〈）NATHEOREM OF GALLOT−MEYER−TACHIBANA
IN RIEMANNIAN MANIFOLDS．O・F POSITrVE
CURVATURE OPERATOR
BYSEIIcHr、YAMAGUCHI
・1．血trOdudion． Let．M％（n＞2）be a compact orientabale Riemannian manifold
−whose metric tensor is 9iven by gαb（a， b，．．．＝1，2，．．．， n）血terms of local coordi− nates ｛xa｝．Throughout this paper， manifolds are assumed to be connected and the differ− 。entiab血ty be C°°． Denote by Rabcd the curvature tensor． ff there eXists a positive constant ．．k such that ・ ’＜＊） ： −R。b。d uabued≧2ku励u“b
−holds good for any skew symmetric tensor uab at any poir並， then Mn is ca皿ed to be of posi− tive curvature operator． About its Betti numbers， b2（ハの＝Ohas been proved by M Ber− ，ger［1］， a血d bゼ（M）＝Oby 1）． Meyer【5］for i＝1，2，．．．，n−1． S．Tachibana口1］has proved the fb皿ow血g： The・rem A・恥c卿砺・rientable・Riemannian manifo”Mn・fp・sitive c〃rvat・・’e・・pera・Xorぷatisfies 』 1 ’‘ ’
▽αR伽ば＝0， ‘ ’ −th￠πハ4n is a space of constant curvature． 1βt△＝∂δ十δd denotes the Laplacian operatgr・If a non−zero、ρ一fb皿usatisfies△u＝ ，Zu for a constant Z， it is called a Prope士fbr【n△corresponding to a proper valueλ． S． Ga1− Iot and D． Meyer【4］have discussed the p｝oper value of Laplacian operator△fbr p−fbm1 ．u in such manifolds and get its lower boumd which is the正杷st possible， that is， they have 元Proved the fb皿owing： Theorem B． Inαco〃rpact Rie〃mnnian〃励ウわ〃ハぜ狗satisiンing（＊）the proper valueλ ．ofムプ’or Pプform u（〃−1≧P≧1）ぷatisjies λ≧P（n−P＋1）k， if du＝0， λ≧⑦＋1）（〃−P）k，がδ〃＝0． Recently， S． Tachiballa［10］11as i皿vestigated in the case when Z actUa皿y takes the possi・ fble min口11um values and obtained the f（）皿ow丘1g interesti血g theorems： Theorem C． betハ〃3 bθαco〃㌘αc’1砲〃螂αη㎜蓼わld satiSXンing（＊）and Z the proper Received June 20，197517
18
S．YAMAGucm
value ・f△f・r P−f・rm u（n−1≧P≧1）． lf du＝Oandλ＝P（n−P＋1）〃，’舵η賠c。n＿ for〃url Killing． Theorem D・Let M” be a compaプi R励辺η〃ian〃iam：t70〃ぷα域か’㎎（＊）andλthe prop−，θ・v・1…ゾム伽ク吻励（n−1≧ク≧1）．∬δ〃＝Oandzi（ρ＋1）（・一ρ）瓦．’乃。。ば
瓦蹴㌘． Theorem E・Let．M” be a c・mpaet Rieman吻n脚nifoldsatisXアing（＊）．∬」14n admits a’・1・s吻・⑳・・1−f・・m・・fク・・q…』鴫功・πMπ蕊・m・t・i・・with〃吻…わ妬1鋤…一
ぷη㈹with radius 1／！t． Theorem F・ Lε’・ルln be a eompact si，〃rply eonneetedRie〃urnnii n manifoldぷατ句ンing（＊）． IfMn admits a cocloぷed、proper 1プわr〃10f△corresponding to theproper value 2（n−1）斥， then− Mn iS either Sasakian or i，ぷo〃tetrie砿h Sn（k）． REMARK 1． S． Gallot has stated Theorem E in【3］． 111this paper， we shall genera五ze Theorem C， D． E and’F as follows： Theorem 1・Let M” be a c・mpaet．Riemannian uamtTo∼d satisfying（＊）・ndλ the pr〈）per value ofムプわ’、ρrプbrm〃．∬吻＝＝0．λ＝、ρ（η一」ρ十1）たand n−1≧p≧2， thenδu is a・・ Ψθ碗11ロ侮9（クー1）．form with constant k． Theorem 2・Lθ’M” be a c・mpact．Rた％〃nian〃tanifo〃ぷ碗かing（＊）and Z． the proper−． value ofムプbrρづ「orm u．万δ〃＝0，λ＝（ク十1）（n一ク）k and n−2≧、ρ≧1， the〃u丞r a spe−＿ eia1 Killing・P〔ノ’orm with constant k． 皿1eorem 3・Let Mnゐe a compactぷimply connectedRie〃mnnian manifo〃satiSLb，ing（＊）．．