On equivariant isometric embeddings Math. Z. 173, 119-i33 (1980) Mathematische Zeitschrift �9 by Springer-Verlag 1980 On Equivariant Isometric Embeddings J o h n D o u g l a s M o o r e 1 , . a n d R o g e r Schlafly z 1 Department of Mathematics, University of California, Santa Barbara, California 93106, USA 2 Department of Mathematics, University of California, Berkeley, California 94720, USA 1. Introduction T h e N a s h e m b e d d i n g t h e o r e m [13] asserts t h a t a n y R i e m a n n i a n m a n i f o l d possesses a n i s o m e t r i c e m b e d d i n g into a Euclidean space o f sufficiently large dimension. This article is d e v o t e d to a p r o o f o f a n e q u i v a r i a n t version o f N a s h ' s t h e o r e m , M a i n Theorem. I f M is a compact Riemannian manifold and G is a compact Lie group which acts on M by isometries, there is an orthogonal representation p o f G on some Euclidean space 115 N and an isometric embedding f r o m M into IE N which is equivariant with respect to p. T h e r e p r e s e n t a t i o n p can be r e g a r d e d as a Lie g r o u p h o m o m o r p h i s m f r o m G into the o r t h o g o n a l g r o u p O(N) which acts o n IE N b y r o t a t i o n s a n d reflections; a s m o o t h m a p X : M ~ I E N is e q u i v a r i a n t with respect to p if a n d only if X ( ~ p ) =p(cr) X(p), for all ~r~G, p E M . T h e m a i n t h e o r e m is true in b o t h the C ~ a n d real analytic categories. W e will w o r k in the C ~ c a t e g o r y for the t i m e being, a n d r e t u r n to the real analytic case in w 4. M o r e o v e r , the t h e o r e m h o l d s for m a n i f o l d s with b o u n d a r y . T h e m a i n a n a l y t i c t o o l used b y N a s h to p r o v e his i s o m e t r i c e m b e d d i n g t h e o r e m is an implicit function t h e o r e m b a s e d u p o n the N e w t o n i t e r a t i o n m e t h o d . T h e implicit function t h e o r e m applies to the e q u i v a r i a n t case with virtually n o change. I n o r d e r to a p p l y the implicit function t h e o r e m we need to a p p r o x i m a t e a given G - i n v a r i a n t R i e m a n n i a n m e t r i c o n M b y a m e t r i c induced b y an e q u i v a r i a n t e m b e d d i n g ; we will d o this b y using the t h e o r y o f the L a p l a c e o p e r a t o r o n c o m p a c t R i e m a n n i a n manifolds. A c c o r d i n g to G r o m o v a n d R o k h l i n [7], a n y n - d i m e n s i o n a l c o m p a c t Rie- m a n n i a n m a n i f o l d can be isometrically e m b e d d e d in IF, N, where N = (1/2) n(n + 1) + 3 n + 5. N o such universal b o u n d is possible in the e q u i v a r i a n t case, a n d in fact, given a n y positive integer N , it is possible to c o n s t r u c t a left i n v a r i a n t * Partially supported by National Science Foundation grant MCS 77-01843 0025-5874/80/0173/0119/$03,00 120 J.D. Moore and R. Schlafly metric on the group S 3 of unit quaternion s which is n o t induced b y any equivariant embedding in IE" for n __< N, as described in w 6 of this article. Moreover, the equivariant isometric embedding t h e o r e m does n o t h o l d without the assumption that M be compact. Indeed, Bieberbach [1] shows t h at the Poincar6 disc with the hyperbolic metric o f constant cu rv at u re - 1 t o g et h er with the circle group of r o t a t i o n s a b o u t the origin possesses no equivariant isometric embedding in any finite-dimensional Euclidean space. It suffices to p r