RIEMANNIAN MANIFOLDS ADMITTING A CONFORMAL TRANSFORMATION GROUP

Transcription

1 J. DIFFERENTIAL GEOETRY 2 (1968) RIEANNIAN ANIFOLDS ADITTING A CONFORAL TRANSFORATION GROUP KENTARO YANO & SUIO SAWAKI 1. Introduction The purpose of the present paper is to generalize some of the known results on Riemannian manifolds with constant scalar curvature admitting a group of nonisometric conformai transformations. Let be a connected Riemannian manifold of dimension n, and g Ji9 V u K kjί hy Kjt = Km* and K = K gi\ respectively, the positive definite fundamental metric tensor, the operator of covariant differentiation with respect to H the Levi-Civita connection, the curvature tensor, the Ricci tensor and the scalar curvature of, where and in the sequel the indices h, i, j,k, run over the range 1,, n. If we put (1.1) G* = X;* --Sϋ, n (1.2) Z w< * = K kj K Z w< K kji n(n 1) we have (1.3) ZtjS^Gjt, When admits an infinitesimal transformation v h, we denote by if the operator of Lie derivation with respect to v h. Thus, if admits an infinitesimal conformai transformation v h, we have (1.4) seg i1t = v^ + p t vj = 2pg su for a certain scalar field p. We denote the gradient of p by p t = P t p. For an infinitesimal conformai transformation v h in, we have [5] (1.5) sekw* = - δw jpi + dψ kpί - (1-6) JδfJfyt = - (n - Communicated October 19, 1967.

2 KENTARO YANO & SUIO SAWAKI (1.7) SeK= -2{n-\)Δp-2pK, where Thus, in with K = const, we have (1.9) Δp =-- p. n 1 We also have (1.10) < G n = - (n~2){f jpi - ~^ (1.11) 2 ι -V k p% t + V)p"g ki Thus, in with K = const, we have n (1.12) SPG lt = - (n - 2)[p jpt + w ( n^ υ pgjt]. We denote by C 0 () the largest connected group of conformal transformations of and by I 0 () that of isometries of. We first state here known results on Riemannian manifolds with K = const, admitting a conformal transformation group, and then try to generalize them. Theorem A (Lichnerowicz [3]). // is a compact Riemannian manifold of dimension n > 2, K = const., K jt K jί = const., and C 0 () Φ / 0 (), then is isometric to a sphere. Theorem B {Lichnerowicz [3], Yano & Obata [7]). // a compact Riemannian manifold of dimension n>2 with K = const, admits an infinitesimal nonisometric conformal transformation v h : Sfgji = 2pg ju p Φ const., and if one of the following conditions is satisfied, then is isometric to a sphere: (1) The vector field v h is a gradient of a scalar. (2) Kfp 1 = kp h, k being a constant. (3) g'kji = ag ju a being a scalar field. Theorem C {Hsiung [1]). If is compact and of dimension n > 2, K = const., K kjih K k J ih = const., and C 0 () Φ I Q (), then is isometric to a sphere. Theorem D {Yano [6]). // is compact orientable and of dimension n>2 with K == const., and admits an infinitesimal nonisometric conformal

3 ANIFOLDS ADITTING A TRANSFORATION GROUP 163 transformation v h : Seg n 2pg jt, p Φ const., such that [GjiptpW is nonnegative, dv being the volume element of, then is isometric to a sphere. Theorem E {Yano [6]). // is a compact and of dimension n > 2 with K = const., and admits an infinitesimal nonisometric conformal transformation v h : Jfgji = 2pgji, p Φ const., such that e(g si G^) = const, or &(Z kjih Z k J tk ) = const., then is isometric to a sphere. Theorem F (Hsiung [2]). Suppose that a compact Riemannian manifold of dimension n>2 with K = const, admits an infinitesimal nonhomothetic conformal transformation v h. If (1.13) a 2 &(Z kjih Z k J iλ ) + (2a + nb)b^g jt G^) = const., where a and b are constants such that (1.14) c = 4α 2 + 2(n - 2)ab 4- n{n - 2)fe 3 > 0, then is isometric to a sphere. To prove and generalize these theorems, we need the following Theorem G (Obata [4]). // a complete Riemannian manifold of dimension n > 2 admits a nonconstant function p such that (1.15) Pjΐip = - c*pg jt, where c is a positive constant, then is isometric to a sphere of radius \jc in (n + \ydimensional Euclidean space. We also need following integral formulas proved in [6]. If a compact orientable Riemannian manifold of dimension n > 2 with K = const, admits an infinitesimal nonhomothetic conformal transformation v h : Sέgji 2ρgj U p Φ const., then we have (1.16) (1.17) J GjtptftdV = J \± 2. Generalization of Theorem B, (2), (3) Theorem 2.1. If a compact orientable of dimension n > 2 with K = const, admits an infinitesimal nonhomothetic conformal transformation v h: &2ji = 2pgj U p Φ const., such that

