TRU Mathemat ics 16−2 〔1980）
ON PARACONTACT RI EMANNIAN MANI FOLDS
Dedicated to Professor Y．C．Wong of the University of Hong Kong on his 65th Birthday Shigeo SASAKI （Received Novernber 14，1980〕 §1． IntrotuctiOll In 1976 L Sato ［2］ introduced the concepts of almost paracontact ㎜nifold， almost paracontact Riemamian manifold as analogues of those of almost contact manifold and almost contact Riemannian manifold． In the same year I・ Sato and K・Matsumoto ［3］ introduced the concepts of paracontact Riemamian manifold and para−Sasakian manifold． After that they and T． Adati， T・Miyazawa published many papers which concern these manifolds． In§2 we first give definitions of paracontact Riemamian manifold， para− Sasakian manifold and so on and introduce the concept of t）？es for these manifolds． And then we show that most of paracontact Riemannian manifolds are not compact． 〔Theorem 1〕 In §3 we derive some exa町）1es of para−Sasakian n協nifolds．他far as I㎞ow， there are only a few examples of para−Sasakian manifolds given by I． Sato ［3］． Cbntrary to these， our examples contain several arbitrary functions． 〔Theorem 4〕 In §4 we studγthe automoI〕phism groups of para−Sasa1（ian manifolds and show that it is a Lie group （Theorem 5） and the possible maximum dimension for automorphism groups of para−Sasakian m・・f・・d・・f・yp・〔・，・〕・・÷｛・〔…〕・，〔，・・）｝・・．…5we g・ve an example of para−Sasakian manifold of type （ρ，〈7〕 whose automorphism group hasthe㎜xi㎜dimension．
§2．Paracontact Riemannian manifolds． 1・Let rdn be a differentable manifold of dirnension n 〔n＞2〕． If there exist three tensor fieldsφ，ξ，n over fwith types （1，1〕， 〔1，0〕 and （0，1〕 respectively which satisfy the tensor equations （…〕｛凝：：紘， then we say the triple an almost paracontact structure and♂with this structure an almost paracontact manifold． 75In this case the relations ． ■ 〔・・2）φ；ξ3… 〔2．3） ・、φ1… 1 一 π ＝ ︺ ．τ．﹂ φ
パ
励
hold good too and there exists a positive definite Reiemannian metricθ with the properties タ 〔・・4）・￠ゴξ3・ni コ （…）碗ゴφ；φ1・・hズnh・k・ Such a Riemal皿ian metric g is called an associated metric to the almost paracontact structure（φ，ξ，n〕and the qUadrttple（φ，ξ，n，θ）an a11ぴDst para− contact metric stuエ℃ture． Anゾτwith such a sturcture is called an almost paracontact Riemannian manifbld． 〔1． Sato ［2］） We put 〔…）φピ・仇φ；・ぴ㎝φ￡5i・a・声・i・t…fi・・d・n f・f・）Pe（・・2）・㎞・ing・・h・
