Bochner曲率テンソルが消えるケーラー多様体について
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(2) Journal of Hokkaido University of Education (Section II A) Vol. 31, No. 1 BgfQ 55 -^ 9 ^. ^•^it^t^^^S (^ 2 @RA) H 31 ^ ^ 1 -^ September; 1980. Remarks on Kaehlerian Manifolds with Vanishing Bochner Curvature Tensor. Izumi HASEGAWA and Toshiyuki NAKANE Mathematics Laboratory, Sapporo College, Hokkaido University of Education,. Sapporo 064. :^J?H • >W ^c^ : Bochner?^T>y^^m^&^--7-^^?^^^T. »»w^^L??wa^ Abstract Let At be a Kaehlerian manifold and let B={Bhuk} be its Bochner curvature tensor. If M is of constant holomorphic sectional curvature, we have. {^) B=0. There is a question: Under what additional condition does this condition ( * ) imply that M is of constant holomorphic sectional curvature ?. In this paper, we offer to prove the best possible pinching theorem on the length of the Ricci tensor for Bochner-Kaehlerian manifold (i.e.., Kaehlerian manifold satisfying the condition (*)) with constant scalar curvature to be of constant holomorphic sectional curvature.. § 1. Statement of results Recently, S. I. Goldberg and M. Okumura [3] proved THEOREM A. Let M. be an n-dimensional compact conformally flat Riemannian manifold with constant scalar curvature R. If the length of the Ricci tensor is less than R/Vn—1, n s^3, then M is a space of constant curvature.. For a Kaehlerian manifold, Y. Kubo [6] proved THEOREM B. Let M. be a real n-dimensional Bochner-Kaehlerian manifold with constant. scalar curvatnre R. If the length of the Ricci tensor is not greater than R/\/n—2, n ^4, then M. is a space of constant holomorphic sectional curvature.. Note that the square of the length of the Ricci tensor is greater than or equal to R2/n, so the Ricci tensor has been "pinched" The inequality in Theorem A is the best possible. But the inequality in Theorem B is less than perfect. So we improve the inequality in Theorem B and obtain the following theorems.. (1.
(3) IZUMI HASEGAWA and Toshiyuki NAKANE. THEOREM 1. Let M be a real n(>4:)-dimensional Bochner-Kaehlerian manifold with constant scalar curvature R. If the square of the length of the Ricci tensor is less than {n3-—2n2. +32)7?2/(%—4)2(%+2)2 then M is of constant holomorphic sectional curvature. THEOREM 2. Let M be a real ^-dimensional Bochner-Kaehlerian manifold. If the scalar curvature is non-zero constant, then M is of constant holomorphic sectional curvature.. §2. Bochner-Kaehlerian manifolds Let M be a real %(^4)-dimensional Kaehlerian manifold. Then the Riemannian metric gij and the almost complex structure Jih satisfy the following equations: (2.1) JiaJah=-Sf, ^/,a//=^,,, F,/,/l=0, ^gi,=0, where V k denotes the operator of co variant differentiation with respect to gij. Let Rhu be the Riemannian curvature tensor and put Rij'.=Raija (Ricci tensor), R:= gabRab (scalar curvature) and Hij: = /,a Ra.,-. We define the Bochner curvature tensor of M [8] by I. 2) Bhijk- = Rhijk~ ._ T ^ {Rij ghk~ RikghJJT gij R hk~ gik RhjJT H ijj hk~ HikJ hjJT Jij H hk — Jih Hhj — 2HhiJjk ~ 2fhiHjk)+ i „ i ov „ , ^\ (<grt'j g'Aft — gikghj + Jijjhk ~ Jikjhj. ~2JhiJjk). M is called a Bochner-Kaehlerian manifold if the Bochner curvature tensor vanishes.. By the straightforward computation, we have LEMMA 1 [8]. If a Bochner-Kaehlerian manifold M is an Einstein one, then M is of constant holomorphic sectional curvature.. On the other hand, Y. Kubo [6] proved LEMMA 2. If a Bochner-Kaehlerian manifold M. has the constant scalar curvature, then we have (2.3) n(n+2)RabRbcRca-2{n+l)RRa,,Rab+R3=0.. § 3. Proofs of theorems We define the Einstein tensor Su by ij- =~ Kij ~j^gij. Then we have Siafa3 =JiaSaj since RiaJaj=JiaRaj. Moreover we see that (3.1) trace S:=Saa=0, (3.2) trace S2:=Sai,Sab= RabRab —^^0, (3.3) trace 53:= Sab S,c Sca= Rab Ri,c Rca-^RSa,Sab--1, R3. /yi. 2). yi.