4 164 KENTARO YANO & SUIO SAWAKI (2.1) seip^seg^ < o, then is isometric to a sphere. We need the following Lemma 2.1. // a compact orientable admits an infinitesimal conformal tsansformation v h : S^g όi = 2ρg Ji9 then we have (2.2) JpFdV = - I for any function F. Proof. Since p = F s v s, we have, by Green's theorem, n J = - 1 (vψ s FdV nj - - ι Γ SeFdV. Proof of the Theorem. Substituting (2.3) XiQjtGi*) into integral formula (1.16), we find (2.4) Consequently, by Lemma 2.1 and the assumption of the theorem, we have - / X ^ (jfig^gjjdv > 0. n\n I) J χ Thus is isometric to a sphere by Theorem D. Remark 2.1. Since (2.5) i n 2 the condition (2.1) of the theorem can be replaced by (2.6) <?{Z^h<?Z kji h) < 0.

5 ANIFOLDS ADITTING A TRANSFORATION GROUP 165 Remark 2.2. replaced by As the proof of the theorem shows, condition (2.1) can be (2.7) f(g^sfg Jt ) = λ, JλdV J < 0. The same remark applies to Theorems 4.1, 4.3, 5.1, 6.1, 6.2 and 6.4. Remark 2.3. Theorem 2.1 generalizes Theorem B, (2). In fact, using Kfp* = kp h, PjKJ* = 0, VjV t + F i v j = 2pgjι and P t v* n p, we have Fj(K»pv t ) = KiipjVt + = k Pi v ( + = kf t (pv<) - from which, by integration, - nk)(?dv = 0 and consequently k = K/n. Thus, from Ki h p l = kp h we have and consequently, by virtue of (1.10), = 0. Remark 2.4. Theorem 2.1 generalizes Theorem B, (3). In fact, from (1.6) and SeK μ = ag μ we find from which and consequently - («- 2)V jpi - a = 2(n that is, ifg^ = 0.

6 166 KENTARO YANO & SUIO SAWAKI Remark 2.5. If G^^G n = const., then (2.1) is automatically satisfied, but under our assumption the constant must be zero. In fact, making use of (1.3) and VjG^ = 0, from (1.10) we have and consequently by integration over we find / Thus, if G ji &Gji = const, the constant must be zero. 3. Decomposition of a conformal Killing vector Theorem 3.1. // a compact orientable of dimension n>2 with K = const, admits a conformal Killing vector field (3.1) v h = p h + q k, where p h is a Killing vector field and q h = V h q, q Φ const, is a gradient conformal Killing vector field, then is isometric to a sphere. Conversely, if a sphere of dimension n > 2 admits a conformal Killing vector field v h, then v h is decomposed into the form (3.1) where p h is a Killing vector field and q h a gradient conformal Killing vector field. Proof. Suppose that a compact orientable with K = const, admits a conformal Killing vector v h. Then we have (3.2) Seg n = V^vi + FiVj = 2pgji, and (3.3) Δp=? p. n l We note here that K is a positive constant [6], If v h is the sum of a Killing vector p h and a gradient conformal Killing vector q h = V h q, substituting (3.1) into (3.2), we find (3.4) PjPiq = pgji, from which (3.5) Δq^np. From (3.3) and (3.5), we find

7 ANIFOLDS ADITTING A TRANSFORATION GROUP 167 from which, by Bochner's lemma, + K q) = 0, n(n 1) / (3.6) H q = constant. n(n - 1) Substituting (3.6) into (3.4), we find VjVάq + c) = - K {q + c)g Jt, n(w 1) where c is a constant. Thus, # being not a constant, Λ/ is isometric to a sphere. Conversely, suppose that, isometric to a sphere, admits a conformal Killing vector v h. It is known that v h can be decomposed into where (3.7) ViV 1 = 0, ς Λ = F Λ g. From ^u = ϊ 7^* + ί 7^./ = 2pg H, we have (3.8) Γ f ; = F^, + FiP, + 2V ό V L q - 2^/: = 0. Forming Γ^Γ'*, we find (3.9) 4-4 (F, ί 7^ - ~ On the other hand, we have + WV*qW,Pt) = 0. because of

