SASAKIAN MANIFOLDS WITH CYCLIC-PARALLEL RICCI TENSOR

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1 Bull. Korean Math. Soc. 33 (1996), No. 2, pp SASAKIAN MANIFOLDS WITH CYCLIC-PARALLEL RICCI TENSOR SUNG-BAIK LEE, NAM-GIL KIM, SEUNG-GOOK HAN AND SEONG-SOO AHN Introduction In a Sasakian manifold, a C-Bochner curvature tensor is constructed from the Bochner curvature tensor in a Kaehlefian manifold by the fibering of Boothby-wang[2]. Many subjects for vanishing C-Bocher curvature tensor with constant scalar curvature were studied in [3], [6], [7], [9], [10], [11] and so on. One of those, done by Choi, Ki and Takano([3]), asserts that the following: THEOREM A. Let M n (n 5) be a Sasakian manifold with vanishing C- Bochner curvature tensor. Then the scalar curvature R is constant if and only if TrRic (m) is constant for an integer m( 2). Further, they proved THEOREM B. Let M n (n 5) be a Sasakian manifold with vanishing C- Bochner curvature tensor. If TrRic (m) is constant for a positive integer m and if the length of the η Einstein tensor is less than 2(R n+1) (n 1)(n 3),thenMis a space of constant φ-sectional curvature, when R is the scalar curvature of M. In [7], we see that the inequality with respect to the η-einstein tensor in Theorem B is the best possible. Received December 1, AMS Subject Classification: 53C15. Key words and phrases: Sasakian manifold, cyclic-parallel, C-Bochner curvature tensor. This paper was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation, Supported by TGRC-KOSEF, and partially supported by BSRI , the Korea Ministry of Education

2 Sung-Baik Lee, Nam-Gil Kim, Seung-Gook Han and Seong-Soo Ahn The purpose of this paper is to develope the above Theorems A and B. At first in section 2, we prove that THEOREM 1. Let M n (n 5) be a Sasakin manifold. Then the C-Bochner curvature satisfies r B kji r and TrRic (m) is constant, for a positive in if and only if the Ricci tensor is cyclic-parallel, where B kji h is a C-Bochner curvature tensor. We remark that according to [4] and [12] there exists a Riemannian manifold whose Ricci tensor is cyclic-parallel but not parallel. Also, it is easily seen that a Sasakian η-einstein manifold is of cyclic-parallel Ricci tensor but is not in general Ricci-parallel. In section 3, we prove THEOREM 2. Let M n (n 5) be a Sasakian manifold with parallel C- Bochner curvature tensor. If TrRic (m) is constant for a positive integer m and if the length of the η-einstein tensor is less than 2(R n+1) (n 1)(n 2),thenMis an η-einstein manifold. From Theorem 2, we have immediately Theorem B as a corollary. 1. Preliminaries Let M be an n-dimensional Riemannian manifold. Throughout this paper, we assume that manifolds are connected and of class C. Denoting respectively by g ji, R kji h, R ji = R rji r and R the metric tensor, the curvature tensor, the Ricci tensor and the scalar curvature of M in terms of local coordinates {x h }, where Latin indices run over the range {1, 2,,n}. An n-dimensional Riemannian manifold is called a Sasakian manifold if there exists a unit Killing vector field ξ h satisfying (1.1) { ηi = g ir ξ r,φ ji = j η i, φ ji +φ ij = 0, φ r h ξ r = 0, φ j r η r = 0, φ i r φ r h = δ h i +η i ξ h, k φ ji = g kj η i + g ki η j, where denotes the operator of the Riemannian covariant derivative. It is well known that in a Sasakian manifold the following equations hold; (1.2) R jr ξ r = (n 1)η j, R kjir ξ r = η k g ji η j g ki, 244

3 Sasakian manifolds with cyclic-parallel Ricci tensor (1.3) H ji + H ij = 0, (1.4) R ji = R rs φ j r φ i s + (n 1)η j η i, (1.5) k R ji j R ki = ( s R kr )φ j r φ i s η j {H ki (n 1)φ ki } 2η i {H kj (n 1)φ kj }, (1.6) k R ji ( k R rs )φ j r φ i s = η i {H kj (n 1)φ kj } η j {H ki (n 1)φ ki }, (1.7) ξ r r R kji h = 0. where we put H ji = φ j r R ri. We denoted a tensor field W (l) with components W ji (l) and a function W (l) as follow; W ji (l) = W ji1 W i2 i1 W i i i 1, W (l) =TrW (l) = g ji W ji (l). Also we define the η-einstein tensor T ji by (1.8) T ji = R ji ( ) ( ) R R n 1 1 g ji + n 1 n η j η i. If the η-einstein tensor vanishes, then M is called an η-einstein manifold. From (1.2) and (1.3), we have (1.9) TrT = 0, (1.10) T jr ξ r = 0, (1.11) T jr φ i r + T ir φ j r =