o v e the equivariant isometric em b ed d i n g t h e o r e m in the special case where M is an n-dimensional sphere S" with a R i e m a n n i a n metric invariant under a Lie subgroup G of O(n+ 1). Indeed, b y a t h e o r e m o f M o s t o w and Palais [-2, p. 315], any c o m p a c t G-manifold possesses an equivariant embed- ding in a sphere S" of sufficiently large dimension, even if the manifold has b o u n d a r y , and by a p a r t i t i o n of unity argumen t o n e easily extends a G-invariant metric on M to a G-invariant metric on S" which makes this equivariant embedding isometric. It will be convenient to formulate the equivariant isometric em b ed d i n g p r o b l e m in terms of certain Frhchet spaces. If G is a given c o m p a c t Lie g ro u p acting on a c o m p a c t manifold M and p: G ~ O ( N ) is a given representation, let C~176 IEN)= {C 00 maps X: M ~ I E N } , C~, p(M, IE u) = {XE Coo (M, IEN)I X is equivariant with respect to p}, Metoo(M)= {Coo symmetric r a n k two covarian t tensors o n M}, M e t ~ (M) = {g e Met OO (M) I g is G-invariant}. These vector spaces b e c o m e Fr6chet spaces with the usual family of C k norms, (Notation: ILX[Ik and [IgHk will denote the C k n o r m s of elements XsCOO(M, IEN), g ~ Met OO (M).) Finally let C~ (M, IE N) = ~ { C~, p(M, IEN) I p: G ~ O(N) a representation}. We define a m a p F: C~(M, I E N ) ~ M e t g ( M ) by letting F(X) be the metric induced on M by X. In terms of local coordinates (u 1 . . . . , u") defined o n an o p en subset U of M, F(X) I U = y (X). (X) du i duJ. j ~ "5 T o p r o v e the theorem, we need to show t h at given a positive-definite g ~ M e t ~ ( M ) , there is some representation p: G-+O(N) and some em b ed d i n g X e Ca~ p(M, IE N) such that F(X) = g. We say that an element g ~ M e t ~ ( M ) is realizable if there is a m a p p i n g (not necessarily an embedding) X ~ C ~ ( M , IF. N) for some N such t h at F ( X ) = g . T h e set of realizable metrics is closed under addition an d multiplication b y positive scalars. T h e r e are two steps to the p r o o f of the equivariant isometric em b ed d i n g theorem. T h e first step consists of constructing a specific " p e r t u r b a b l e " embed- ding X o E C ~ ( M , IE N~) such that if g o = F ( X o ) , t h e n any g l ~ M e t ~ ( M ) which is On Equivariant Isometric Embeddings 121 sufficiently close to go can be realized as F(X 0 for some em b ed d i n g XI~ C~(M, IENI). This step relies on the Nash implicit function t h e o r e m an d o u r t r e a t m e n t o f it will be based u p o n the w o r k o f Schwartz [14] and Sergeraert El5], [16]. T h e second step consists of showing t h at an y given positive-definite g e M e t ~ ( M ) can be a p p r o x i m a t e d arbitrarily closely by realizable metrics. T h e two steps of the p r o o f are p u t togeth er in the following way: If g is a positive-definite element of M e t e ( M ) , choose a co n st an t c > 0 so t h at g - c g o is positive-definite, where go is the metric induced by the p e r t u r b a b l e em b ed d i n g X o constructed in step 1. (The metric c g o will be induced by the p e r t u r b a b l e embedding ~ c X o . ) Use step 2 to a p p r o x i m a t e g - c g o by a realizable metric g2; g2 =F(X2) for some X z E C ~ ( M , ]EN2). T h e n g - g 2 will be close to cg o a n d hence ~ M by step 1, there will be an element XleCG( , IE N1) such t h at F ( X 1 ) = g - g z. H e n c e X=(X~, X2) will be an embedding in C~(M, IE