8 168 KENTARO YANO & SUIO SAWAKI or PΨ jpi = K Jt p<. K. Taking account of K όi = g jt we then have n = - ϊ = o. Thus from (3.9), by integration we find from which (3.10) ( J jj«ϋ) («qgή^v = 0 (3.11) ^ showing that p Λ is a Killing vector field and q h a gradient confoπnal Killing vector field. Remark 3.1. Theorem 3.1 generalizes Theorem B, (1). Remark 3.2. We can see in the following way the fact that a sphere admits a gradient confoπnal Killing vector field. Let (3.12) X Λ = X A {x h ), Σ χλχa = r 2 be the equations of Λ-dimensional sphere of radius r in an (n + ^-dimensional Euclidean space, where A = 1, - - -, n + 1. The equations of Gauss and those of Weingarten of the shpere are, respectively, (3.13) PjBS = -gjin-\ r and

9 ANIFOLDS ADITTING A TRANSFORATION GROUP 169 (3.14) FjN* = - - /, A where B t = V t X A and N A are components of the unit normal to the sphere. Considering a parallel vector field Bfu 1 + ocn A in the Euclidean space along the sphere, we have from which Fj&Su* + an A ) = 0, and consequently i- UjN A + B^FjW* + (Pja)N A - 5/ = 0, thus giving I r that is, We introduce here the notations: 4. Generalizations of Theorem E (4.1) f = G Ji G», g = Z khh Z*i»'. Theorem 4.1. // a compact orientable of dimension n > 2 with K = const, admits an infinitesimal nonhomothetic conjormal transformation v h such that (4.2) / and m being nonnegative integers, and a k and β k constants such that the I m sums 2 cc k and 2 βt are nonnegative and not both zero, then is isometric to a sphere.

10 170 KENTARO YANO & SUIO SAWAKI We need the following Lemma 4.1. // a compact orientable with K = const, admits an infinitesimal conformal transformation v h : Sfgμ = 2ρg ju then we have (4.3) for any function F and any nonnegative integer L Proof. Remembering that is, Λ- 1 r n- 1 we have, for any scalar field F, that is, Repeating the same process, we hence obtain (4.3). Proof of the theorem. We have, from (1.16) and Lemma 4.1, X

11 ANIFOLDS ADITTING A TRANSFORATION GROUP 171 We also have, from (1.17) and Lemma 4.1, jtptp'dv = jp 2 gdv + j From these equations, we have 2(β 0 + β βj + (β o + β ί β m )g)dv + 4-, and consequently, by Lemma 2.1, Thus, if the conditions of the theorem are satisfied, we have

12 172 KENTARO YANO 8c SUIO SAWAKI 3 Ji ptp t dv > 0, and consequently, by Theorem D, is isometric to a sphere. Theorem 4.2. Suppose that a compact orientable of dimension n > 2 with K = co/wί. satisfies (4.4) wλtfre <* 0, α l9 β 0, β 1 are nonnegative constants not all zero such that, if n > 6, (4.5) _i*l~ >(*6K>0, J ^ f t ^ Λ 1 n 1 // Λf admits an infinitesimal nonhomothetic conformal transformation v h : J&gn = 2^ i 9 0 ^fc constant, then is isometric to a sphere. To prove the theorem, we need the following Lemma 4.2. For a conformal Killing vector v h in, that is, for a vector field v h satisfying we heve n j + (4.6) Δ(SeF)^se(ΔF) + 2pΔF^n - 2)pΨ i F for any scalar field F. Proof. Since v Λ is a conformal Killing vector field, we have (4.7) giψjviv*- + KW + {n - 2)p* = 0, (see [6] for example). We also have, for an arbitrary scalar field F, (4.8) g^v 5 ViV h F - K k Ψ t F = F h {ΔF). Thus we have Δ(&F) = and consequently, by using (4.7) and (4.8), -(n. that is,

