・TRU MathematiCs 15−2 〔1979）
ON A SASAKIAN MANIFOLD WITH O−BOCHNER（㎜TURE TENSOR
Gor6 CHUMAN 〔Received November 16，1979） 1． Introduction， M．Matsumoto has proved the f6110wing ［2］： THEOREM A． 1アαKahleriαn spαeθwithρα㌘αZZθZ Boehne㌘α妙α加゜θカ●ηsor hαseonstCtnt scαzα㌘cur・vαtu）e． th￠ηほisαsymmetrdσsρaee． ’ ‘ Moreover， M． Matsumto and S． Tamo have shown the fbllowing theorem［4】： THEOR日M B． 五θカ〃Z）θαKZihZer￠Ctn spαoθzuith PCtr，αZZeZ Boσhner etαmフα勧1？θ カθηsor． Then〃isαZoeaZZy symmetric spα0θ0㌘α8Pαeθntth Vαnishing Boehne㌘ OL的α力辺9θ カθηεOr◆ M．Matsumto and the present author have血tro血ced the O−Bochner curvature tensor in a Sasakian space ［3］． ・In 1975， T． Takahashi has introduced the notion of a Sasakian locally、φ一symmetric space砲idh is an analogous notion of Hemiti｛m synmetric space［6］．脆㎞㎝that the condition of a sy㎜etric space is too strong fbr a Sasa虹飢㎜ifbld． It seems that the notion of Sasakiallφ一symmetric space is good cσndition for a Sasa】（iall manifbld． The purpose of this paper is to prove the fbllo砿ng theor㎝which is ・ analogy of Theorem A by using the notion of Sasakianφ一symmetric space・ 1肥OR】an． rf a Sasakiαn m励∫bZ4㎡肪Pα1・αZZeZ C−Boehner emoαtUl・e tensor hasαeonstαnt soαzα㌘omoαtur・e． then it isαSasαkian loeαZZyφ一synvnetrio spα已θ・ The author wishes to express his hearty thanks to Prof． S． YaitLaguchi fbr his 1くind， various pieces of advice and constant encouragement・ 2． Sasa】dan manifbld． An n−dimensional Riemannian manifOld f is called a Sasakian manifold 〔or no頂mal contact metric manifold） if it ad皿its a unit Killing vector field na★），u、in・that 〔2・1〕 ▽。▽bn。 ＝ nb9。。 一 n。9。わ・ 5 ／6
G．CHUMAN
血ereθ。． i・th・me・・i・・㎝・・r・f f ・nd▽。一・th・・perat・r・f・・vari皿・ differentiation． It is we11㎞own that f is orientable…mdη〔Z3）is odd． If we defineφαゐby φαゐ＝▽αηb・ then 〔2．1） is rewritten as ▽αφわ。＝nb9。。一η。g。ゲ and we have ．〔…） φα「φ。ゐ一δ。カ・・。・ゐ，φα「・。・・，φ。、・一φ、。． On a Sasakia！i manifold the fbllowing identities are we11㎞own［7］： R、zz，。ア・。・η。9ゐσ一ηbga。・ R。カ。。φ。sφ！・R。カ。d・・。。・bd−％．・ゴφ。。φゐ∂・φb。φ。∂， φs㌔藺＝−2・。ゐ・2（・−2）φ訪， 舳e R、zb。㌦・R。、 are・h・③rv・・ure…n・・一・趾cc・t・n・・r，胆…p・・ 5。ゐ・S㌦・
・na…』㎜・f・・d・・h・・−B・chner−・・碇・t・n・・r・。、。d・・d・f血・d by Bαb。∂＝R。ゐ。∂ ＋ l n＋3 免。9bd−Rb。9，，d＋％。㌦一9ヵ。R。d＋5。。φヵ4−Sb。φ。d ＋φαoSbd一φZ）oSαd＋25αZ）φed＋2φabSod 一Ra已nbnd＋Rb。η。nd一η♂。Rhd＋ηゐη。R。d ．、 ＋ρ鑑鵠1）（一φ。。tl、d・φ、。φ。d−2φczbφ。、） ・（謡et3X，（一・。。・bd・・、。・。d） ・（ P＋n−ln＋1）（n＋3）〔・。。・、・d・・。・。・、d−・、。・。・d−・、・。9。d〕， whereρis the scalar curvature． The O−Boc㎞er curvature t㎝sor is constructed from the Bochner curvature tensor in a K狙erian space by the fibering l of Boothby−wang ［1，3，5】・ By straight forward co酊唖）utations， we can verify fbllow− ing identities： ・。、。d＝−Bbaad， ・、zb。d・B鋤、， ・。、。∂・Bb．αd・B。abd・・， ・ON A SASAKIAN MANIFOLD 7 ・。、。α・…。、。d・、・P．φαア・。、。∂・φ、㌦∂ φαカ・。、。d’・・． ASasakian manifold is called Sasakian locallyφ一s）㎜etric space if its R。、。d・ati・f…th・f・・・・…、・・nd・・…［・］・ （2・3〕 ▽。R。・。d＝・。〔一φ。。9bd・・、。φ。∂・φ。㌔。d）・・、〔φ。。・。d−・。eφ。∂一φ為，。d〕 ＋n。（9b．φ。ガ9αeφbd・φd”4。カ．。）・nd（・。。φ加一％。φ。。一φ。「R。b。。）・ 3．Proof of Theorem Consider a Sasakian manifold with paralle1σ一Boc㎞er curvature tensor． Then，血aSasak迦．one， the following fact is we11㎞om：the tensor T α1◆．●αρ ’th’ch sat’sf’es▽hZ・、…ap＝°and T・、…・芦＝°…’・h・・id・n・ica・・・・・… we can see that a Sqsakian manifbld withσ一Bodmer curvature tensor has vanish− ing one． So， we may assu酊re that a Sasakian manifOld has vanishing C−Bochner curvature tensor and the scalar curvature is constant． After long con享）1icated con耳）utations， it fOllows that （3．1） ρ＋（n−1）（n＋2） ▽eRabod＝nα［ （n＋1）（η＋3） （−9dbφeo＋9cカφθ∂−9 ・。…，（φ。』一φ。♂。、・ぽ。。一φ。ゐ・。d・・φd。・。、 oeφ（ib＋9deφcb−29beφde） ＋・。。9dゲ5。♂。b＋5、ih9。e−5。ゐ9。d＋2Sd．．q。b）］ ・・、［ρ炎增`㍑il）（一・。。φ。d・・daφ．。躍。α＋・。．φ、、・一 2・。．φ。d〕 ・。圭，〔φ。♂。α一φ。eRdn・φcαR。d−tp、、aR。。・・φ。轟α ＋SeEeα一Se已9dU＋5（3αxged−Sdnge已＋2Segeα〕］ ・・。［ρ＋（n−1）（n＋2in＋1）（n＋3））〔一％。thdn・％。φdb−・。d・bbe・9bdφ。。一・・。、φ、。〕 ・。…，（φぬ。一・thdbR。、・φ、。・dn一φ。。Rdb・2φ、。Rde ＋Sd。9be−5ゐ9αθ＋％。9da−5。。9de＋25b。9de〕］ ・・d［ρｼ瓢1辛；2）（一％。φ。．・・、。φ。α一・、。φ。e＋・。。φ、、−2・。。φ。、） ・。圭，〔φ。bR。。一φ。αRb。＋φ。。R。ガ暢α・・φ。bR。θ
8 G．（HUMtUsl 、 ＋5。b9。。−3。。％．＋5。。b。わ一Sb。9。α＋2s。あ9。。）］・ ．