(4) Kaehlerian Manifolds with Vanishing Bochner Curvature Tensor M is an Einstein space if and only if trace S2 vanishes,. Substituting (3. 2) and (3. 3) in (2. 3), we have (3.4) n{n+2)trace 53+(^+4 )R trace S2=0. M. Okumura [7] proved LEMMA 3. Let a, i= 1,2, •••, m, be real numbers satisfying m. m. 1=1. 1=1. 2 d=0 and 2 c?=A;2 (k^O).. Then we have m.~2 ^ A-3<S c3< ,w-^—^3. /m(m—l) " - ^ -- v/m(m—l). We put f2:=trace 52(/^0). From the commutativity of 5/ and Ji, we see that every characteristic root of S, is multiple one. Combining this fact with Lemma 3, we have (3. 5) —=^=4=^f3^ trace S3^ ^n~^—fs /2%(%-2) -/ ~ " m" " - V2n(n-2) J '. Applying the above inequality, (3. 4) yields the following inequality: -zf n+4 ^ n-4: ^^^ ^ %+4 p, n-4. fz\n{n+2) R~^2n(n-2} /^0^/21 n(^2) 7?+/2^2T/tUnder the assumption of Theorem 1, we see that /2=0, that is, M is a Einstein space. Therefore, from Lemma 1, Theorem 1 has been proved. If dim M. is 4, we see that trace 53=0 since the characteristic roots of S/ are K,K, -/c and —ic. From (3. 4), we have R trace 52=0. Therefore, if R is non-zero constant, then M. is an Einstein space, whence M. is of constant holomorphic sectional curvature. Thus Theorem 2 has been proved.. § 4. Exaples of the Bochner-Kaehlerian manifolds Let Mi be a real (w—2)-dimensional (%^4) Kaehlerian manifold of constant holomorphic sectional curvature c{4:0) and Mz be a real 2-dimensional Kaehlerian manifold of constant holomorphic sectional curvature —c. The product manifold M=MiXMz is a typical example of Bochner-Kaehlerian manifold with constant scalar curvature which is not of constant holomorphic sectional curvature. Moreover we have. (4.1) ^<"-W"+2)_, and ,3_9^2.. (4.2) Ra.Rab= n~~^^d.
(5) IZUMI HASEGAWA and Toshiyuki NAKANE. Therfore, if n > 4. we obtain 'ab _ n Ln ~t-6z, p2. RabRaD=(n-mn+2Y ^ This shows that the inequality in Theorem 1 is the best possible. If w=4, scalar curvature of M is zero constant. This shows that there exists a real 4-. dimensional Bochner-Kaehlerian manifold with constant scalar curvature 0 which is not of cnostant holomorphic sectional curvature.. References. [1] S. I. Goldberg:0n conformally flat spaces with definite Ricci curvature, Kodai Math. Sem. Rep., 21(1969), 226232.. [2] S.I. Goldberg:0n conformally flat spaces with definite Ricci curvature H, Kodai Math. Sem. Rep.,27(1976), 445-448. [3] S.I. Goldberg and M. Okumura:Conformally flat manifolds and a pinching problem on the Ricci tensor, Proc. Amer. Math. Soc., 58(1976), 234-236. [4] I. Hasegawa and T. Nakane:0n Sasakian manifolds with vanishing contact Bochner curvature tensor, to appear. in Hokkaido Math. J. [5] I. Hasegawa and T. Nakane:0n Sasakian manifolds with vanishing contact Bochner curvature tensor II, in manuscript.. [6] Y. Kubo:Kaehlerian manifolds with vanishing Bochner curvature tensor, Kodai Math. Sem. Rep., 28(1976), 8589. [7] M. Okumura:Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math., 96. (1974), 207-212. [8] S. Tachibana:0n the Bochner curvature tensor, Natur. Sci. Rep. Ochanomizu Univ., 18(1967), 15-19.. (4).
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