13 ANIFOLDS ADITTING A TRANSFORATION GROUP 173 J(jS?F) = &ΔF + 2pΔF - (Λ - 2)pΨ h F. Lemma 4.3. For any scalar field F and a scalar field p satisfying Δp = kp, k being a constant, in a compact orientable we have (4.9) CppΨ k FdV = - ± J Z (4.10) Cp%ΔF)dV = 2kC P ΨdV + Proof. Integral formula (4.9) follows from VKp*V h F) = 2ppΨ h F + p 2 ΔF by integration. On the other hand, we have which proves (4.10). Proof of the theorem. jp\δf)df = J(Δp 2 )FdV = 2J(pΔp + pip^fdv = 2k f (?FdV + 2 Γ pφψdv, J J AT ΛΓ From (1.16), (4.3), (4.6) and (4.9), we find n-2 2 Thus, we have

14 174 KENTARO YANO & SUIO SAWAKI n 2 n 2 2 Γ i -" 2 C ipw - J^/dF + l fptffdv, itfάv \ p 2 fdv -\- ΓρS \ n ~ Similarly, we find < /» iv = fp'gdv + IJpSfgdV, X If X X From the above four equations, we obtain (4.11) * (n _!)(«+2) -fpxa'jf C» lnlλ4^ + V b'δg)dv, / 8/s: - bg-^lb'δg\\dv a, a', b, b' being nonnegative constants. Now we choose α, a', b, b' in such a way that we have (4.12) a o a, «x Then we have, from (4.4), (4.13) af - ILzA-a'Δf + bg - w ~ 2 ^^g = c (const.) and

15 ANIFOLDS ADITTING A TRANSFORATION GROUP 175 and consequently, from (4.11), a'δf + b'δg = ± (af + bg) - -~, n 1 n 1 jz!_(a + a') + 2(6 + ί a')f +(b + b')g that is, (α/ +, g) OΛ 1/2 1 Λ 1 (4.14) $P 2 [{* a ' ( n 6)α}/ + {86' - (n - 6)ft}* + (π Now, constants are both nonnegative for n < 6. Since 8α' - (n - 6)a, δfc 7 - (n - 6)6 8α' - (n - 6)α = δ A Λl - (n - 6)α 0, w 1 8d' - ( R - 6)fr = -_ ft - («- 6)^0, /I I they are nonnegative also for n :> 6 by virtue of the assumption. oreover, we have, from (4.13), ajfdv + bjgdv = cjdv, and consequently c is nonnegative. Thus we have, from (4.14), and consequently, by Theorem D, is isometric to a sphere.

16 176 KENTARO YANO & SUIO SAWAKI Theorem 4.3. // a compact orientable of dimension n > 2 with K const, admits an infinitesimal nonhomothetic conformal transformation v h such that (4.15) XXiaj + ctjf + βog + β.δg) < 0, <*OJ oc l9 β 0, β 1 being constants not all zero such that (4.16) 4(n ~ 1) 4 ( a 0 >(n + 6)a 1 >0, n ~ l) β 0 > (n + 6)ft > 0? K. K. then hi is isometric to a sphere. To prove this theorem, we need the following Lemma 4.4. // a compact orientable of dimension n > 2 with K = const, admits an infinitesimal nonhomothetic conformal transformation v h, then (4.17) 2 f p >fdv - JL=j-L (4.18) * Proof. From (1.16), we have π > 1 Substituting p = J/? into the last term of the second member of K. this equation, we find *7 4K and consequently, by (4.6), if

17 ANIFOLDS ADITTING A TRANSFORATION GROUP ^Λ Γ Thus by (4.9) and (4.10) with k = =Jp*fdV - ±^L -, we find n 1 We can similarly prove (4.18). Proo/ of the theorem. We first write down (1.16), (4.17), (1.17) and (4.18): =f(?fdv + λ = fp*gdv + λ J from which we obtain

18 '78 KENTARO YANO & SUIO SAWAKI!L=jL(a _ a') + 2(b - b')\ [G^pipHV 2 ) J = l{4a - {n + 6)a'} \ fidv +l{4b-(n + 6)b'} 4 J 4 or by Lemma 2.1, = l{4α - (n + 6)α'} JV/dK + 1{46 - («+ 6)6'} (4.19) * je#af + ^ Δ i + bg α, α', b, b f being constants. Now we choose these constants so as to have (4.20) a Q = a, * 1 =.ZL=J-β', β 0 = b, βl = llz_±b'. Then from (4.16) we find 4a ~ (w + 6)Λ' > 0, ^ > 0, 4^, _ ( W + 6)^r > 0, Z? r > 0, and and consequently 4(α - a') > (π + 2K > 0, 4(fe - *0 > (Λ + 2W > 0, n - 2 (β - aπ + 2(* - 60 > 0, the equality sign occurring when and only when a = a! = b = 6' = 0, that is, α 0 = αi = ft = ft = 0.