covariant derivative with respect to the Riemannian metric g by a co㎜， we consider the conditi㎝ 〔…）・ちゴ・φ萄〔ξt・ゴ・φ；）・ An almost paracontact metric structure satisfying （2・7〕 is called paracontact metric structure and iU「z with such a structure is called a paracontact Riemannian manifbld． 〔The concept of paracontact metric structure here is narrow than the one given by I． Sato 【2】． It is nothing but special paTacontact metTic structure by him．〕 If a paracontact metric structure satisfies fUrther the condition （2・8）φゴ瓦．z＝（一㍗z＋・ゴ・z）ηた＋←θたz＋・k・z〕・ゴ・ カ then it is called para−Sasakian structure． Adifferentiable manifbld〃顧th this structure is called a para−Sasa1（ian manifbld． 〔T． A己ati and K Miyazawa ［1】） Let〃be a Riemannian manifbld with the metric tensorσ． If there exists a （0，1〕 type tensor field TI with norm l whidh satisfies the differential equation （2・9）・ちゴ＝ε〔−9ib’＋・i・ゴ）・〔ε＝1・・−1〕 then we say that♂is a special para−Sasakian manifbld， because we can easily verify that φ，ξdefined by （2．7）2， 〔2．4） and n，g constitute a para− Sasakian structure．〔1． Sato【3D Pr㎝〔2．9）we getON PARAOONTACr RIEMANNIAN MANIFOLDS 〔・…）∼．i＝一・〔・一・〕 2． When 〔φ，ξ，η，g） is an almost paracontact metric structure the tensor fi・ld・O・ha・c・n・t・nt・ig・nvalu…nf．（in・・f・th・m・i・th・・麺1e r・・t O 。nd others are l or−1． We denote the I皿iltiplicity of the eigenvalues l and−1 by p and q 〔0≦p，（1≦n− 1） respectively． Then we see that （2・11） φ1＝P−q， P＋q＝n−1． W・ call・f・with・u・h・1皿・・t p・・ac・ntact皿・t・i・・tru・ture．・．parac・ntact Riemannian ma皿ifold of type （p，q）． In what follows we assume that the indices ㎜always through the following ranges： i，」，k＝1，2，．●・，n， a，Z），o＝1，2，．・・，P， 「 λ，1」，v＝P＋1 ，・・◆，P＋（7， △＝n・ 陥・nfi・an・㎞・t p・・ac・ntact・Ri・㎜i孤㎜ifdld， w。・can・t。k。。n each・ang・nt・pac・Mp〔・・f） ・n…h…㎜・f−｛・。，・λ，・△｝・・tha・ φθ。＝・。・ φ・λ＝一・λ θ△＝ξ， （φe△＝0〕 ・ We call such frame an adapted frame at P of the a1皿ost paracontact Riemannian manifold in cons ideration． 3．Fbr paracontact Riemannia n manifold we get first the fo11㎝ing theorem： THEOREM 1． Ever21ραrαooπカαθカRiemannian manifo Z〈元o了type （P3q） （lp f（7） oαnπo古ゐeoompαot． PROOF． First we remark tha亡 コ リ ξ∵φ；・P−・ by （2．7〕． ぷ）w assume that♂is co㎎）act and orientable and denote the vo1ロne e1㎝ent・f th・趾㎝・mni㎝㎜ifOld byゐ・Th・nr by Green’・th…㎝w・g・t
… L・畑・〔・一か≠・（pfq）
噸・his a c・ntradi・ti・n． H・nce♂・an・・t be c・叩・・t ・nd・ri・nt・b1。． ． Next， supPose that Mz is co町）act but not orientable． Then there is a tw・−f・1d・・ve血9㎜i制d解・f♂砲i・h is c・mPact ・nd。ri。n励1。．（迦防we
