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1 (Received March 30, 1961 Revised July 21, 1961) A. Lichnerowicz1) has proved that the Matsushima's theorem2) in a compact Kahler-Einstein space holds good in a compact Kahlerian space with constant curvature scalar. In the previous paper [4], we have shown that the Matsushima's theorem is valid also in a compact almost-kahler-einstein space. The purpos of this paper is to show that it holds equally well in a compact Einstein K-space. In 1 we shall give definitions and propositions. In 2 we shall give well known identities in a K-space. In 3 we shall prepare some lemmas on contravariant almost-analytic vectors in a K-space. The last 4 will be devoted to the proof of the main theorem. 1. Preliminaries. We consider a 2n-dimensional almost-hermitian space X2n which admits an almost complex structure qz3) and positive definite Riemannian metric tensor gjti satisfying (1.2) grsqjrqis=gji. By (1.1) and (1.2), we have (1.3) ji=-qhi7 Oa-1(Si0h-qq1), Oh-1(imhl+gamPhi) and a tensor is called pure (hybrid) in two indices if it is annihilated by 1) A. Lichnerowicz [1]. The number in brackets refers to Bibliography at the. end of this paper. 2) Y. Matsushima [2]. 3) As to the notations we follow S. Sawaki [3]. Indices run over 1, 2,..., 2n.

2 456 S. SAWAKI transvection of *0(0) following PROPOSITION 1. Ohpab-0, on these indices. From the definition, we have easily the 0bVMab-0. PROPOSITION 2. For two tensors T and S, if is pure in j, i and Si is hybrid in ji then TS vanishes. A vector v is called a contravariant almost-analytic vector if its contravariant components satisfy (1.4) t-rvrvt+prpvr=04) where is the operator of Lie derivative. From (1.4) we have (1.5) Vv+aPbtavb-vr(Or,J,l)TL=0 which is equivalent to (1.4). Lastly multiplying (1.5) by 1/21zpk, we have (1.6) vr(or7l)oktl+qrl(ki)vr=0. In this place, if pvr=prv, kq being anti-symmetric in j, r, we have Thus from (1.6) we get grl(okq,jl)0vr=0. (1.7) vorql=0. 2. Identities in a K-space. An almost-hermitian space X2,3 is called a K-space5) if it satisfies (2.1) Viih+VPjh=0 from which we have easily (2.2) jij=0, (2.3) Oabh=0.6) put Hereafter we shall consider only a K-space X2n. Let Rkand R=Rrr be Riemannian and Ricci tensor respectively and 4) S. Tachibana [5]. 5) S. Tachibana [5]. 6) S. Sawaki [3].

3 (2.4) abrabrirr*t=rrg Applying the Ricci's identity to we get DkVghDiVkin=Rkrhpir-Rkirprh. Transvecting the last equation with and using (2.2) and the Bianchi's identity, we have (2.5) DTDcrh-1pgRP9jh+Rjrp or using (2.1) (2.6) DDrPjhRjrh If we notice the anti-symmetry w. r. t. j and h in (2.6), we find that Rrrh+Rhrpr=0 from which we have i. e. R jh is hybrid in j, h. OI4Rab=0, On the other hand, in a K-space we know that (2.7) Rt=R, (DjTab)DiqabRatRi) and transvecting (2.5) with phk we get (2.8) ThkVrVqrh=R*;k-Rjk. Since by (2.3), (Vjqab)Vjq ab is hybrid in j, i, from the last equation of (2.7). it follows that R*, is also hybrid in j, i. In this place, since (2.6) can be written as DTDrPjh=Rrhjr-Rjrcorh, Again by the Ricci's identity shv svhkg-(vvshkl-qhvsgkl) (Rshkapat-RBhlka) 7) S. Tachibana [5].