19 ANIFOLDS ADITTING A TRANSFORATION GROUP 179 Thus from the assumption and (4.19), we have jipiptdv > 0, and consequently, by theorem D, is isometric to a sphere. Remark 4.1. If &{.ccts + a^f + βog + β 1 Jg) = constant, then (4.15) is automatically satisfied. But if S h = constant for a scalar field h in a compact space, the constant must be zero, because h attains an extreme value at a certain point of the space at which Jί?h = vψji = 0. The same remark applies to Theorems E, 4.1, 5.1, 6.1, 6.2 and A theorem similar to that of Hsiαng To obtain Theorem F, Hsiung [2] used the tensor but we would like to use here the tensor (5.1) W kjih = az kjih + _*_(g AΛ σ i4 - g jh.g ki + G kh8ji - G jh8ki ) n 2 a and b being constants. It is easily seen that (5.2)»W** = <α + b)g μ, and that, when a + b = 0, (5.3) W kjih = ac km, where 2 + (H _ iχ w _ 2 ) is the covariant Weyl conformal curvature tensor. In general, we have

20 180 KENTARO YANO & SUIO SAWAKI (5.5) W k Ww = c?zzw + 4 ( 2 a n 2 and for the case a + b = 0 we have (5.6) W t Using the tensor W kjίh defined above we can obtain Theorem 5.1. Suppose that a compact orientable of dimension n>2 with K = const, admits an infinitesimal nonhomothetic conjormal transformation v h. If (5.7) XX(W* iih WW) < 0, or equivalently, (5.8) (n - 2)a*&&{Z kjih ZW) + 4(2a + tybxxipjig'*) < 0, a and b being constants such that a + b = 0, is isometric to a sphere. To prove this theorem, we need the following Lemma 5.1. For an infinitesimal conformal transformation v h in : = 2ρg ju we have h = 2apZ kjuι + {g kh Gji g jh G ki + G kh gji G jh g ki ) n 2 (5.9) - (Λ + 2(α + b) n Proof. This follows from (1.10), (1.11) and (5.10) &Z kjih = X{Z kji 'g ιh ) = <jez ki f)g ih + 2pZ kjih. Lemma 5.2. For an infinitesimal conformal transformation v h in : = 2pg JU we have (5.11) (&W kίih }wv ih = 2pW kjih W*'«- 4(a + bfg Proof. This follows from (5.5) and (5.9). Lemma 5.3. For an infinitesimal conformal transformation v h have in, we (5.12) &{W kλth WW*) = - 4pW khh WW* - 8(α Proof. This follows from (5.11) and - SpW kjih W** ih.

21 ANIFOLDS ADITTING A TRANSFORATION GROUP 181 Lemma 5.4. Tor an infinitesimal conformal transformation v h K = constant, we have in with (5 13) 8(α = 8(α Proof. This follows from V j G jt = 0 and (5.12). Lemma 5.5. // a compact orientable of dimension n > 2 with K = const, admits an infinitesimal nonhomothetic conformal transformation v h, then we have (5.14) Proof. This follows from (5.13) by integrating both sides over and using Lemma 2.1. Proof of the theorem. If &&QV kjih W** ih ) < 0, and a + b Φ 0, then from (5.14) we have > 0. Thus by Theorem D, is isometric to a sphere. 6. Characterizations of conformaliy flat spaces Theorem 6.1. // a compact orientable of dimension n > 3 admits an infinitesimal conformal transformation v h : &gji = 2pgji such that p does not vanish on any n-dimensional domain and (6.1) se<eh < o, h = c kjih c^ih, then is conformaliy fiat. Proof. ultiplying (5.12), with a + b = 0, by p and integrating the resulting equation over, we find or by Lemma 2.1, 0 = 4 Cp 2 hdv + Jp&hdV, id H

