KkifhK jih = const., and Co(M) $ 1o(M), then M is isometric to a sphere, where Kkiih is the curvature tensor of M.

Transcription

1 ON RIEMANNIAN MANIFOLDS WITH CONSTANT SCALAR CURVATURE ADMITTING A CONFORMAL TRANSFORMATION GROUP* BY ENTARO YANO DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERELEY Communicated by S. S. Chern, January 3, Introduction.The following is a wellknown conjecture. A compact Riemannian manifold with constant scalar curvature admitting a oneparameter group of conformal transformations which is not that of isometries is isometric to a sphere. The results known up to now which support this conjecture are the following. We denote by Co(M) the largest connected group of conformal transformations of a Riemannian manifold M, and by Io(M) the largest connected group of isometries of M. THEOREM A (Yano and Nagano8). If M is a complete Einstein space of dimension n > 2 and Co(M) 5 Io(M), then M is isometric to a sphere. THEOREM B (Nagano5). If M is a complete Riemannian manifold of dimension n > 2 with parallel Ricci tensor and Co(M) 0 lo(m), then M is isometric to a sphere. This generalizes Theorem A. THEOREM C (Goldberg and obayashi'). A compact homogeneous Riemannian manifold M for which Co(M) $ lo(m) is a rational homology sphere. THEOREM D (Goldberg and obayashi2). If M is a compact homogeneous Riemannian manifold of dimension n > 3 and Co(M) $ Io(M), then M is isometric to a sphere. This is stronger than Theorem C. THEOREM E (Lichnerowicz4). If M is a compact Riemannian manifold of dimension n > 2, = const., jji = const., and Co(M) $ Io(M), then M is isometric to a sphere, where is the scalar curvature and it the Ricci tensor of M. This generalizes Theorem D. THEOREM F (Lichnerowicz,4 Yano and Obata9). If a compact Riemannian manifold M of dimension of n _ 2 admits an infinitesimal conformal transformation vh which is not an isometry: ggj = 2pgi,, p $ const., and if one of the following conditions is satisfied, then M is isometric to a sphere. (1) The vector field vh is a gradient of a scalar, (2) ihpi = kp", k being a constant, (3),1 = agii, a being a scalar field, where is the operator of Lie derivation with respect to Vh, gii the fundamental metric tensor, and ph the gradient of p. THEOREM G (Hsiung3). If M is compact, of dimension n > 2, = const., kifh jih = const., and Co(M) $ 1o(M), then M is isometric to a sphere, where kiih is the curvature tensor of M. This generalizes Theorem D. Now the assumptions = const. and jj1 = const. in Theorem E are equivalent to = const. and GjGJI = const., where 472

2 VOL. 55, 1966 MATHEMATICS:. YANO 473 Gji = gji. n The assumptions = const. and kjihk1ih = const. in Theorem G are equivalent to = const. and Zkjih Zkjih = const., where Zkj = k~ih n(n 1) (bkgji fjgki) is the socalled concircular curvature tensor. The purpose of the present paper is to prove first of all the following theorem. THEOREM 1. If M is compact orientable, of dimension n > 2, = const., and admits an infinitesimal conformal transformation 0h which is not an isometry: gji = 2pgjj p # const. such that fm GjipJpidV is nonnegative, dv being the volume element of M, then M is isometric to a sphere. As a corollary to this theorem, we can deduce the following theorem which covers Theorems E and G. THEOREM 2. If M is compact, of dimension n > 2, = const., and admits an infinitesimal conformal transformation vh which is not an isometry: gji = 2pgii p # const. such that (GjGJ1) = const. or (ZkjihZkjih) = const., then M is isometric to a sphere. In the next two sections we state some lemmas and integral formulas from which the theorems above follow by virtue of the following theorem. THEOREM H (Obata6). If a complete Riemannian manifold M of dimension n _ 2 admits a nonconstant function p such that VjVip = c2pgji, where Vj is the operator of covariant differentiation with respect to g9j and c a positive constant, then M is isometric to a sphere of radius 1/c in (n + 1)dimensional Euclidean space. 2. Lemmas. LEMMA 1. For a vector field vh in a compact orientable Riemannian manifold M of dimension n > 2, we have the integral formula: IM (gjivjvivh + ihvi + n2 VhViv) vhdv 2 ~~~~~~2 + 1/2 f (Vivi + Vi1v vtvgii)(vjv + Vvj Vv'g, dv = 0. n ~~~~n This can be verified by a straightforward computation (see, e.g., ref. 7). In the sequel we denote by M a compact orientable Riemannian manifold of dimension n > 2. LEMMA 2. If a nonconstant scalar field p in M satisfies Ap = kp, where Ap = glivvip, k = const., then the constant k is negative. 0 = J Ap2dV =2 [(Ap)p + gjipjpi]dv = 2 [kp2 + gjip pi]dv. LEMMA 3. If a scalar field p in M satisfies Alp = kp with a constant k, then we have