4 and therefore by (2. 7) we have Moreover, making use of (2.7) and Proposition 2, we have R=V(VabVi) (0-ab)ab+(jq)v3jqab (Qqab)Oviab identity and the Bianchi's identity the last equation turns to from which we have R*=Dj'pab(iVfqab+R1satpsb+R1sbas) CjabQQjab+2Cvgab)Risab 1CQabQab)(CYb9a)Ratssb R=v(R-R*)-(ob7a)Rassb, i. e. (2.10) (bja)iasbobsar=ofr*1oir*-(vr1) where R=Rzg and R*=R*5g. (2.11) (vba)rjai/psb=-1vir*. And by the Bianchi's identity the left hand side of (2.11) can be written as CvbjaRisvbja(Ris-st)q -Obj But as we have by virtue of (2.3) a(s-rsti) vbtjasb=(vspba)pjb, (VbPja)Psb=Cvsjb)ab, the above equation becomes 8) K. Yano and S. Bochner [7], p. 19.

5 (vbqia)is-(djb)sia (2.12) 3(DbPja)Rais)sb=0. Consequently from (2. 11) we have (2.13) R*=1DR*. And by the Bianchi's identity and (2.1) r(dkjt)r kjirrkji q(vq.3)rkijr-ar(djik)rjikr, i. e. Thus multiplying VkV/Pih-OjOkh=-RkjtirPrh-Rkjhrpir (2.15) 2(7k9i)kpjPih=Oki(Rkjirrh+RkjhrPjr) QkjiRkj4rTrh+(ikj)Rkjhrri because of (2.1) (2.12) and (2.14). On the other hand, taking account of (2.1) and (2.7), we get 1k(R-R*)=(V5q)VkvjPrs =Q(QjVkrs-Rkjrtpts-RkjsTG)=0 That is, we see that in a K-space (2.16) R-R*=constant.9) For the Nijenhuis tensor, by (2.1), becomes lvjih=jl(lih-viq-lh)til(qljh-qjglh) (2.17) Njih=4qjQlgih. Finally, for any vector vi we have (2.18) qvavbi=1laq(vavbvi-qbvavi) 9) S. Tachibana [6].

6 LabtsVsR*lr 3. Contravariant almost-analytic vectors in a K-space. In a K-space, we know the following lemma. LEMMA ) In a compact K-s pace, a necessary and sufficient condition that a contravariant vector v1 be almost-analytic is that it satisfies analytic. Suppose that for a contravariant almost-analytic vector vk in a K-space, vk is also almost-analytic, then we have from (1.5) or using (2.3) (3.1) (VjVa)ak-pjT Orvk+Va(2VjPak-Va)=0. (3.2) Osv+gjrqkrvkkva(2VHakDajk)=0. From (1. 5) and (3. 2) we have 2pkjVaV7akQajkl=0, i. e. VV/Jrk=0, or vrprk=0. Thus we have LEMMA 3.2. When a contravariant vector vk in a K-space is almostanalytic, a necessary and sufficient condition that vk be almost-analytic is that it satisfies 4. A generalization of the Matsushima's theorem. almost-analytic vector vi is decomposed in the form vi=pi+prigr decomposition stated above is unique. 10) S. Tachibana [5].

7 PROOF. Let v1 be a contravariant almost-analytic vector in a compact Einstein K-space, then from Lemma 3.1 we have (4.1) ppwt+xv=0. From this equation, we can easily deduce (4.2) vvivr=yr+v and (4.3) vvcvvrvr+rvvrvr=0. If we put (44) p=vh+nih (4.5) phph=phvh+nphphprr=0 and by (4. 1) and (4.3) we have (4.6) ppcp+xpf=0. But since (4.5) and (4.6) is a necessary and sufficient condition that pi in a compact Einstein space be a Killing vector,11) it follows that If is a Killing vector. I=Qjlk+1j6kVr1 and writing out the square of Pjk, we get 1/21k1 (Djk)D/k+tk(Vi-lk)OrTj+ar(TTb)nqjb. Consequently, we have (4.7) 1PkPk+v(Pfkrk)=1PkPx+(vPkW+PJkVj)k 11) K, Yano and S. Bochner [7], p, 56.