N) (where N = N 1 +N2) such that F(X)= g. T h e above t h e o r e m extends the main result o f an earlier article [10]. T h e two steps of the p r o o f of the main t h e o r e m will be given in w167 2 and 3. In w we will discuss the modifications necessary for p ro v i n g the real analytic version of the main theorem. In w 5 we will show t h at even if M is n o t c o m p a c t it possesses an equivariant isometric embeddi n g into H i l b ert space, o r into a finite-dimensional pseudo-Euclidean space if it has finitely m a n y orbit types. Finally, in w 6 we will discuss nonexistence theorems, including an extension o f Bieberbach's example m e n t i o n e d above. 2. Step 1. Nash's Implicit Function Theorem Except for a few details, this is just like the c o r r e s p o n d i n g step in the non- equivariant case. F o r each choice of representation p, the m a p F : C~,p(M, 1 E N ) ~ M e t ~ ( M ) defined previously is Fr6chet differentiable. Indeed, if X, AX are elements o f C~G, o( M, IEN), F(X + AX)IU= ~ o@(X) o@(X)dui duJ i , j = l i,j= 1 \ Ouz OUJ {71AJ GU~ / + - - (AX) (AX) du i du J, i, j= 1 ~ui from which we see that the Fr6chet derivatives o f F are given by the formulae i, j= 1 \OU OIA ~ F " ( X ) ( A I X , A 2 X ) I U = 2 ~ ~ ( A i X ) 2 ( d 2 X ) , i , j = l F (n) (X) = 0 for n > 3. 122 J.D. M o o r e a n d R. Schlafly Let g be a given positive-definite element of M e t e ( M ) an d suppose t h at we can find an embedding X~C~,p(M, IE N) such t h at F(X) is close to g. W e might then try to construct an embedding X + A X such that F(X+AX) is closer to g by solving the linearized equation If F'(X)(AX)=Ag, where Ag=g-F(X). AgIU= f (Ag)ijdu idu j, i , j = I the linearized equation is cqui (X). �9 ~--~ (AX) = (A g)ij. (1) Following Nash, we impose the additional con d i t i o n t h at AX be p erp en d i cu l ar to X ( M ) : c~u, (X). AX = 0 . (2) Integrating by parts shows that Eqs. (1) an d (2) are equivalent to the linear system au' (x). (AX)=0 (~2 1 A ~u'~uJ (x). (AX)= - ~ ( g),j (3) Recall that we are given Ag and we wish to solve for AX. It is n at u ral to restrict a t t e n t i o n to embeddings for which this linear system is nondegenerate. Definition. An embedding X : M ~ I E u is said to be perturbable if for every peM there are local coordinates (u 1,...,u") defined o n some n e i g h b o r h o o d U of p such that the matrix of c o l u m n vectors has r a n k n+�89 ( N o t e that this condition does n o t d ep en d u p o n the choice of local coordinates.) Any G-manifold M possesses a p e r t u r b a b l e equivariant embedding into a Euclidean space of sufficiently large dimension. Indeed, by the equivariant embedding t h e o r e m of M o s t o w a n d Palais m e n t i o n e d in the i n t ro d u ct i o n , there is an equivariant embedding X=(x 1, ...,xm): M - - , I E "~ On Equivariant Isometric Embeddings 123 into a Euclidean space of some dimension m, fro m which we can c ~ a p e r t u r b a b l e equivariant e m b e d d i n g (x,) 2, 7 x1 xo, into a Euclidean space of dimension m + �89 m(m + 1). We can rewrite the linear system (3) in the form If X is perturbable, tAA will be a nonsingular square m at ri x an d A X = A ( A A ) - : ( _ ~ A g ) (4) will be a solution to (3), a n d in fact the a r g u m e n t given in Schwartz [-14, p. 48] or G r e e n e [4, pp. 32, 33] shows that it is the unique solution o f smallest length. Thus even t h o u g h A is defined in terms o f local coordinates (ul, ..., u") on M, the solution AX given by (4) does not depend on the choice o f local coordinates. AX is a globally defined s m o o t h m a p p i n g from M into IE N, which as we will next check, is also equivariant. If p~M a n d ~r~G we can choose c o o r d i n a t e systems (u' . . . . ,u ~) defined o n a n e i g h b o r h o o d U o f p and (if1, . . . , f t , ) d e f i n e d on aU so that 72~(crq)=ui(q) for q~U. Since X is equivariant the matrices 0 2 ~ 0 2 are related by the e q u a t i o n A ( ~ p ) = 0 ( 0 ) ~4 (p), and since p(~) is orthogonal, 0 AX(ap)=A-(crP)(tA(aP) "4(':rP))-I ( - � 8 9 A g ( a p ) ) / \ 0 = p ( a ) A(p)(tA(p) A(p)) -1 [-�89 Ag(p)) = p(a) AX (p). Thus AX is indeed an element of C~ ~, p(M, IF, U). H e n c e we can define (0) L(X): M e t ~ ( M ) ~ C a ~ p ( M , IE N) by L(X)(Ag)= A('AA)- : -�89 Ag ' 124 J.D. Moore and R. Schlafly so t h a t F'(X) o L ( X ) = identity. Suppose t h a t X o is a fixed p e r t u r b a b l e e m b e d d i n g with F ( X o ) = g o. I f g is a m e t r i c which is close to go, we m i g h t try to solve the n o n l i n e a r e q u a t i o n F ( X ) = g b y the N e w t o n iteration X n = X n _ l q - L ( X n _ l ) ( g - F ( X n _ l ) ) , n>=l. I f the X , ' s were to c o n v e r g e in the C ~ t o p o l o g y to a n element XooeC~,,p(M, lElV), it would follow i m m e d i a t e l y t h a t F ( X ~ ) = g , b u t unfor- t u n a t e l y this s t r a i g h t f o r w a r d iteration does n o t c o n v e r g e since the e s t i m a t e we o b t a i n for L(X), IIL(X)(g-- F(X))[Ik_ 2 ~ Mk IIXII~ IIg-- F(X)llk, (5) M k a constant, involves a loss of two derivatives. T o c i r c u m v e n t this difficulty, we m a k e use o f s m o o t h i n g o p e r a t o r s S(t): CS, o(M, IF, N) --, C~, p(M, IE N) which satisfy the estimates II S(t) X i l k + , < A k , , t ~ IlXll,, (6) ][(I-- S(t)) XNk <=Bk, lt -I []XNk +l, where Ak, t, Bk, ~ are suitable constants. I n the n o n e q u i v a r i a n t case, s m o o t h i n g o p e r a t o r s S(t): C ~ (M, IE N) ~ C ~ ( M , IE N) satisfying (6) are c o n s t r u c t e d b y L a n g [9] or Schwartz [14, pp. 38, 39]. I n the e q u i v a r i a n t case, we set S ( t ) = P o S ( t ) , where P: C~176 IEN) --, C~,p(M, IE N) is the p r o j e c t i o n o p e r a t o r defined b y P ( X ) (p) = ~ . ( a - 1) X ( a p ) d . , G the integral being t a k e n with respect to the H a a r m e a s u r e o n G. (It is easily checked t h a t P is b o u n d e d in each C k norm.) O n c e we h a v e defined these s m o o t h i n g o p e r a t o r s we can consider the modified N e w t o n iteration: X n = X , _ l + S ( t n 1) o L ( X , _ I ) ( g - F ( X , _ I ) ) , n > l , where { t > t 2 , . . . , t , _ 1, t , , . . . } is a suitably chosen increasing sequence of real n u m b e r s . T h e r e m a r k a b l e fact is t h a t the t~'s can be chosen so t h a t the m o d i f i e d N e w t o n iteration converges. This is essentially the c o n t e n t o f N a s h ' s implicit function t h e o r e m . I t is easiest to p r o v e convergence when it is a s s u m e d (without loss o f generality) t h a t M is the n-sphere a n d G is a s u b g r o u p o f O(n + 1). W e can use the fact t h a t F a n d L are i n v a r i a n t u n d e r r o t a t i o n s o f the sphere (just as Schwartz [-14, pp. 40, 41] uses the t r a n s l a t i o n - i n v a r i a n t n a t u r e of F a n d L on the t o r u s in his t r e a t m e n t o f the n o n e q u i v a r i a n t case) to establish estimates