@can naturally induce paracontact metric structure of the same type （p，〈1〕． And so we meet again with a contradicition・ Q．E．D． 77COROLLARY． SpeoiαZ pαrα一Sasαkiαn mani∫foZds oαn mot わθ oompaot・ ・ ・…i・ev・d・n・，・ecaus・φ1＝一ε（・一・〕by（…）・ It is an open question to know if there exist co㎎act paracontact Riemaエmian mallifolds 〔especially co叩pact para−Sasakian manifolds） of type （p，p） or not． And if they exist theoretically， it is desirable to find examples of them． §3． Exan4）1es of paracolrヒact Riemannian manifblds． 4．Fi・・t， d…ting th…al n迦b・r・pace with・…dinates ・1，x2，…，xn by五7τ， 1et us intエ・oduce an almot paracontact metric structure of type （p，q〕 〔0≦P，q≦n−1）・n♂． L・tθ（・1，…鋪ノ＋1，…，ψ＋q）be a differenti・b1・ f皿。ti。n・。nd・d。fi。。 th。 t。n，。。 fi。1d，as th・9r。di・nt・fθ・。，△，i．・． 〔…）・。・、農，・、・蒜・△… We also define the tensor fieldφandξby ． 〔3．2〕 （3．3） and the tensor field g by 〔3．4〕 φz・・； ￠1 ・ ・一δ1 △ △ φわ＝−nb φμ＝nμ other components of φ are O， ξ阜・δ； whe「e Ah qnd Bv a「e differentiable p easily see that the relation 〔3．5） Making use of these relations we can easily verify that ．good． If X i・avect・r field・n♂， w・get 9ゐわ＝刀E％。＝η♂。 （b≠・〕・．％。＝nカ∼・ 9v。＝・1．1∼〔pfv〕9uv＝Bv σb△＝ηゐ 9v△＝∼9△△＝1 0sitive fmctions on Rn． Then we c…m （2．1）holds’go6d． By virtue of 〔2．6〕we have φbわ＝Ab−nb・φvv ＝−Bμ＋∼ othe「co㎎）onents of φiゴa「e ze「o・ 〔2．4） ’and 〔2．5〕 hold
ON PARAOON工A⊂汀 RIEMANNIAN MANIFOLDS 9i Xゴ子＝Σ〔∠｛α＿n2）〔YCi）2＋Σ〔Bλ＿η；）〔f）2〔ΣηαXα＋ΣnλX）L＋X△）2， 〔…）d・t1・、・、1 ＝・（・。−n；）・・〔・ズη1〕・ Therefore we see that our tensor field g is positive definite if and only if （…）・。一・：・… ズ・貧… Hence if A。鋤d Bλ・ati・fy（3・7）・〔φ・ξ・η・9〕d・fin・d・b・ve c・n・titut・an・㎞・・t paracontact metric structure of type 〔P，q〕 onガτ． The inve「se mat「i・・f〔gZゴ〕i・v・ry・麺1…ti・gi・・n by ． 〔3．8） So we have the relations （3．9） 5．Next， we shall contact metrlc structure given above reduces structure・ For that purpose， ηゴ。た 砲ich are obtained by by 〔3．4〕，〔2．7〕 reduces by 〔3．10〕 恥ting t｝ロtφゴたi・ 、 ・αα・ロα一・…）一’・・λλ・〔・λ一n；ブ’， 9α△＝¶。〔・α一n2）一’．9λ△＝一η、〔Bズη；「モ 9△△＝1＋Σn：〔Aα一η…）−1＋Σηそ〔Bλ一nf〕−1 ． ・ther・卿・n・nt・・fσ一l are・er。． σαλ一・，9α`・・．・λλφλλ＝一・． ・α△・・αθ・。・… λ△・・λρ・，… ・必 ﾓαα＝一・。．・λ△φλλ・・λ． study the conditions under砲ich the almost para− to a paracontact metric we first note that 9η． ＝3≠一［ゴた・△］ 〔2．4〕 aIld 〔3．3）． As ［ゴ…］・÷〔纂・豊一寮・ゴ、） 〔3．1）to ・ゴ・一
黶Eゴ、・
given by 〔3．5），we see that 〔3．10〕 reduces to 79〔3．11）． iil・・Ab 一 ・・：− 9B f，”−2B・ ・ 2n・2・ The solutions of these differential equations are given by 〔3．12） where ・b ・nd bu theorem： ％・α、（xl．…ノ＋q）・2虚△・・；， B1．1・ゐμ〔xl．…ノ＋q）・一鋤㌔…・ are differentiable functions． Hence we get the following 班0剛2．th。αZm。．カρ。・。…カα・t m・t・i・・t・・u・鋤・・n Rn gi・・n by〔3．ヱ〕
一（3．4）鋤C・・…繭‥・カ・i・if・nd・鳩Zf％唖苧・（3・4）av・㌦力伽S