4 Sung-Baik Lee, Nam-Gil Kim, Seung-Gook Han and Seong-Soo Ahn A Sasakian manifold M is called a space of constant φ-holomorphic sectional curvature c if the curvature tensor of M has the form: R hji h = c +3 4 (g jiδ k h g ki δ j h ) + c 1 4 (g kiη j ξ h g ji η k ξ h +η k η i δ j h η j η i δ k h φ ki φ j h + φ ji φ k h 2φ kj φ i h ). Matsumoto and Chuman ([10]) introduced the C-Bohner curvature tensor B kji h defined by (1.12) B kji h =R kji h + 1 n +3 (R kiδ j h R ji δ k h + g ki R j h g ji R k h + H ki φ j h H ji φ k h +φ ki H j h φ ji H k h +2H kj φ i h +2φ kj H i h R ki η j ξ h + R ji η k ξ h η k η i R h j +η j η i R h j ) k +n 1 n +3 (φ kiφ h j φ ji φ h k +2φ kj φ h i ) k 4 n +3 (g kiδ h j g ji δ h k ) + k (n +3) (g kiη j ξ h g ji η k ξ h +η k η i δ h j η j η i δ h k ), where k = R+n 1. It is well-known that if a Sasakian manifold with vanishing n+1 C-Bochner curvature tensor is an η-einstein manifold, then it is a space of constant φ-holomorphic sectional curvature. By a straightforward computation, we can prove (1.13) n + 3 n 1 r B r kji = k R ji j R ki η k {H ji (n 1)φ ji } +η j {H ki (n 1)φ ki }+2η i {H kj (n 1)φ kj } 1 + 2(n +1) {(g ki η k η i )δ r r j (g ji η j η i )δ k +φ ki φ j r φ ji φ k r +2φ kj φ i r }R r, where we put R j = j R. 246

5 Sasakian manifolds with cyclic-parallel Ricci tensor 2. Sasakian manifold with cyclic-parallel Ricci tensor The Ricci tensor Ric of a Riemannian manifold is said to be cyclic parallel if C Ric = 0, namely k R ji + j R ik + i R kj = 0. From this and the second Bianchi identity it is easily seen that the scalar curvature R of the manifold is constant. Suppose that the Ricci tensor of a Sasakian manifold M is of cyclic-parallel. Then by the Ricci formula for R ji,wefind (2.1) k k R ji = 2(R kjih R kh R ji (2) ) because the scalar curvature of M is constant. On the other hand, (1.6) is reduced to k R ji +( r R ks )φ j r φ i s + ( s R kr )φ j r φ i s = η i {H kj (n 1)φ kj } η j {H ki (n 1)φ ki }, which together with (1.5) implies that 2 k R ji j R ki + ( r R ks )φ j r φ i s Thus, it follows that we obtain =3{H jk (n 1)φ jk }η i +2{H ik (n 1)φ ik }η j. (2.2) k R ji = H jk η j + H ik η j + (n 1)(φ kj η i +φ ki η j ), where we have used (1.6) and hence the right hand side of (1.13) vanishes identically since the scalar curvature of M is constant. If we apply R ji(m) to (2.2) and make use of (1.1) and (1.2), the we also see that R (m) is constant for any integer m 2. Summing up, we have LEMMA 1. For a Sasakian manifold with cyclic-parallel Ricci tensor, we have r R kji r = 0. Futhermore TrRic (m) is constant for any positive integer m. Let M be an n( 5)-dimensional Sasakian manifold with r B kji r = 0and TrRic (m) is constant for a positive m. By (1.1), (1.2), (1.5) (1.7) and (1.13), 247

6 we then obtain Sung-Baik Lee, Nam-Gil Kim, Seung-Gook Han and Seong-Soo Ahn (2.3) k R ji ={R kr (n 1)g kr }(φ r j η i + φ r i η j ) 1 + 2(n + 1) {2R k(g ji η j η i )+ R j (g ki η k η i ) +R i (g kj η k η j ) φ kj φ r i R r φ ki φ r j R r }. Applying R ji(m) to (2.3) and owing to (1.1) (1.3) and (1.7), we easily verify that (n + 1) k R (m+1) = (m + 1){2R kr R r + (R (m) (n 1) m )R k }. From Theorem A, we see that the scalar curvature R is constant, this R (m) is constant for any integer m 2. Therefore (2.3) becomes (2.2) and hence the Ricci tensor of M is of cyclicparallel. Consequently, together with Lemma 1, we have proved Theorem Parallel C-Bochner curvature tensor Applying k to (2.2) and owing to (1.1) and (1.2), we get k k R ji = 2{R ji (n 1)g ji Rη j η i n(n 1)}η j η i }. Combining this with (2.1), we obtain (3.1) R kjih R kh = R (2) ji R ji +(n 1)g ji +{R n(n 1)}η j η i. On the other hand, because of (1.1) and (1.2) and (2.2), it is clear that R kjih l R kh = l R ji. Thus, if we differentiate (3.1) covariantly, we get (3.2) ( l R kjih )R kh = l R (2) ji +{R n(n 1)}(φ lj η i +φ li η j ). 248