which are s o m e w h a t s t r o n g e r t h a n (5): I f ] [ A X I ] 2 < 1, [IL(X o + A X ) Agllk_ 2 --<__ Adk(1 q- [1AXI[k)H AgHk, ][L(Xo + A X ) F ( X o + AX)Hk_ z < Mk(1 + [IAXHk), O n E q u i v a r i a n t I s o m e t r i c E m b e d d i n g s 125 where X 0 is the perturbable embedding fixed earlier. These inequalities yield the hypotheses for the elegant version of the implicit function theorem proven in the short note of Sergeraert [15], which can be applied without any change to our situation to establish convergence. It follows from Sergeraert's theorem that there exist k and ~ such that if ][g--g011k<8, then g can be realized by an equivariant isometric embedding, and Step 1 is finished. 3. Step 2. Approximation by Realizable Metrics Thus the p r o o f of the theorem reduces to showing that any positive-definite G- invariant metric can be approximated by realizable metrics. It will suffice to show that if g is any positive-definite element of M e t e ( M ) , k a positive integer, and e a small positive number, there is a realizable metric F(X) such that Pig - - F ( X ) l l k < a It follows from the nonequivariant version of the Nash embedding theorem that any positive-definite metric g on M can be expressed as a finite sum g = ~ dyi@dY i, i = l where each yi: M ~ R is a smooth function. (Instead of relying on the entire p r o o f of the Nash embedding theorem the following a r g u m e n t could rest on the weaker assertion that g can be Ck-approximated by a sum ~dy~| ~, an assertion which is proven in Nash [13, p. 58] or Greene [4, p. 39] by a direct construction.) According to the theory of elliptic operators on compact manifolds (as presented in [12, Chap. 3] for example), each yg can be approximated in the C k topology by a finite linear combination of eigenfunctions for the Laplace operator on M. It follows that there exists a finite sum ni f,= Y, fij, j = l where each f~j is an element of some eigenspace V~j for the Laplace operator, such that i i tidy | -df~| <~/n, and hence Ng- dfi| llk< . i = 1 Since G acts on M via isometries it acts on each eigenspace V~ via a linear representation (as described in [17, p. 257]). Let Vii be the minimal G-invariant ni subspace of ~ V/j which contains f/, and define a s m o o t h equivariant m a p j = l Xi: M ~ V i * by (Xi(P))(f)=f(p) for f ~ V i. 126 J.D. M o o r e and R. Schlafly Here we are using the fact that Vii is a linear space of real-valued functions defined on M. We now construct a suitable G-invariant Euclidean metric on Vii*. By differentiation under the integral sign it can be shown that defined by P" Met ~ (M) ~ MetS (M) P(g) (P) = S (a* g) (p) da, G where d a is H a a r measure, is n o r m decreasing in each C k norm. We have P(dJi| ) = ~ a* (dfi| da, G where fi is regarded as a real-valued function on M. But f~ also determines a linear functional J~: Vi*~ IR which satisfies the equation f~o X i = f i . We give V~* the Euclidean metric if-- S a* (dJ~| da. G (This metric is positive definite because as a ranges t h r o u g h o u t G, a(fl) gen- erates V/.) With this choice of Euclidean metric on V/* it is easily verified that F(X,)=-P(df~| The Xi's fit together to give an equivariant mapping X = ( X 1 , . . . , X n ) into a Euclidean space of large dimension such that Jig- F(X) LIk = IIg- ~ F(X31Lk = []g--ZP(df~| I[g-Zdf~| This shows that g can be Ck-approximated by a realizable G-invariant metric and finishes the p r o o f of the main theorem. 