ON A tf-space OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE

Transcription

1 WATANABE, Y. AND K. TAKAMATSU KODAI MATH. SEM. REP 25 (1973) ON A tf-space OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE BY YOSHIYUKI WATANABE AND KICHIRO TAKAMATSU 1. Introduction. It is well known that a Kahler space of constant holomorphic sectional curvature is an Einstein space (Yano [13]). The main purpose of the present paper is to generalize this result to a Zf-space (Theorem 4. 4). Preliminary facts will be given in 2. In 3, we shall prepare some lemmas for the proof of main theorem. Particularly, we shall prove some lemmas about the iγ-space of constant holomorphic sectional curvature. In 4, we shall prove the main theorem and some related theorems on the generalized Chern-form. 2. Preliminaries. Let M be an w-dimensional (n>2) almost Hermitian manifold with Hermitian structure (F/, gjt), i.e., with an almost complex structure F/ and a positive definite Riemannian metric tensor gμ satisfying (2.1) FfFJ^-ty (2.2) 0uFSF % 9 =Qj t. (2.3) If an almost Hermitian structure satisfies where Fj denotes the operator of covariant differentiation with respect to the Riemannian connection and Fji=F/g t i, then the manifold is called a K-space (or Tachibana space, or nearly Kahler manifold). Now, in a if-space let l?*yλ Rβ=Rhji h and R=g Jί Rji be Riemannian curvature tensor, Ricci tensor and scalar curvature respectively. Then we have the following identities [7], [8]: (2.4) (2.5) F hk FΨ t Fj h =:R kj -R*j k, or VΨ t F 3h =F h \R jl -R* lj \ Received September 13,

2 298 YOSHIYUKI WATANABE AND KICHIRO TAKAMATSU where P = g ία f α and R* Jt = (l/2)f δα i? δαίί F/, (2. 6) ^=Fj b F t ar ba, R* μ =F»F«R* ba, (2.7) R*ji=R* %j9 (2.8) FjF ts (F ί F ts ) = Rj i -R* Jί, where F ji = F t ιg ίj \ and (2. 9) FjF ih (FJF ίh ) = R-R*=constant>0, where R*= Since we have T kji F k R jih i=0 for any skew-symmetric tensor Γ* y<, we get (2.10) As (ll2)fir* = FJR*ji in a ϋγ-space [5], we have (2.11) F*(R ίk -R* ik )= -ί- F 4 (Λ-Λ*)=0. 3. Some lemmas. In general, it is well known that the differential form is closed in any almost Hermitian manifold, where K ji =2R jir t F t r -F s KFjF r ψ i F t r > which is called the generalized Chern-form [8]. In a iγ-space, taking account of (2. 8), we have (3. 1) &ji = Recently, A. Gray proved the following LEMMA (Gray [1]) In a K-space, the relation (3.2) R jist -R bast F Jb F z -=-(FjF ιr )F s F trf (3.3) RjiHt-FSFtFSFSRa**^ hold good. Moreover, the present authors proved the following LEMMA (Takamatsu and Watanabe [11]) In a K space, we have

3 K-SPACE OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE 299 (3. 4)?nS n =- -γ{sr where S^Rβ-R*^. Now we have the following LEMMA In a K-space, the relation (3. 5) S*\R hjih - 5R km Fi b F a h )=0 holds good. Proof. If we transvect (3. 2) with F h F Ji, then we have, taking account of (2. 4) and (2. 8), 2(F h F^)R Jist =-S h r F s F tr. Applying F h to this equation, we have, by (2.10) and (2.11), or by Ricci's identity and S hr F s F h F tr -0, 2(F*F h FV)R m = -S h r F h F s F tr, (3.6) 2{F h F h F^)R m =S/( - F s F h F tr +R h st a F ar +R\r a F ta ) =S h r (R\ at F r «-R\ ar F t «). Transvecting (3. 6) with F k \ we have, (3. 7) 2F k \F h F h F^)R jist =S hr (R hs jcr -Λwα/VF r β ). Substituting (2. 5) into (3. 7), we obtain 2F k Ψ a j S ai R m = S Λr (i?^r - RlutaFlΐFr*), or by Bianchi's identity, (3. 8) 2F k Ψa J S ai (R iφ +Rtjis)=S Λr (i? Λsλr - R hst af k Ψ ra ). Moreover, making use of F^S^-FΛS"-', (3.8) reduces to (3. 9) 4F k Ψa'RtfuS Λi = & r (Rh.tr-Rh«Λ F k Ψr a ), from which we have (3. 5). Transvecting (3. 5) with g kh, we have the following LEMMA (Takamatsu [10]) In a K-space, the relation (3.10)

