundary C, and therefore the conditions (4.2), (4.3) are satisfied for an arbitrary point X0 on the boundary C. In particular, if X and X0 are arbitrary points on the boundary C and e3, • • •, e^+2 are normal to the manifold M2 at the point X, then (X — X 0 )'e r = 0 for r = 3, • • •, N+ 2, and therefore the boundary C is a curve along which e3, • • •, eN+2 are constant. From the last part of Theorem 4.1 it thus follows that the boundary C is a circle of radius L/(2?r), and hence by the above remark the manifold M2 is the circular disc bounded by the circle C. For the case in which N = 1, Theorem 4.2 is due to Carleman [3]. THEOREM

MATHEMATICS RESEARCH CENTER, MADISON, WISCONSIN, AND LEHIGH UNIVERSITY BIBLIOGRAPHY

1. E. F. BECKENBACH and T. RADO, Subharmonic functions and surfaces of negative curvature, Trans. Amer. Math. Soc, 35 (1933), 662-674. 2. O. BOLZA, Vorlesungen tiber Variationsrechnung, Teubner, Leipzig, 1909.

220

CHUAN CHIH-HSIUNG

3. T. CAELEMAN, Zur Theorie der Minimalflachen, Math. Zeit., 9 (1921), 154-160. 4. L. P. ElSENHART, Riemannian Geometry, Princeton University Press, 1926. 5. F. FIALA, Le probleme des isoperimetres sur les surfaces ouvertes a courbure positive, Comment. Math. Helv., 13 (1940-41), 293-346. 6. G. H. HARDY, J. E. LlTTLEWOOD, G. POLYA, Inequalities, Cambridge University Press, 1934. 7. E. HOPF, Elementare Bemerkungen iiber die Lbsungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungsber. Preuss. Akad. Wiss., 19 (1927), 147-152. 8. C. C. HSIUNG, A uniqueness theorem on two-dimensional Riemannian manifolds with boundary, Michigan Math. J., 5 (1958), 25-30. 9. A. HUBER, On the isoperimetric inequality on surfaces of variable Gaussian curvature, Ann. of Math., 60 (1954), 237-247. 10. S. LOZINSKY, On subharmonic functions and their application to the theory of surfaces, Bull. Acad. Sci., SSSR. Ser. Math., 8 (1944), 175-194 (in Russian). 11. W. T. REID, The isoperimetric inequality and associated boundary problems, J. Math. Mech., 8 (1959), 897-906.

CHUAN-CHIH

HSIUNG

A NOTE OF CORRECTION

The purpose of this note is to announce that a correction should be made on a previous paper by the author « Curvature and Betti numbers oj compact Riemannian manifolds with boundary, this Rendiconti, vol. 17 (1957-58), pp. 95-131 ». In the introduction of this paper it was assumed throughout the whole paper that the (n — l)-dimensional boundary Bnl of every compact re-dimensional Riemannian manifold Mn is connected and that the manifold M'1 is not homeomorphic to an 7z-dimensional Euclidean manifold with an (n—l)-dimensional boundary. However, after the mistake made in the paper was pointed out by Professor M. B E R G E R in the following communication, for which the author is greatly indebted to him, the author has been able to show that all theorems in the paper are still true, if we merely replace the above two extraneous conditions by the convexity or concavity of the boundary Bnl, with respect to its inner normal at every point in the tangent space of the manifold M", according as the theorems are concerned with p-dimensional ( p ~ l ) harmonic tensor fields or Killing (or conformal Killing) tensor fields. The details of this correction will be published later. The remainder of this note is the original communication of Professor B E R G E R to the author concerning the paper mentioned above. « Cet article nous semble appeler quelques remarques; nous les «presentons ci-dessous dans le cas precis du theoreme 5.5A, les « autres theoremes etant justiciables de remarques analogues. Les « references seront celles de l'article. « Le theoreme 5.5A repose sur le theoreme 5.4; la demonstration « d e celui-ci est la suivante (lignes 15 a 2 4 de la page 1 1 3 ) : le

— 128 —

«champ de vecteurs harmonique £ de composante tangentielle « (resp. normale) nulle a la frontiere verifie

J(RifM

+

P*£iti)dAn=0,

«parce que dans la formule (4.8) (resp. (4.15)) l'integrale de «droite qui porte sur la frontiere est nulle, le terme a integrer « etant nul parce que In.n = 0 (resp. fi; n = fB. t = 0). Ces der«nieres egalites sont fausses en general. En effet, supposons £ a « composante tangentielle nulle a la frontiere, c'est a dire que Ton «sait que, sur Bnl: f* = 0 quel que soit i = 1, 2,..., n — 1. Comme « f est harmonique: i

£* :i =0

ein=-n2

done

P.i .