・ア吻f・・vn〔3・ヱ2）・』θαゐ・nd bu・r・ diff・r・ntiabZ・㌦・㌦・・励微力 αb（tcl・…ノ＋q）＞0・bu〔xl・…ノ＋q）＞0・ コ Since p・・ac・n・ac・趾・−i・n㎜if・・d・wi・hφ1−・〔・一・〕・・e・peci・・ para−Sasakian manifolds （T． Adati and T． Miyazawa ［1D we have the fbllowing Theorem： THEORE“｛3．1ZVze pa”aoon古αθヵmetrio 8tr・uotur？θθ乞ven in三Vheorem 2 is pα㌘α一 Sasαk鋤if it is of type〔n−1．0〕o㌘（0，n−1）・ 6． Our next proble冨n is then to study the conditions under砲ich the paracontact metric structuビe of type 〔ρ，q〕 〔p，q≠0） given in Theorem 2 is para−Sasaldan． T（）solve this problem we have to put the paracontact metric ・t・u・ture i・〔2・8〕and d・t・mi…b ・nd by・・that th・diffe「飽tial equati°ns thus obtained are satisfied． However the calculation is s麺1e but lengthy． So， we state here only the key pt）int and㎝it the detai1． φ、。，、＝一｛蒜｝φw−｛畠｝φ、、〔…n・t−・d〕 ＝一〔θvv［bdブリ］＋gvd［bd．ム］）φvv あ△ わわ ［vd・△Dφカゐ ［vd，∼）］＋9 −〔θON PARACONTACT RIFMANN圧AN MANIR）工J〕S ＝［ゐ姻一∼［bd・△］一［vd・b］・nb ［・d・△］ ＝一顎4・÷争。・嘉・嘉 S・・if●≠d・（2・8〕・・duces t・an・id・ntity as gbd・ηわnd f・rカ≠d・｝1・w・ver， if Z）＝〈1， 〔2．8〕 reduces to −iiξ・嘱〕∼・2・、 il；… by （3．11〕． Putting （3．12） into the last equation， we ha▽e
’ i3−・・、・∪・・
㎜・・ip・i・d by。2x△D・。・。9r。ti。、・h。・。。t diff。。。nt・。、。qu。t、。n。e。ee th。t ・h・fm・ti・yS・b（x’，…，ノ＋q）i・（3．2）皿・t b・・f・h・f・rm 〔…3〕、・、・c・b・（x’・…ノ）・2θ（・．・・〕 where θ is the function given in 〔3．1）． Si皿ilarly， studying the． caseゴ＝b， k＝・・Z＝σi・（2・8〕・w・・see・・ha…h・f皿・・i・…，〔x’・…ノ＋q）i・〔…2） ㎜st be of the fom 〔…3）、・、・・、（ψ＋’・…・zF’＋q）・−2θ（・，・・）・ For any other set of values of indices ゴ，た，Z， 〔2・8〕 reduces to an idelltity． Sur皿arizing the above results， we get the fOllowing theorem： THEOREFf 4・ 批e set oアブ句Ul， tensorβθZ4sφ⊃ξ⊃η，σon ge with oonlF）onents （3．ヱ）一〔3．4）．where Aガ・、（xl．…ノ〕・2（x4’θ）・・；．〔・、・・） 〔3．ヱ4） ・・，〔r”＋’・…ノ＋q）・−2（°eA’e ）・・，2。〔・，・・） ・・nst鋤tesαP・蹴一Sa＄akian s鋤・雄θ・f type（P．q〕〔P．4≠0）． Noteworthy is that the metric tensor g contains functions．P＋q＋1arbitrary
§4．Automorphism groups of para−Sasaki㎝㎜愉1ds． 7・L・t♂be a para−S・・aki飢㎜if・・d with・tru・t・・e t・ns…〔φ，ξ，，，θ）． Adiff・・…phi蹴f・f f・nt・it・elf is said t・b・飢・ut・。・Tphi。m。f th。 para−Sasakian manifold ifアsatisfies the relations 81（φ・4・なφ・ 〔4’1）
戟c．
なξ＝ξ・ 戸9＝σ．而…f。i・th・differ・㎡i・1・f知d戸iS th・du・1㎜P・f 4・血e・set°f
・11・ut・m・埴i・鵬・f・p・・a−S・・aki皿㎜if・1d♂・・n・tit・tes ・ gr・up砲i・h we call the aut㎝・rphi・皿9r・up． W・d…t・it・by・A〔♂〕． On the other hand， a vector field X on Mτis said to be an infinitesimal automorphisln of a para−Sasakian manifbld if it satisfies 早＝o・ 早＝o・ 〔4．2） Lxrt ＝ O・ 阜＝o・ 砲・・gLX d・mt・・th・Li・der迦i・・with・e・pect t・X・lt is equivalent t° the fact that exptX ←。。〈t＜。。） is an autolnorphis皿． The set of all infinitesima1 autpm・rphisms c・n・titut・・aLi・a19・bra乙・ver the real number field R． It is clear that the dimension ofヱ｝is not greater than n（n＋1）／2． Now，．we have the fo11σwing 四EOREM 5． ． サ ZS α ヱ｝tθ ≦r？oup・ th・aut。m・rphi・m gr・up A〔め。f any para−Sasαkictn manifioZd PROOF． By〔4．