4 300 YOSHIYUKI WATANABE AND KICHIRO TAKAMATSU LEMMA In a K space, we have (3.11) (Rw-R t j ta FW)S' t = -j-(3#* r +i?** r )S ft r. Proof. By Ricci's identity, we have (3.12) P h FiSsi - PiF h S Λ =R Mm,Sι m +Rm mt S m s. Transvecting (3.12) with g iι, we have, by virture of (2.11), (3.13) i? fet «S»=R^Sr - PΨtSn. Now, applying V i to (3. 4), we have (3.14) Let us calculate the right hand side of (3.14). By Ricci's identity and (3. 2), we have F,F mr - F s F t F mr = Rua r F ma - Rum>F t r (3.15) =F ma {Rua r -RumF r =F Ma (-F i F«)P a F rt. Multiplying both sides of (3.15) by S^F^, we have Thus, the first term of the right hand side of (3.14) reduces to (3.16) S r{ (PΨ r F >m )F^= - VΨ s ψ τ F M )S n. For the third term, by (3. 2), (3. 4), (3. 9) and (VΨ mr )F hm =(P τ F h )F m \ we have F l S rs (F i F m r)fn m = - ~ (3.17) = - -i- {S ts S h '+S tr (F t F m )F h F r m

5 i -SPACE OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE 301 The fourth term, making use of (2. 5), reduces to (3. 18) Srs^ΨiF^F^^SrsS^. Substituting (3. 16), (3. 17) and (3. 18) into (3. 14), we have (3. 19) FΨ h S sι = -j- S r s S hr - FH'FJRWSS* 1. Thus, making use of (3. 19), from (3. 13), we have 1 or from which we have (3. 11). LEMMA In a K-space, we have (3.20) SR kjih = lb (3. 21) S^RwaFtFn" = 4" T kh, (3. 22) S»F hr F k *R rjiq = 4- Tu, Λ = T hk = (3i? fcr + i?*a;r)s,λ Proo/. First, by (3.11) and Sji=S i3} T kh = T hk is easily verified. (3. 20) and (3. 21) immediately follow from simultaneous equation (3. 5) and (3.11) with respect to S Ji R kjih, S Jί R kjba Fr b F h a. For (3. 22), making use of Bianchi's identity, (2. 6) and (3. 21), we have O F k '( - Rjir Q - R %rjq ) (3. 23) =S h Ψ t /F k (ιr τj r q LEMMA In a K-space, we have (3. 24) rt 4

6 302 YOSHIYUKI WATANABE AND KICHIRO TAKAMATSU Proof. Calculating the square of both sides of (3. 4), we have (3.25) 2P k S β (F k S^)=S\S H (F lι F ]m )P b F^+S M S>>K! 7 ι ί F jm )F b F ι "'. Substituting (2. 8) and (3. 2) into (3. 25), by (3. 20) and (3. 21), we obtain -s«( Ί - τ M - -ί- \ ID ID Lastly, we shall state some lemmas for a if-space of constant holomorphic sectional curvature. Let k be a holomorphic sectional curvature of an almost Hermitian space with respect to a vector X h, that is, If &=constant with respect to any vector at any point of the space, then the space is called a space of constant holomorphic sectional curvature. We know the following LEMMA (Mizusawa and Koto [4]) If an almost Hermitian space is a space of constant holomorphic sectional curvature, then the curvature tensor of the space satisfies the following relation: Rkjih + Rtjkh F]c q Fh\Rljίq + Rijlq) (3. 27) kp+r khmp ) - F k qf/(r imq + Rι^ = - AkF k qfi p (g qp g hj + g qh g JP + g qj g P h). For a Tί-space of constant holomorphic sectional curvature, we know the following LEMMA (Koto [2], Takamatsu [9]) In a K-space of constant holomorphic sectional curvature, we have (3. 28) R kr + 3i?* fcr = (n + 2)kg kr,

7 /ί-space OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE 303 or (3. 29) R kr + 3R* kr = where k is positive constant. LEMMA In a K-space of constant holomorphic sectional curvature k, we have (3. 30) (R kr - 5R* kr )Si r S kι =2k(S 2 - ns kι S u ), where S=R-R*. Proof. Transvecting (3. 27) with S hj and making use of (3. 3), we have, (R kjih + R %jkh )&> - F kq F h \R ljiq + R (3. 31) - F kq F/(R ihlq + R lhtq )&" + (R JkM Then, substituting (3. 20), (3. 21) and (3. 22) into (3. 31) and making use of (2. 6), we have "(π Tu+ Jβ Tik ) - {Ύ TU+ Ίe Tu ) (3. 32) - (i, T kl+ j- T k ) - (i- Tu+ ± T k ) - (-1 T U+ 4- Γ tt From (3. 32), by T kι =T ik) we have i.e. (3. 33) (3i? ftr + Λ**r)Si r - 2k(Sg ki +2S kz ). Then forming (3. 33)-2S/X(3. 28), we have (R kr - 5R* kr )Si r - 2k{Sg ki - ns kt ). Thus, transvecting this last equation with S k \ we have