一 ∬M％α㎞ゴ’ぷαcloぷε4ρrρクθr、ρづeorm u（η一1≧ρ≧2）of」ρrqρθr valueλ＝p（π一ρ十1）k and the length ofu or 6〃Zぷη0’CO噺碗砺η．M力緬30〃te〃ic・with・S・㈹． Theorem 4・Letルtn be a compact吻ply eonnected Riemannian manifold satisfying（＊）． IfMn acimits a cloぷedproper 2−form〈ヅρrqρθr value 2（n−1）k， then Mn‘ぷeithei Sasakiare・ or伽’netric・withぷn（k）． Theorem 5・Lε’」lfn be a compactぷ‘〃⑳co〃〃ec’edRiε〃励痂〃manifo〃ぷatistVing（＊）．． 〃’五（nα伽its a cocloぷed P’qρθr P：ノわrm u（n−2≧ク≧1）ρアproper valueλ＝：（P十1＞・ ・（n−P）k and the length of u or・du is not￠・πぷ励’勲β〃」匪・緬ぷomβτr’C W乃S・（k）． 2．Conformal Killing pform．血an nrdirnensional．Riemamian manifold Mn， we cal主之 ap＿fbrm w＝（1／p！）wcvl．．．αp dUk．．．dUaP a K皿ingρ一fbrm jf it satis丘es （2．1） ． ▽bWα1αz．．．ap十▽alWbdl．．．ap＝0， where▽means the operator of covariant derivative with respect to the Riemannian con−一一 nection． Then we haveON A THEOREM OF GALLOTI：−MEYER□A．CHIBANA
1’9 イ2．2） 〈吻）bal．．．・ap＝（ク・十1）▽δwα1．．．．αφ・．、 ：1。［8］，S． Tachib。。。・and T． K。shiw・d・hav・p・・v・d th・t醐藏㎞tss晒・i・・t・m・・yK迅一 『ingρ一fbrms（〃≧ク≧2）tllen丑ぞ司s a ’space of conStant curvature a血d ，the converse ls 才rue． If a Killing p−fbrm w satisfies ＜2．3） ▽。嘱．．．。，＋・（ ρ9・bWal．．．αP＋Σ（−1）・。，αi・、。1＿a．、．。，）一・， i‘’ 』where a is ・constant and ai means that ai is omitted， then it is called a sp㏄ial Killingクーfb皿 with constant a． Then it is㎞own that Theorem G P】． Let Mn be a completeぷ」〃rply connected Rieman〃ian mo〃ifold ut加itting 理θC‘α1疏〃ing p−forms u⑳idッW匡吻0ぷiガve constant k．ガオ乃θ加研クro伽τ（u，v）血η0τCO〃一 ぷ伽ち沈θη』M銘垣ぷ・〃ie’trie with S”（k）・ R．EMARK 2． To prove Theorem G， tht’ y have msed tbe foll（柄ing M． Obata，s theorem stated in［7］without proof： 皿eorem・H．．Lθ’Mn ．be a complete．simply conneeted Rie．〃tannian〃Utmifold． ln orderノ「or ．Mn・．to・admit −a・．non一τrゴ吻1 solution ￠ of the ld」fferential equation ▽α▽bφc十（1／2）β（2φα8・bC十φb9・碗十iPcgab）：＝0， ・W乃θ・θφ。＝▽。φαn4β畑・sitiv・C励砺， it・is・nec・∬ary and sufici・nt that M” be is・met・・ic ”t，ithぷn（fiノ． ’ We sha皿call a Ki皿ing 1ヰblm whiCh is sp㏄ial with positiveαand of constant length（≠0） ・aSasakian stmcture and a Riemannian manifold admitting such a structure is ca皿ed ：Sasakian． Next we are gOing to recaU the de麺tion of a confbrmal Killing p・fbrm． In Mカwe・窃皿a 汐一f（）rm〃with coe伍cient〃殉．．．ap a conf（）nnal Ki1血g、ρ一f（）皿， if there eXists a（p−1）−fbnnθsuch that
X2．4） ▽bUal．．．zじP十▽a1Ubα2．．．αP P ＝20吻．．．αPgba1−．Σ（−1）i〈6ai．．．iD．．．ap 9あai t＝2 ＋’ebeq＿ai＿qp 9・iq）・ This form e is called the associated form of〃． For a◎onformal Killingクーfbrm〃， the fb110w− ing identities are’㎞own I6】： （2．5） δ〃＝一（n一ク十1）θ， P 【2・a 低）・・…・・＝・ip＋1）（▽晦＿吻＋Σ〈一・1 z＝1）‘θ・・…翁一’・・p gb・・）・ ● 3． Proof of Theorem l． By・o1rr assump60ns， we have fo’ r P−f（）rm況 吻＝0，△・u＝λ〃andλ＝ク（π一ρ十1）た． ＜）perating△toδμ， it fbllows加m△δ＝δ△that△δ〃t． ・nδu． As we血ay use層Theorem D for a coclosed（P−1）−fbrmδ〃， we can ’fi’ fid that 6H is a Killing（ρ一1）−fbrm， which showsり