1），A〔めis a・ub．qr・up・f th・i・㎝・t・y g・・XP 1（め・f the Riemannian manifold Pt4n．with the same fundamental tensor≦7． ｝k）reover， A（めi・c1・sed in・r（f〕． H・nc・， by・th…㎝・n・Li・9r・up・， A（f）i・a Lie group． 8． Next， we shall study the possible maxi皿u！n dimension fOr automorphism grolrps of para−Sasakian manifolds of type ⑫，（r〕． For that purpose， it is sufficient to study the maximum dimension of the Lie algebra L of infinites㎞al autanorphisms． The method used here is s㎞ilar to the one in ’S． Taエmo ［5］． The equation 〔4．2〕 can be written precisely as 碗・φ；ノーり；・♂，ゴφ；… 〔，．，〕・ピ・∼ノー子プ… ・P凸ノ・・’z．ゴ・h… ・砺・♂．、9hゴ・・’z．ゴ・、k…ON PARACONTfACr RIEMANNIAN・MANIrOIDS by（2・4〕・・（4・3）、孤d（4・3）3 f…㎝f・㎝（4・3〕3，巴d〔4・3）、バe・p・e・i・・1y・ 血砺〔2・7〕2・（4・3）、f・11㎝・f・・m〔4・3）2，Uご〔4・3）k麺1ies t』「elati°n 〔…〕・X｛義｝・〆。〆㌦・Z… 廠・re磁、・・are c・輌戚・・f・h・田・一・ar・・−t・・e t・鵬・r・・・・…t equation Call l）e written as 〔4．S〕 お
舞・一｛よ呼・
豊1一鶴・・よ｝縁ピ・
Therefbre， 〔4．3） can be identified with a system of partial differentエa1 タ の ・q皿・i・ns（…〕・・th血・w…fU・・ti・n・ X’．・nd Xt Cゴ植・h・・et・f access・ry conditions 〔4．3）． Hence、4〔f〕 has the maxi皿】皿dimension if and onユy if the syst㎝．of partial differential equations 〔4．5〕with accessory conditions （4．3）is c卿letely integrable．・ And in that case， the㎜㎞d㎞e輌 の ■ ・・eq・a…th・・㎜ber・f・i・d・p・㎡・n…9由ra・・・・・・…鵬ち子。ゴ・f〔…）atap・血tp・f」れ
M・，・・t｛・。・・λ，・△｝b−・dap・・d f・alne・・t・p・i・…ff・th・…h・ four tensorsφ，ξ，n，≦7 at P with respect to the adapted fTame are given by （3・13〕・（3・14∈dgジδZゴ・．Fi・・g we・ha11・・n・ider s°1uti°ns°f〔4・3）at ・・u・t・・ha・Xt〔・）…th・n〔4・・〕、，3，U・t P・e血ce t・ ．〆．、〔・〕一・・‘・fムメ・〕… ． ． 〆．〔P）・ジ．（P〕・o． 3t ，∂ From 〔4．3〕， we get「also ピ．、（・〕一・・ S…h・ma・・x xtC」〔・〕ha・th・f・rm ． （ピ．〔P）〕・ ，9 ♂。、〔・〕…000
0ρμO
AOO
， where A，B are skew s）㎜etric square matrices of order p and〈7 respectively． ゆ th・ref・re・X’Bゴ（・）・an・b・・r・gard・d・・・…㎝・n・・f・h・L・e a・9めra・f・he orthogonal group O〔P）xO（q〕×1． ロ ゆ 』X’ ・nd K’．ゴare・…！・・n・・f（…〕・i・h access・ry・・㎡i・i・ns〔4・・〕・ ・he・・…i・y・f xtBゴ（・〕 with Xt〔P）＝Oconstitutes the Lie algdbra．’，．of重he ・mea・i・・t・・py g・・up LTp・f A（f〕・Since・A〔めi・a・ubg・・XP・f・h・ 83i・㎝・try・9・・upエ（め， th・．acti・ns・f A（f）・n f are・・flect・d f・ithfUlly． zrP・n・he acti・ns・f A（め・n−the輌・・d f・an・｛・。・・λ・・△｝・… H・nc・・ we see that the dimension of the aUtomorphism group is equal to or slnaller than ． ・ dim・・L。・・≦÷｛・（P−・）・q（q−・）｝… CbnseqUently， we get the following theorent： 田…0剛6．吻励・η・伽・f・th・娠・㌘均碗・m．gr・zrp・A〔f〕・τ・卿ρ…a− Sasakian rmzifoZゐ．0アtyPe〔P，q〕isαカmost ｛p（p＋1）＋q（〈1＋1）｝／2十1． In §5， wel．Shall give’an example of para−Sasakian manifolds of type （P，q） 曲ose automorphism group actually has the maximum dimension． §5． An exa㎎）1e of para−Sasakian manifolds whose automorphism group has the maximrn di皿ension． ． 9・’lt i・ ・uffi・i・nt t・c・nsid・r the ca・e・fθ三〇・・ゐ三1・βla≡1 in the para−Sasa1（ian manifold stated in Theorems 3 and 4． Namely，1we have the following theor㎝．： ． THEOREM 7． L・t u…n・磁r拗P鋤一sd・αkia・ manif・Zd・π♂協力 str，uoture tensors φ3ξ3n，9 deゴ：’ined by 〔3．ヱ3）