8 462 S. SAWAKI k1+r(o)ojb+ bjokga/jlvab and therefore by virtue of Green's theorem we have (4.8) rfq5pjk+1ir(rqjb)kb+bjcokaj)=0, In this place, by using (2.1), (2.2), (2.7), (2.8), (2.17) and (2.18), we have and hence GjPk-7V7k-17+tr(ijrgal)glk-r +vjr(vrji)lk+ir(vrqj)vjlk vjvjrlk3r*ks+r(r*kr-rkr)jr(drl)vjlk +vj1r(orjl)lk+r(vrjl)jlk DjPjk+r(Drjb)vkb+bj(vkPa)Dab -frr rk+3rjovflk1vrpjb)vk(vrhl) Thus (4.8) turns to Djpj'lk-rRrk+NrlkVr17l+1r(Rrk-R*rk). (4.9) vvi1k-rhe+3nrtkvr7+3irrrk-r*rk -1PjkPjkdh=0. Substituting k=rpk-rvk, in (4.9) and using (4.3), we have (4.10) I3Rk{Nrlkvr(pl-vc)+2(pr-V)9-1PjkPjkdo=0. On the other hand vk being almost-analytic, from Lemma 3.1 we have Hence (4.10) becomes Nrvl2r(Rrlkv+Zvgrk-2nRrk=0.

9 (4.11) fi1krrlkdrpl+2prgrk-rrkfpikpjdojx=0. Furthermore (4.11) can be written as (4.12) [RDkj(VIVI)NrlkVrPt+2(Divl)P9rkRrk -1PkPjkdo=0. In fact, taking account of (2.17), we have Dk(Nrlkvrpl)=4vx(vrplq-ktvtqrl) =4{(VVrpl)kIVITLk+(vrpl)ktDkvcrl =4(DkVrpl)ktVtrl because of (2.9). Here by (2. 1), (2.3) and (2. 12), we have Consequently, we have On the other hand 4(DkDrpl)ktDcrl=4lt(Dkrc)OkDrpl -2ltCDkrt)CDkvrpi-vrvpl) 2ltCDkrt)RkrBlps pk(nrckprp)=0. pkpr(j9rk-r*rk=pkprr9rk-r*rk)-prpxr*rk vanishes. k, r, the first term of the right hand side vanishes. For the second term, from (2.13) and (2.16) we have 2pkR*rx=prR*=prR=0. Thus again by Green's theorem from (4.12), we have PkPkdcr=0

10 464 S. SAWAKI analytic vector. (4.13) rorc=0 (4.14) qh-naha then qh is a contravariant almost-analytic vector and a Killing vector. In fact vhgh=fltahvhla=0 and From (4.4) and (4.14) we have Ocgh+Kqh=0. vh=ph+qrhgr, rhr=h. Finally we shall prove that such a decomposition is unique. If we have vh=ph+qrhgr, vh=kph-qrhgr where qrqr and rqr are both gradient vectors, then (4.15) ph-fph=grh(qr-qr), Since the left hand side of (4.15) is a Killing vector and the right hand is a gradient vector, we have side Vih=0 Hence by the Ricci's identity we have VVOt1h=Rjtshs=0 from which we get Ryas=x=0. Thus we have ph=fph and qh=qh. q, e. d.

11 ON THE MATSUSHIMA'S THEOREM 465 BIBLIOGRAPHY [1] A. LICHNEROWICZ, Geometrie des groupes de transformations, Paris, (1958). 12] Y. MATSUSHIMA, Sur la structure du groupe d'homeomorphismes analytiques d'une certaine variete kahlerienne, Nagoya Math. Jour., 11(1957), [3] S. SAWAKI, On almost-analytic tensors in *0-spaces, Tohoku Math. Jour., 13(1961), [4] S. SAWAKI, A generalization of the Matsushima's theorem, to appear, in Math. Ann. [5] S. TACHIBANA, On almost-analytic vectors in certain almost-he rmitian manifolds, Tohoku Math. Jour., 11(1959), [6] S. TACHIBANA, On infinitesimal conformal and projective transformations of compact K-spaces, to appear in Tohoku Math. Jour. [7] K. YANO AND S. BOCHNER, Curvature 32(1953). and Betti numbers, Ann, of Math. Studies, NIIGATA UNIVERSITY.

Sufficient conditions for an almost-hermitian

Sufficient conditions for an almost-hermitian manifold to be K\"ahlerian Dedicated to Professor Y Katsurada on her 60th birthday By Sumio SAWAKI \S 0 Introduction If an almost-hermitian manifold M is a