« Mais il n'y a pas de raison pour que f \ 4 = 0 (i = 1,2,..., /i — 1); « car, de £' = 0 ( i = 1,...,ra— 1) a la frontiere on peut seulement «deduire: d£l/8xt-=0 (i= l,...,n — 1). Mais £*.i est une derivee « covariante, on a done en fait:

8?

«qui se reduit ici a »-i

«

-2" J T ^

n

.,

I*. f = T ^ f" (i = 1, ..., M — 1).

n'est pas nul en general le long de B"1

Done

(remarque

« analogue pour un champ a composante normale nulle a la fron«tiere). »-i

«En fait

2 r*n t=i

est lie a la convexite de Bnl

en tant

« qu'hypersurf ace dans Mn; un theoreme tel que 5.5A ne peut pas «etre vrai sans de telles hypotheses de convexite sur Bnl. II est « aise de fabriquer des centre-exemples en partant d'une sphere S", «munie de sa structure riemannienne canonique qui est a. courbure « de RICCI definie positive et en otant un ouvert V de sorte que le «complement Mn == 5" — U, qui est une variete a bord de bord «egal a. la frontiere de U, ait un premier nombre de BETTI non

— 129 —

« nul; ainsi si U est le voisinage tubulaire ouvert d'un cercle de S?' « (remarquer que la frontiere est un tore a deux dimensions done « connexe et que l'interieur de M 3 n'est pas homeomorphe a Rs: «note (2) du bas de la page 95). « Par contre, avec en plus des conditions geometriques sur la «frontiere Bnl, il est possible d'obtenir un theoreme de conclusion « analogue a celle du theoreme 5.5A; ceci a ete fait dans l'article «posterieur de K. YANO: "Harmonic and Killing vector fields in «compact orientable Riemannian spaces with boundary', (Annals « of Mathematics, 69, 1959, p. 588-597), theoremes \T et 2N, pour « une frontiere tournant sa concavite vers l'interieur. « Ces theoremes lr et 2 ^ impliquent que le premier nombre de « BETTI de la variete considered est nul, lorsque la courbure de « Ricci est definie positive et que la frontiere tourne sa concavite «vers l'interieur; on peut obtenir ce resultat de facon geometrique « en generalisant la demonstration de MYERS dans le cas sans front i e r e (S. MYERS : "Riemannian manifolds with positive mean cur«vature", Duke Math. Journal, 8, 1941, p. 401-404). Precise«ment: si Mn est une variete complete a courbure de RICCI supe« rieure ou egale a c > 0 et dont la frontiere tourne sa concavite vers «l'interieur, le diametre de Mn est inferieur ou egal a ir/(c/ « (n—1)) Vi (d'ou il resultera que tout revetement de Mn aura la «meme propriete et sera a fortiori compact, d'ou la finitude du «groupe fondamental de Mn puis l'assertion sur le nombre de « BETTI). Ce resultat est prouve par MYERS dans le cas ou deux « points quelconques de Mn sont joints par un segment geodesique «tout entier dans l'interieur de M°; mais il doit toujours en etre « ainsi parce sinon une courbe realisant le minimum de la distance «entre ces deux points serait composee de morceaux tout entier « dans la frontiere et de segments geodesiques arrivant tangentiel«lement a la frontiere, deux cas qui sont proscrits par la concavite « de cette derniere ».