岡｛嘘よ㍍φ＿’
〔…）ξ’ ・・’△・・、・・；・ （5．3） SVten the dimθn8ion ｛p（ρ＋1）＋q〔q＋1）｝／2＋1 9αz）＝θ2鐙△ ﾂαカ ・、、・θ一n△ﾂλい・鎚・・
other oomponents Oτσαrθaero O了it3 automo鞭phiSm group is θq祐zZ カ0 ■ PROOF． It is sufficient to study the number of linearly independent コ ・・1uti・n・・X㌔f〔4．3）．品鴨v・・，鳩αm・・1v・〔4．3）．c・ncret・1y． Fi・・t， by （3．14），〔4．3）．ﾋ・ies t』t〆i・血d・典・nt卿ψ㎝d（4．3〕麺・i。。
・ha・fi・ac・n・t孤・，・by・△． S血ce a・…卿㎝…fφare c・n・taP・，（）NPARACON工A〔汀RI正べAN MANIFOI、DS の ． 〔4・3〕、t・…u・t繊h・㎜・…〔・舳∂）・・一・・t・・・…h・h・ma・…（φ多）・ So we get ill−・・ llttb−・ by（・．・〕．・・。。ef。re， X・k。。e・・un。t・。n。轍・。h．d。p。nd。仙，。n。・，＿，rF’1 ≠獅 ☆・il・f皿・ti・n・・whi・h d。p。nd。n・y。n♂＋1，…ノ＋q． Next・putting i＝・・ゴ＝b in（4・3）Uw・g・t （…〕2δ。、f・i鐸… PUtting 2）＝αand integrating the resulting eqμation we have 、（55〕xa・．。△。αザ〔。，ユ，…，苛，…，め， where￡αin the right hand side means that we have omitted theα一th variable xαDW・th・n・・n・id・r the caseゐ≠・in（5．4）・nd p・t（5．5〕in it g・tting a〔xl．…．sα．…ノ ∂xb）・9〔・’”えlb”°ri’）… th・1・・t・q・・ti・n・㎞・・t｝・・t∂戸／e。b〔。，ゐfi。・d）d。e， n。t．d。p。。d。p。n 。b 〔bfa〕．So， we see that，，（xIs are fmctions of the fo㎝ プμ＝Σ・αbxz）＋oα 〔Oab＝−Oz，α） and oα are constant． Thus we havewhere o ab ピ＝−o△xa＋Σeαbxb＋oα ． In the・am・way・P・tti・gトλ・ゴ＝・1・1血（4・3）b・w・g・t 」yN＝o△苫λ＋ Σoλvxv＋θλ・ When we put other set of values forちゴ in （4．3）we get no new relations． Hence， any infinitesimal automorphism of the para−Sasakian manifold in cons ideration has the form ・＝Σ・α㌔・Σ・λへ・・△XA・。1、・・b・…λi，・λ衙・ where we have put Xa−∂iiα ， Xx−9…λ 3 ． XA…α、i…λ、i・・、1・・ 85
xαゐ一￠⑥∂…α 一toa∂…わ 3 〔α〈b） へ〆、iλ一・λ、き・（…）・ ［［he set・f vect・r fi・1d・｛X。・XX・XA・X。が㌔｝（α＜ゐ・λ＜1・L〕・・n・tit・t・・ab・・i・ of the Lie algebra五〇f infinitesimal automorphism． They consists of ｛p〔P＋1）＋q〔4＋1）｝／2＋1 vectors． This coiXpletes the proof． COROLLARY． Theρara−Sasαkinn mαn乞fc）Z〈l of type 〔P．9〕 given in Theorem 6is homogeneou8． PROOF． AS is eaSily seen， the finite e（luations of the automorphism grolrp which has〔5．6〕as its infinites㎞al transformations are given by 〔5．7） where（AZ〕・・（P），・ α，・λ，t ar・p・am・ter・…d・mt・th・・9r・WP by G（P，q）． By 〔5．7） the origin O of♂ is transfonned to a point with 。。。rdinat。，（sα，。λ，t）． S。 G（P，q〕i。 tran。itiv。． dS th・i・Qt・・py．9r・up・t O is O（p）×0〔q〕， the para−Sasakian manifold in consideration can be regarded as the homogeneous space G（P，q）／o（P）×o〔q）． Q・E・D・ 一．ｿ 一tαb α