3 474 MATHEMATICS:. YANO PROC. N. A. S. giivjvph ihpi = kph. VhAp = Aph = gjiviph iphi. LEMMA 4. If a scalar field p in M satisfies Ap = kp with a constant k, then we have the integral formula: IM (~ + n 1kgi) p'pdv + IM (Vii kpgj) X VjP ± kpgii) dv = 0. Lemma 1 in which vh = ph and Lemma 3. LEMMA 5. For an infinitesimal conformal transformation v" in M: gj1 = 2pgji, we have kjih = SkVipi + 5V&kPi Vkphgji + V gkpogks Yji = (n 2)Vjpi Apgji, Y = 2(n 1)Ap 2p. We can verify these relations by a straightforward computation (see, e.g., ref. 7). LEMMA 6. For an infinitesimal conformal transformation vh in M: gjj = 2pgji, we have G = (n 2)(Vjp 1 I Pgji n~~~~~ Y Zkji = kv jpi + &kvpi VkPhgji + V jphgki + A (k~j J*ki) Lemma 5. LEMMA 7. If M has = const., and admits an infinitesimal conformal transformation Vh which is not an isometry: gf, = 2pgj,, p 5 const., then we have AP= n p, >0, and f pdv=o. The first follows from the last equation in Lemma 5, the second from Lemma 2, and the third from the first and the second. LEMMA 8. Under the assumptions the same as those in Lemma 7, we have I'M gpdp'dv = n 1 IM p2dv. 3.2I 1/2 Ap2 = (Ap)p jp\ + = iipjp n p 2 + gjjp p. 3. Integral Formulas.If Mhas = const., is of dimension n > 2, and admits

4 VOL. 55, 1966 MATHEMATICS:. YANO 475 an infinitesimal conformal transformation vh which is not an isometry: j, = 2pgj,, p 5 const., then we have G1ipp'pdV = [22jiGipl + 112P (GjiGJ')]dV. IM () Proof: First of all, we have,(gjigj) = 2( Gji)GJ' 4GjiGJ'p = 2 (n 2) V jpi gi) G' 4jiGJGp, by virtue of the first equation in Lemma 6, or 2 1 GjiVp'p G=GJ p YGGJ2 n2 2(n2) by virtue of gjigj' = 0. On the other hand, we have VJ(Gjipp') by virtue of VJGj, = 0, that is, = Gjjpipi + pgjivjp9, Vj(Gjipp') = Gipjp' n2 GjijGp2 2 (GjtO'). n 2 2(n 2) Thus, integrating this on whole M, we get the integral formula (1). Under the assumptions the same as those for integral formula (1), we have Proof: IA G,,p'p dv = [l/2zkjihzkjihp2 + 1/8 PY(ZkjihZ"Jih) Id V. (2) We have (ZkjihZkjih) 2 ( Zkji)ZkJih 4ZkjihZkjihp = 2[bkVjpi ajvkpi + Vkphgji Vjphgki hgji 5~g i) ]Zkj,, 4Zk jhzkiihp, by virtue of the second equation of Lemma 6, or GjiV'P' = 1/2ZkjihZkiihp '/8 (ZkiZkjih) by virtue of Zkjigkh = Gj, and Gj gji = 0. On the other hand, we have VJ(Gjippi) by virtue of VjGj, = 0, that is, V'(Gippi) = Gjipipi 1/2ZkjthZkjihp2 1/8P (ZkjihZk)ih Thus, integrating this on whole M, we get the integral formula (2). 4. Proof of Theorems.Proof of Theorem 1: Since = const., we have from Lemma 7, = Gjipjpt + pgjivjpi,

5 476 MATHEMATICS:. YANO PROC. N. A. S. AP = p, > 0. n 1 Thus, assuming M to be orientable without restricting any generality, we have from Lemma 4, 6' GjipjpidV + 6' (V'pi + l) JM JM \ P VjPi + n(n 1) /n(n 1 from which f GjipjpidV being nonnegative, g)itdv) =O n(n 1) Consequently, following Theorem H, M is isometric to a sphere of radius Vn(n 1)/. Proof of Theorem 2: Since = const., we have from Lemma 7 JopdV = 0. IM On the other hand, since (GjjGJ') = const., or (ZkihZkjih) = const., we have from integral formulas in section 3, or GjippidV = 2 GjiGJ'p2dV _ 0, I GjipJpzdV = 1/2 I ZkihZhjiZp2dV > 0, respectively, from which Theorem 2 follows by virtue of Theorem 1. * 1 Supported by National Science Foundation grant GP3990. Goldberg, S. I., and S. obayashi, these PROCEEDINGS, 48,2526 (1962); Goldberg, S. I., and S. obayashi, Am. J. Math., 84, (1962). 2 Goldberg, S. I., and S. obayashi, Bull. Am. Math. Soc., 68, (1962). 3 Hsiung, C.C., these PROCEEDINGS, 54, (1965). 4 Lichnerowicz, A., Compt. Rend., 259, (1964). Nagano, T., J. Math. Soc. Japan, 11, 1014 (1959). 6 Obata, M., J. Math. Soc. Japan, 14, (1962). 'Yano,., Theory of Lie Derivatives and Its Applications (Amsterdam: NorthHolland Pub. Co., 1957). 8 Yano,., and T. Nagano, Ann. Math., 69, (1959). 9 Yano,., and M. Obata, Compt. Rend., 260, (1965).