4. Real Analytic Embeddings If the R i e m a n n i a n manifold M is real-analytic, then the embedding in the equivariant isometric embedding theorem may be chosen to be real-analytic. The p r o o f of this follows, as before, from the following two steps. Step1. Given an analytic perturbable X I ~ C ~ , p ( M , IE N) and an analytic g c M e t ~ ( M ) sufficiently close to F(X1) in the C ~~ topology, then there exists an analytic X ~ Co~ p(M, IE N) such that F ( X ) = g . Step 2. The analytic realizable metrics are dense in M e t e ( M ) . F o r the p r o o f of step 1, we follow the proof of the analytic isometric embedding theorem in Greene-Jacobowitz [-5]. We remark that their p r o o f also O n E q u i v a l e n t I s o m e t r i c E m b e d d i n g s 127 holds for c o m p a c t manifolds with b o u n d a r y . See also the m o r e general treat- m e n t by G r o m o v [6]. As before we use the N e w t o n iteration X , = X , _ I + L ( X , _ O ( g - F ( X , _ O ) only this time we don't have to use s m o o t h i n g operators. Each X , is evidently ff oo equivariant (i.e., X , CG, p(M, IEN)) and analytic. By extending everything to a complex analytic extension o f M and using C a u c h y estimates, G r e e n e and J a c o b o w i t z are able to show that X = lira X . n ~ o o exists, is real-analytic, and satisfies F ( X ) = g . Step 1 follows from this. If M has no b o u n d a r y , the p r o o f o f step 2 is exactly as before, because the eigenfunctions of A on M are real-analytic. If M has b o u n d a r y we can analyti- cally continue M to some slightly larger open R i e m a n n i a n manifold N. L e t M~ = {x e N [ dist(x, M) < e} for small enough e > 0 for this to be a c o m p a c t manifold with b o u n d a r y . We can suppose that the G-action has been analytically c o n t i n u e d to M~. N o t e t h at M e is real-analytic, and G acts on it by isometries. T h e eigenfunctions o f A o n M~ with Dirichlet b o u n d a r y conditions are real-analytic. F u r t h e r m o r e linear com- binations o f eigenfunctions (when restricted to M) are dense in Ck(M) for all k. T h e p r o o f of step 2 n o w proceeds exactly as before. 5. Embeddings into Other Spaces T h e examples in w 6 show that we c a n n o t d r o p the condition that M be c o m p a c t in o u r main theorem. However, we do have the following Theorem. Let G be a compact Lie group acting by isometries on some Riemannian manifold M, where M may have boundary and need not be compact. 7hen there exists a continuous orthogonal representation p on some real separable Hilbert space ~ and an isometric embedding from M into H which is equivariant with respect to p. Proof L e t f : M - M E N be an isometric embedding. L e t ~ be the direct sum o f N copies of LZ(G), i.e., = {~ = (~1, --., ~,)] each ~,eL2(G)}, where the integration is with respect to n o r m a l i z e d H a a r measure. Define a c o n t i n u o u s o r t h o g o n a l r e p r e s e n t a t i o n p o f G o n ~ by (p(rr)~)(6) = ~(6rr) a, 6 e G , 6 ' e ~ . 