22 182 KENTARO YANO 8c SUIO SAWAKI (6.2) (6.2) implies 0 = 4 Cffhdv - [sesehdv. J n J JfhdV < 0, from which p 2 h = 0, or by the assumption of the theorem, h = 0, that is, C kjih = 0, which shows that is conformally fiat. Remark 6.1. If S h = constant in a compact space, we have &h = 0. On the other hand, if S h = 0 in a general Riemannian space, from S h H- 4ph = 0 we find h = 0, which shows that the space is conformally flat. Theorem 6.2. Under the same assumptions as in Theorem 6.1, // K = const, and (6.1) is replaced by (6.3) 2\za k (- -5LJZ-L) k ΔK&h)j < 0, 2 ing α nonnegative integer and a k constants such that 2 a k > 0, ίλen fe conformally flat. Proof. Similarly, as in the proof of Theorem 4.1 we can obtain 0 = 4 Γ(α 0 + a x adfhdv or, by virtue of Lemma 2.1, 0 = 4 j(a 0 + «! (6.4) - 1 J\s?(ob^A + «i(- iz-l) J(^A) + α 0, «!,»««bein S constants such that 2 α Λ > 0. Thus by (6.3), we have fc-0 Γ' ffhdv = 0, from which h = 0 and consequently C ΛjiΛ = 0.

23 ANIFOLDS ADITTING A TRANSFORATION GROUP 183 Theorem 6.3. Under the same assumptions as in Theorem 6.1, // K = const, and (6.1) is replaced by (6.5) a o h a x Δh c (constant), <x 0 and a x being positive constants such that (6.6) - ^ r *i X" - 6 > > 0, for n > 6, w 1 //ze«w conformally flat. Proof. Similarly, as in the proof of Theorem 4.2 we can obtain (6.7) 0 = JV[{8α" - (n - 6)α}/z + (π + 2)c]rfK. Now, the constant 8a 7 (n 6)α is positive for Λ < 6. Since 8α' - (Λ - 6)α = - ^ - ^ - (it - 6K, n 1 by (6.6) this constant is also positive for n > 6. On the other hand, from (6.5) we have aλhdv = c\dv, which shows that c is a nonnegative constant. Thus from (6.7) we see that h = 0 and consequently C kjiiι = 0. Theorem 6.4. Under the same assumption as in Theorem 6.1, if K = const, and (6.1) is replaced by (6.8) XSedxJi + α Jft) < 0, α 0 and <*! being constants such that (6.9) 4(n ~ Ό «0 >(ιi + 6) Λl > 0, then is conformally flat. Proof. Similarly, as in the proof of Theorem 4.3 we can obtain 0 = {4a - (it + 6)α'} (fhdv - [sestiah + n ~ 1 a'δh\dv n X K (6.10) * ί I + ( "-^W + 2) J PiP <(a>h)dv,

24 184 KENTARO YANO & SUIO SAWAKI a and a! being constants. Now we choose these constants such that (6.11) cc o = a, aι = ILz±a'. K Then from (6.9) we have K 4a - (n + 6)a' = 4α 0 - (π + 6) a x > 0. n 1 We also have Waft + / ϊ " 1, a'δh\ = Sf(ajι + A) = constant. Thus, from (6.10), we have h = 0 and consequently C kjih = 0. Bibliography [ 1 ] C. C. Hsiung, On the group of conformal transformations of a compact Riemannian manifold, Proc. Nat. Acad. Sci. U.S.A. 54 (1965) [ 2 ], On the group of conformal transformations of a compact Riemannian manifold. II, Duke ath. J. 34 (1967) [ 3 ] A. Lichnerowicz, Sur ϊes transformations conformes d'une variete riemannienne compacte, C. R. Acad. Sci. Paris 259 (1964) [4]. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. ath. Soc. Japan, 14 (1962) [ 5 ] K. Yano, The theory of Lie derivatives and its applications, North-Holland, Amsterdam, [6], On Riemannian manifolds with constant scalar curvature admitting a conformal transformation group, Proc. Nat. Acad. Sci. U.S.A. 55 (1966) [7 ] K. Yano, &. Obata, Sur le groupe de transformations conformes d'une variete de Riemann dont le scalaire de courbure est constant, C. R. Acad. Sci. Paris 260 (1965) TOKYO INSTITUTE OF TECHNOLOGY NΠGATA UNIVERSITY