CHERN, SHIING-SHEN, AND CHUAN-CHIH HSIUNG

Math. Annalen 149, 278-285 (1963)

On the Isometry of Compact Submanifolds in Euclidean Space* By SHIING-SHEN CHERN

and

CHUAN-CHIH HSIUNG

in Berkeley, California

Introduction The question as to the conditions on an isometry between compact hypersurfaces in euclidean space in order that it be a congruence has been the subject of extensive researches in differential geometry. In this paper we will prove a theorem which states that a volume-preserving diffeomorphism between two compact submanifolds in euclidean space satisfying certain conditions (see § 2) is an isometry. Our main tool consists of some integral formulas. The latter should be of independent interest. § 1. Preliminaries on Algebra Let V be a real vector space of dimension n, and let G be a bilinear realvalued function over V x V. G is completely determined by the values 9ik — G(e{, ek), 1 :g i, k gL n, where e( form a basis of V. Under a change of basis e,- -^ ef = JJ t\ek, the matrix \\gilc\\ is changed to T\gik\*T, where T = ||
and *T is the transpose of T. Let H be another bilinear real-valued function over V x V, and let hik = H(eit ek). Consider the determinant (1)

det(gik + hikX) = detfat) + nlP(gik,

hik) +••• + !« det(hik) .

2

Since it will be multiplied by (detT) under a change of basis, the ratio of any two coefficients in the polynomial (1) in A is independent of the choice of basis. In particular, if G is non-singular, i.e., if det(gik) =f= 0, the quotient (2)

Ha=

P(gik,hik)ldet(gi]e)

depends only on G and H. For example, if gik = dik ( = 1, i = k; = 0, i 4= k),

then H0 = 2J huln. i

This construction of HG is linear in H. Hence, if H is a bilinear function over V x V with values in a vector space W, the above construction can be generalized and Ha is an element in W. H0 can be called the contraction of H relative to G. In the language of tensors we can express our construction by saying that H0 is a vector in W constructed from a covariant tensor H of *) Work done under National Science Foundation Research Grant NSF G-19137.

Isometry of Compact Submanifolds

279

order two with values in W relative to a non-singular covariant tensor G of order two. I n what follows we will need an inequality, due to LARS GAEDING [2], on HG. I t can be stated as follows: Let G and H be symmetric positive-definite bilinear real-valued functions over V X V. Let (3)

g = det {g{ k),

Then (4)

h= det {ht k) .

HG^(hlg)V\

and the equality sign holds if and only if hik = ogik, for a certain Q. % 2. Preliminaries on Differential Geometry and Statement of Theorem Let M and M* be two C 2 -riemannian manifolds of the same dimension n, and let / : M -> M* be a 0 2 -mapping. I n M there are t h e n two connections: the Levi-Civita connection defined b y its riemannian metric and the connection induced b y the mapping / from the Levi-Civita connection of M*. Their difference is a tensor field A of contravariant order 1 and covariant order 2. If G denotes t h e fundamental tensor field of M, the construction in § 1 gives a vector field AG. We will call / an almost isometry, if Ag = 0. Next consider an immersion x: M -> E, i.e., a 0 2 -mapping x of M into an euclidean space E of dimension n + m, such t h a t the induced linear mapping on t h e tangent spaces is univalent everywhere. Then x(p), p £ M, is a vector in E and will be called the position vector of the submanifold. We will denote b y a parenthesis the scalar product of two vectors in E. If v is a normal vector a t x(p), then (y, d2 x(p)), where d2x is the second differential in the ordinary sense, is t h e second fundamental form relative to v. Since t h e latter is a symmetric covariant tensor of order two, we can form its contraction (v, d2x(p)')G relative to the fundamental tensor G of the induced riemannian metric on M. There is therefore exactly one normal vector field H over x(M) defined b y (5)

(v,d*x(p))G=(v,H).

This normal vector field H is called t h e mean curvature vector. I t generalizes t h e notion of mean curvature in ordinary surface theory. A submanifold is called minimal, if the mean curvature vector is identically zero. Consider now two immersions x, x* of M into E, and a diffeomorphism / as given b y the commutative diagram M~J^^

x{M)

CE

The mapping / is called a n isometry, if (dx*(p), dx*(p))= (dx(p), dx(p)), i.e., if it maps the induced riemannian metric of the one into t h a t of the other.