128 J.D. M o o r e and R. Schlafly F o r x ~ M and a~G define h: M ~ by h(x)(a) = f ( a x ) . h is obviously s m o o t h and injective. We claim that h is an equivariant isometric embedding, h is equivariant because for x e M and o, 6EG, h(crx)(6) = f (6 crx) =h(x)(6a) = {p(a) [h(x)] } (6-) SO h(ax) = p(a) h(x). If v ~ TM, then h,(v)(a) = (fo a), (v) = f , ( a , ( v ) ) SO I h , ( v ) ( ~ r ) l = I f , ( o ' , ( v ) ) l = 1 % ( 0 t = Ivl and ,e = 5 [h,(v)(a)12da G = ff Ivl 2d0- G = l v l 2. H e n c e h is isometric. This completes the proof. We n o w discuss embeddings into pseudo-Euclidean space, i.e., IR N with an indefinite metric. It is shown in the next section t h at the Poiucar6 n-disc does n o t have an equivariant isometric embedding into Euclidean space. H o w e v e r it does have an equivariant isometric embedding into the pseudo-Euclidean space (m "+1, d x ~ + . . + d x . ~ 2 = - 1 . �9 - d x . + l ) as the h y p e r b o l o i d x ~ + ... + x 2 - x.+2 1 L e t G be a compact Lie group acting s m o o t h l y o n some (not necessarily compact) manifold M. Recall that two orbits are said to be equivalent if the c o r r e s p o n d i n g isotropy groups are conjugate�9 An orbit type is an equivalence class of orbits�9 T h e t h e o r e m of M o s t o w and Palais cited earlier possesses a n o n c o m p a c t version due to M o s t o w [11]: M has an equivariant embedding into some Euclidean space if and only if the number of orbit types is finite. N o w suppose that M has a symmetric bilinear form g (such as a R i e m a n n i a n or p s e u d o - R i e m a n n i a n metric) and that the action o f G preserves g. T h e n using Mostow's t h e o r e m and a trick due to G r o m o v [6-1 we n o w prove: Theorem. M has an equivariant isometric embedding into some pseudo-Euclidean space if and only if the number of orbit types is finite. In particular, if M is compact then M has an equivariant isometric embedding into some pseudo- Euclidean space, Proof. In view of Mostow's theorem, it is sufficient to construct an equivariant isometric embedding from a given equivariant embedding X e C ~ , p ( M , IEN). O n Equivariant Isometric E m b e d d i n g s 129 According to w we can suppose that X has been modified to m a k e it perturbable. In the n o t a t i o n of w 2, let g , = ~ (~X ~ X i F ( X ) = L ~u~ui'~uJu ~ du du j be the metric on M induced by X, where (U 1 . . . . ,U n) are local coordinates o n M. Let T h e n and hence Y = L (X) (g - g') ~ C~, p(M, IEN). g -- g' = F ' ( X ) . L ( X ) ( g - g') = F ' ( X ) ( Y ) / ~ X ~ Y OX 3 Y \ =E 0. 7) du'a F ( X -]- Y) = 2 ( X @ Y). ~u j ( X -~- Y) dbl i dig j = F ( X ) + g - g ' + F ( Y ) = g + F ( Y ) . I f ] ' [ is the usual n o r m on 1R N then the m a p ( x + Y, Y): (M, g ) ~ 0 R N, I" 12) O ( ~ N, - I " 12) is an equivariant isometric embedding. 6. N o n e x i s t e n c e Theorems L e t G be a c o m p a c t Lie g r o u p acting on a R i e m a n n i a n manifold M by isometries. In this section we will show that, u n d er certain circumstances, M m a y not have any equivariant isometric embed d i n g into IE N. This means t h at M has no isometric embedding into IE N which is equivariant with respect to some h o m o m o r p h i s m from G to the isometry g r o u p o f IE N. However, an y h o m o m o r - phism G--,Isom(IE N) is conjugate to a h o m o m o r p h i s m G-~O(N)~Isom(IEN), so it is sufficient to consider isometric embeddings which are equivariant with respect to some o r t h o g o n a l r e p r e s e n t a t i o n of G. L e m m a . L e t { X > X 2 , X3} be a basis for ~u(2) satisfying [Xl,X2]=x3 [x2,x3o=xl [x~,xl]=x2. Then f o r any Lie algebra homomorphism p: ~ a ( 2 ) ~ u ( N ) and any v~l12 N we have 2 N - 1 Ip(X3)vl <~(Ip(XOvlZ +lp(Xe)vl2). 