8 304 YOSHIYUKI WATANABE AND KICHIRO TAKAMATSU Moreover, we know the following LEMMA (Sawaki and Yamagata [6]) In a K-space of constant holomorphic sectional curvature, we have (3.34) an (3. 35) (Rn - 5#* n)(r^ - 5R* Jί ) = (R+3#*)(5#* - R), and (3.36) R*<R^5R*. LEMMA Let M be a K-space of constant holomorphic sectional curvature. Then, in order that M is an Einstein space, it is necessary and sufficient that R=5R*. Proof. For sufficiency, taking account of (3. 29) and (3. 35), we can easily see that M is an Einstein space. Conversely, if M i s an Einstein space, then from (3. 29), we have R*ji = (R*ln)gji. Therefore we have Rμ R* μ = {(R R*)jri)g μ, from which we get i?=5i?*, by virture of (3.10). n 4. Theorems. THEOREM 4.1. Let M be a K-space such that Then M is an Einstein space. Proof. Substituting (4. 1) into (3. 5), we have (4.2) Rji = 5R*ji. Transvecting (4. 1) and (4. 2) with g jί, we have (4.3) R-R* = na, R=5R* f respectively. Substituting (4. 2) and (4. 3) into (4. 1), we have - n Since a 6-dimensional if-space satisfies the condition (4. 1) [10], we have

9 K-SPACE OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE 305 COROLLARY (Matsumoto [3]) A ^-dimensional K-space is an Einstein space. If the generalized Chern-form K vanishes, then from (3.1), we have Therefore from (3. 24), we get F k Sji=0. But, we know that if a symmetric tensor Eji is parallel in an irreducible Riemannian space, then Eji=cgji where c is constant. Consequently, by virture of Theorem 4.1, we have COROLLARY An irreducible K-space with vanishing generalized Chern-form K is an Einstein space. THEOREM A K-space of constant holomorphic sectional curvature is an Einstein space. (4.4) Proof. Substituting (3. 34) into (3. 30), we have (R kr - 5R% r )S\S k '=2&(S 2 - ns kt S kί ) =k(r-r*)(r-5r*). On the other hand, making use of (4. 4), from (3. 24), we have (4. 5) (?»)?*&*= -^-k(r-r*xr-5r*). In the above equations (4. 5), since k>0 and R R*>0, we have (4.6) R Comparing (4. 6) with (3. 36), we have R=5R*. Consequently, by virture of Lemma 3.12, the proof of the theorem is complete. COROLLARY The generalized Chern-form K of a K-space of constant holomorphic sectional curvature vanishes. Proof. By Theorem 4.4 and Lemma 3.12, we have R=5R*. (3. 35), we have Rji = 5R*ji which shows that K=0. Hence, from REFERENCES [ 1 ] GRAY, A., Nearly Kahler manifolds. J. Differential Geometry 4 (1970), [ 2 ] KOTO, S M Curvatures m Hermitian spaces. Mem. Fac. of Educ. Niigata Univ. 2 (1960), [ 3 ] MATSUMOTO, M., On 6-dimensional almost Tachibana spaces. Tensor N.S. 23 (1972),

10 306 YOSHIYUKI WATANABE AND KICHIRO TAKAMATSU [ 4 ] MIZUSAWA, H., AND S. KOTO, Holomorphically projective curvature tensors in certain almost Kahlerian spaces. J. Fac. Sci. Niigata Univ. 2 (1960), [ 5 ] SAWAKI, S., On Matsushima's theorem in a compact Einstein ϋγ-space. Tδhoku Math. J. 13 (1961), [ 6 ] SAWAKI, S M AND M. YAMAGATA, Notes on almost Hermitian manifolds. To appear in Tensor N. S. [ 7 ] TACHIBANA, S., On almost-analytic vectors in certain almost Hermitian manifolds. Tδhoku Math. J. 11 (1959), [8], On infinitesimal conformal and projective transformations of compact if-spaces. Tδhoku Math. J. 13 (1961), [ 9 ] TAKAMATSU, K., Some properties of i "-spaces with constant scalar curvature. Bull. Fac. of Educ. Kanazawa Univ. 17 (1968), [10], Some properties of 6-dimensional if-spaces. Kδdai Math. Sem. Rep. 23 (1971), [11] TAKAMATSU, K M AND Y. WATANABE, On conformally flat if-spaces. Differential Geometry in honor of K. Yano, Kinokuniya Tokyo (1972), [12], Classification of a conformally flat if-space. Tδhoku Math. J. 24 (1972), [13] YANO, K., Differential geometry on complex and almost complex spaces. Pergamon Press (1965). TOYAMA UNIVERSITY, AND KANAZAWA UNIVERSITY.