280

SHUNO-SHBN CHBBN and CHTTAN-CHIH HSITTKG:

It is called volume-preserving, if it maps the volume element of one into that of the other. As a consequence of this definition a volume-preserving diffeomorphism exists only if If is oriented and the diffeomorphism is then orientation-preserving. We will denote by G, 0* respectively the fundamental tensors of the two induced metrics. Our main theorem is the following: Let x, x* : M -> E be two immersed compact submanifolds and let f : x (M) -> -> x* (M) be a volume-preserving diffeomorphism. Suppose that f has the properties: 1) It is an almost isometry relative to Q*, i.e. AG, = 0; 2) (x±(p), d2x(p))g ^ (x±(p), d2x(p))Gt, where x~l(p) is the orthogonal projection of the position vector x(p) in the normal space to x(M) at x(p). Then f is an isometry. § 3. Integral Formulas Over the abstract manifold M there are now two induced riemannian metrics with the same volume element, namely (dx(p), dx(p)) and (dx*(p), dx*{p)) = (d{f o x) (p), d(f o x) (p)) . The notion of frames e1; . . ., en having measure 1 and coherently oriented with that of M has thus a sense in both metrics. At a point p £M any such frame can be obtained from a fixed one by a linear transformation of determinant 1. Since the induced linear map x* on tangent spaces is univalent, we identify e( with x# (et). Let co1, . . ., co™ be the coframe dual to ex, . . ., en, so that the volume element of M is (6) dV= co1 A • • • A co". Let en+1, . . ., en+m be a frame in the normal bundle at x(p). We introduce the matrices c o = llco 1 , . . . , « w « | L

e=

(7)

Then we have the matrix equations (8) where

dx= cue,

de=i2e+6a,

fi=Kll, 6 = \\cof+s\\,

1 g: i, k g n, 1 g s g m ,

+s

aft, a)f being linear differential forms in the bundle induced over M from the principal bundle of E. Exterior differentiation of the first equation of (8) gives (10)

dm=whQ,

COA0=O.

Before proceeding, we shall give the geometrical meaning of these equations. It is well-known that the matrix Q gives the connection form of the induced riemannian metric by x. Written explicitly, the second equation of (10) gives (10a)

£ co* A co%+> = 0 .

Isometry of Compact Submanifolds

281

This allows us to put a>f+s= Z Af^co1 >

(11)

i

where A^fs is symmetric in the lower indices: (12) Atf*= Aff • These quantities are the ones which enter into the second fundamental form. In fact, v being a normal vector at x(p), we find (13)

(v, d*x) = S (".

en+s)Ani^(oim',

where the quadratic differential form is in the sense of the symmetric algebra, multiplication being commutative. We put (14) and introduce the matrix

(*>en+3) = yn + s,

\<s^m,

(15) AHISfc+.Hto'*)1). Then we have the one-rowed matrix h*d = 1 Zyn+S^sa>'

(16)

On the other hand, we have (17) x-L =

JZJ

yn + s^n + s •

Thus we find the following explicit expression for the quadratic differential form in the second condition of our theorem: (18)

Q = (x^, d*x) = Z y B + .4?/«a>'o)'. s, i, j

The notation Q will hereafter be used as an abbreviation of this expression. Furthermore the condition that the frames are of measure 1 is (e1

A••

•

A

e„, ex A • • •

A

e„) = 1 ,

where the left-hand side is the scalar product of multivectors defined by the scalar product in E. Differentiating this equation and using the second equation of (8), we get (19) TrQ=Za>i=0. i

To derive our integral formula we have to introduce exterior differential forms globally defined over M. Their study will be facilitated by the adoption of notations developed by H. FLANDERS [1]. This is to consider tensor products of multivectors and exterior differential forms. Differentiation of multivectors will be taken in the sense of (8), while differentiation of exterior differential forms will be exterior differentiation. Multiplication of matrices will be by x

) For an (m X w)-matrix a = ||o^j|| and an (n X j>)-matrix 6 = |&y*||, whose elements are vectors in E, we will use the notation (a, b) to denote the matrix of real numbers given by (a, 6)= 2" (<*„,&;*) . IIJ II

282

SHUNG-SHEN CHBBN and CHUAN-CHIH HSIOTG :

the usual row-by-column law. With this understanding we introduce the following matrices: (20) 0=(e,*e)=*G=lgtil, (21)

G*=(/*(e),/*(«e))=«G* = 1 ^ 1 ,

(22)

A= | K , . . . , © , 1 = 0 ) 0 ,

(23)

u=Ae,

(24) (25)

A*=lcol,...,(o*l=a,a», u* = A*e,

7=(x,*e)=\yv;Vnh r = Ye ,

(26) z = ru*n~1=
}d
M being supposed to be compact. It is therefore our problem to calculate d