130 J.D. Moore and R, Schlafly Proof It is sufficient to prove this for irreducible representations. So suppose that p is an irreducible representation of dimension 2I+I<__N. Then p(X3) has eigenvalues - l l / ~ , ( - l + 1)1/-Z- 1 , . . . , l ] f ~- 1, 3 p(X~) 2 has the value - l(l + 1). i = l It follows that and Hence and the Casimir operator lp(Xa)vl 2 < 12 Ivl 2 3 lp(X~)vL 2 = l(l + 1)lvl z. i = 1 IP(Xl)Vl 2-/-[p(X2)vl 2 ~ llvl 2 ~ l f l ( X 3 ) v l 2. Theorem. Let G be a non-abelian compact Lie group (with Lie algebra g) acting smoothly on M. Suppose that for some point y e M , the derived map g ~ T y M is injective. 7hen for any integer N there exists a G-invariant metric on M such that M has no equivariant isometric embedding into IE s. Proof By hypothesis, there exist nonzero X 1, X 2, X3Eg satisfying [ X 1 , X 2 ] = X 3 [ X 2 , X 3 ] = X i [ X 3 , X 1 ] = X 2 . F u r t h e r m o r e the vectors ~ d t x i X i = ~ e " ylt=oe TyM are linearly independent. Given N, put a metric on {21, 2 2, X3} that satisfies 12312 > N---2-~l (12112 + 12212). Extend this to a G-invariant metric on M. N o w suppose we h a d an isometric embedding f : M-ME N which is equi- variant with respect to p: G ~ O ( N ) . Denote also by p the induced map g ~ o(N). Then f , (Xi) = p(Xi) f (y) and hence Ip(X3) f ( y ) t z = I f,(3;3)1 z =L){3] a N - 1 2 N - 1 2 which contradicts the lemma. > ~ ( 1 2 1 1 2 + 1 2 2 1 2 ) - - - ( I / , ( 2 0 7 + I f , ( 2 2 ) ? ) - - - (Lp(X1) f(y)l 2 + IP(X2) f (y)l z) On Equivariant Isometric Embeddings 131 Thus left-invariant metrics on SU(2) (or any o t h e r c o m p a c t non-abelian Lie group) require arbitrarily m a n y dimensions for an equivariant isometric embed- ding into Euclidean space. In general, if the manifold M is not c o m p a c t there m a y n o t be any equivariant isometric e m b e d d i n g of M into Euclidean space. T h e r e are examples of manifolds with c o m p a c t Lie g r o u p actions having infinitely m a n y o rb i t types. These manifolds do not even have equivariant embeddings into Euclidean space. A somewhat different example was p r o v i d e d by Bieberbach [1] w h o showed that the Poincar6 disc with its obvious circle action has n o equivariant isometric e m b e d d i n g into Euclidean space. T h e following is a mild generalization" Theorem. Let M be a complete simply-connected manifold with sectional curva- tures <--elVo(t)[ for all t e [ 0 , ~ ) . W e 132 easily c o m p u t e J.D. Moore and R. Schlafly s i n h ( ] / 7 0 IYo(t)l=lv'(o)l 1/~ F o r r e [ 0 , oo), let Ct: [ 0 , 2 n ] ~ M be the c l o s e d c u r v e given by C t ( ~ ) = e i~. y(t) 0 _ < ~ < 2 ~ . T h e l e n g t h o f C t is 2~lV'(t)l, so l e n g t h (Ct) > 2 h i V'(0)] s i n h ( l / T t ) F o r t h e sake o f o b t a i n i n g a c o n t r a d i c t i o n , s u p p o s e t h a t f : M ~ I E N is a n i s o m e t r i c e m b e d d i n g w h i c h is e q u i v a r i a n t with r e s p e c t t o ei~-~e ~B e i ~ S a , B E o ( N ) . T h e n the c u r v e f o C~ satisfies ( f o Ct)(a) = e ~B. f (y(t)) a n d h a s l e n g t h l e n g t h ( f o C~) = 2 n ] B f (7 (t))l < 2 n L]Bll (I f (P)I + t). B u t f is isometric, so c o m b i n i n g this with t h e a b o v e i n e q u a l i t y gives 2 n lV'(0)], sinh~l/~t) < 2 n lIB ]l (1 f (P) I + t) w h i c h is a c o n t r a d i c t i o n for large t. R e f e r e n c e s 1. Bieberbach, L.: Eine singularit~itenfreie Fl~iche konstanter negativer KriJmmung im Hilbertschen Raum. Comment. Math. Helv. 4, 248-255 (1932) 2. Bredon, G.E.: Introduction to compact transformation groups. London-New York: Academic Press 1972 3. 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