7 Sasakian manifolds with cyclic-parallel Ricci tensor By the definition of H ji,wehave k H ji = R ki η j R kj η i by virtue of (1.1) and (2.2) and consequently (3.3) ( k H jr )H i r = R kr (2) φ i r η j. Now, suppose that C-Bochner curvature tensor of M is parallel and TrRic (m) is constant for a positive integer m. Then, by Theorem 2 the Ricci tensor is of cyclic-parallel. Thus, all relationships obtained in previous section are valid. We also have from (1.12) (n + 3) l R kjih +( l R ki )g jh ( l R ji )g kh + ( l R jh )g ki ( l R kh )g ji +( l H ki )φ jh + H ki (g lh η j g lj η h ) ( l H ji )φ kh H ji l φ kh +( l H jh )φ ki + H jh (g li η k g lk η i ) ( l H kh )φ ji H kh l φ ji +2( l H kj )φ ih +2H kj (g lh η i g li η h ) + 2( l H ih )φ kj +2H ih (g lj η k g lk η j ) ( l R ki )η j η h R ki (φ lj η h +φ lh η j ) + ( l R ji )η k η h + R ji (φ lk η h + φ lh η k ) ( l R jh )η k η i R jh (φ lh η i + φ li η k ) + ( l R kh )η j η i + R kh (φ lj η i +φ li η j ) (k + n 1){(g li η k g lk η i )φ jh +(g lh η j g lj η h )φ ki ( l φ ji )φ kh φ ji l φ kh +2(g lj η k g lk η j )φ ih + 2(g lh η i g li η h )φ kj }+k{g ki (φ lj eta h + φ lh η j ) g ji (φ lk η h + φ lh η k ) + g jh (φ lk η i + φ li η k ) g kh (φ lj η i +φ li η j )}=0. Applying R kh η i to the last equation and making use of (1.1) and (1.6), we find (n + 3)( l R kjih )ξ i R kh + R r j ( l R ir )ξ i + ( l R jr )R r i ξ i +R{H lj (n 1)φ lj }+3( l H ri )ξ i H r j 3H jr R r l (n 1) 2 φ lj R (2) jr φ r l + R (2) φ lj +3(k +n 1)H jl + k{(n 71 R)φ lj + H lj } =0. or using (1.3) and (3.3) (n+3)( l R kjih )ξ i R kh +{ l R (2) ji }ξ i 5R (2) r lr φ j +{R 3(n 1) 2k}H lj +{R (2) (n 1) 2 (n 1)R +k(n 1 R)}φ lj =

8 Sung-Baik Lee, Nam-Gil Kim, Seung-Gook Han and Seong-Soo Ahn From this and (3.2) it follows that we have (n+4)ξ i l R ji (2) 5R lr (2) φ j r +{R 3(n 1) 2k}H lj +{R (2) + 4R + k(n 1 R) (n 1)(n 2 + 4n 1)}φ lj = 0. which together with (1.6) and the fact that (n + 1)k = R + n 1gives (3.4) R ji (2) = βr ji +γg ji +{(n 1) 2 (n 1)β γ }η j η i, where β and γ are given by (3.5) (n +1)β = R 3n 5, (3.6) (n 1)γ = R (2) R2 n R n 1 n + 1 (n2 + 3n + 4). Transforming (1.8) by R i k and making use of (1.2) and (3.2), we find T jr R r k = (β + 1 R n 1 )R jk +γg jk +{R n+1 (n 1)β γ }η j η k, which together with (1.8) (1.10), (3.3) and (3.4) implies that n 1 (3.7) n + 3 T (3) + R + n 1 T (2) = 0. n + 1 If we suppose that the length of the η-einstein tensor is pinched as that in Theorem 1, then by applying the proof of Theorem in [6] to (3.7), we verify that T (2) vanishes identically. Thus we have proved Theorem 2. Also, from (1.8) and (3.4) we can find (see [9]) (3.8) T (2) ji = n+3 n 2 1 (R+n 1)T ji + T (2) n 1 (g ji η j η i ), where we have used (1.2), (1.10), (3.5) and (3.6). According to the main theorem of [9], we can prove by using (3.8) the following: THEOREM 3. Let M be an n( 5)-dimensional Sasakian manifold with parallel C-Bochner curvatures tensor. If TrRic (m) is constant for a positive integer m, thenmis a space of constant φ-holomorphic sectional curvature 4R (n 1)(3n 1) or M admits a cyclic parallel almost product structure which is (n 1)(n+1) not integrable. 250

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