Watanabe Yoshiyuki

University of Toyama Repository

Mach . Rep . Toya m aUniv .Vol. 4く198 1l, 129 - 1 3 4A C O N STR UC T IO N OF EINSTE IN ME T R IC SBY W AR PED PR ODU C TYo sh iyuk i W AT AN AB EWe c o n str u ct E in stein Tnetric s by m e a n s of w a rped pr o-du ct a nd study their pr ope rtie s.1. Wa rped pylOdu ct.We r e vie w s o m epr ope rtie s of w a rped pr odu c亡くrll,t2コl.Let くB,gl a nd くF,gl be the R ie m a n nia n m a nifol ds a nd fO つa po sitiv e C - fu n ctio n o n a . Co n si de r the pr odu ct m a nトfol d BxF with pr oコe Ctio n s 7T芸BxF- - - ヰ B a nd lJ芸BxF- - ヰ - F .T he w a rped pr odu ct BxfF is the Rie m a n ni an m a nifoldri 己くBxF,gl defin ed byr Vく1 .1l gくX,Yl ニ gくT央X,T 央Y1 + fくTくxll2gくリカX,り央Y1 ,fo r al l ta nge nt v e cto r s X, Y e TxくBxFl . Let u s ide nti fyTxくBx FI with TTfxlくBl + TリくxlくFl . Note that fo r p 色 F,山 pl is total ly ge ode sic in BxfF a nd TIM-1芸u-1くpl - B1 2 92lュ仙RWA T A N ABEs a nis o m etryくcf . r1コl. We c o n si de r th9 C u r v atur e te n so rof the w a rped pr odu ct BXfF with dim B - 1 . Let Fbe a n n- d im e n sio n al Riem an nia n m a ni fol d a nd f a po sitiv eO Oc - fu n ctio n o n ち . Le亡 tdldt , el,. . .,enl be a n ortho n o r malfr a m ef ield of BxfF,wher e dldt is a u nit v e cto r f ieldo n B a nd eia V e CtO r f iel d o n F . T he n w eha v e thefol lo wing 亡w o le m m a sくcf. t2コl, put ting f甘ニd fldt and fy-ニd2fldt2 .LE M M A1. 1.J V Nくい g 糾 ea,eblec,edl - く11fl2盲く青くfea,feblfec,fedl- Cf7Jfl2く6a c6bd- 6ad6bc1 ,fl u Nく2I gく良くea,eblec,dldtl 三 0 ,r V J Vく31 gくRくdldt , ealdldt , ebl デー くf-,lfl6ab,ぴhe p e 盲 i8 the e u rvaiu p ete n s o rof F a nd lSa ,b , c ,din .LE MMA 1. 2 . Le士 S a nd 言 be 舌he 8 e aZap i, u P 1,,atu r,aof F a nd BXfF p e 8Pe e め eZy. The n we ha veNS ニ ー 2nくfHlfl - nくn - 11 くfVlfl2+ く11fl2盲 .八 ■Let Rla nd Rlbe the Ricci te n s o r s of F a ndBxfF r e spe ctiv ely. T hen by Le m m ai. 1, w e ha v e 亡he fol lo w-ingLE M MA 1. 3.ハ Jくい Rlくeb,edl - く1lfl2もcfeb ,fedl- くくn - 11 くfYlf12+ くfy ylfll6b d,13 0W AT A N A BEr URlくea,dldtl ニ 0 ,r URlくdldt ,dldtl ニ ー nくf-ソf ト32. Resul ts .TH EOREM 2. i. Le七 M be a Rie m a r mia n m a nifoヱd ofdi附O De n Bion nン2 a nd f a po 8i七ive c -fu n e士io n o n1 - dim e n 8io n a乙Rie m a n nia n m a nifo乙d. A n e c e 8 8 a Ty a nd 8 ufficie n士 e o ndi舌io nfTo p a u a pped pT Odu e七 BxfM io be Ein 8ieinia n i8 七ha七 Mi8 Ein 8teinia n a nd f 8a士i8fie 8く央I くf-l2- ff7 t ニ SInくn - 1トPR O O F. Sup po s ethat BxfM is Ein steinia n. The n Le m mai.3 impl ie s thath J rへJRlくeb ,eblニ Rlくdldt,dldtl ニ - nfY YIf ,a ndRlくfeb,feblニ くn - 11 くくfY12- ffVり .On the othe r ha nd,w e hav e く央1 by u sing Le m m a1. 2, becau s er V r vslくn +1l ニ Rlくdノdt,dldtl ニ ー nfHJf .The r efo r e w eha v eRlくfeb,febl こ Sln .Th is show s that M is Ein steinia n.Co n v e r s ely s up po s e亡hat 凹 1s a n Eln s亡ein manifold白atisfying く虫l . The n sin c e1 3 14 WATANABERltfebタfedJ こ くSInI6b d三 くn - 11 くtfY12- f fy 716b d,w e ha v eJ■ tl NRlくeb, ebl 芸 Rlくdldt ,dldtl ,tak ing a c c o u nt of く31 of Letrm a1 .3. T h is c o nclude s thatBxfM is E in steinia n8Let I be the i - dim e n sio n al Eu cl i de a n spa c e withst anda rd Rie m an ni an m etric and くM,gl a n a- d im e n sion alEin stein m a ni fol dwith n o npo sitiv e s c ala r c u r v atu r es. LetO Df be a po sitiv e C - fu n ctio n o n E and c o n si de r the wa rpedpr odu ct ExfM . No w put ting k-SInくn - 1l , w e c o n si der o nlythe c a s e Sく0 . Be c a u s ei f Sン0, the n w e c a n n ot ha v ethes olutio n of d i f fe r e ntial equatio n く央I s u ch that fンO o nE . T he n by a n ele m e nta ry c o npu tatio n, w e ha v eく叫 f - く1I21くBexpくcxtl - くkJct2Ble xpトcxtJl ,whe r e cxis a n a rb itr a ry c o n sta nt a nd B an a rb itr a rypo sitiv e c o n staTlt .t f M is complete , the n by a r e s ult of R. L . B ishopa nd B. 0-Neil 1くs e erlコ1, w e ha v e 亡he fol lo wingTHE O R EM 2. 2. Zf M i8 a C O mPZete Ein stein m a nifoldui士h r - po si士ivQ Se al.ar7 c u r,ua+u p e, 士he n a warped pr30du c士ExfM ui士h く叫 i8 a C O mP乙e士e Ein stein m a n.Jfo乙d Withn egative 8 C aZa y3 c u pVatuP e .Mak ing u s e of T he o r e m 2. 2 indu ctiv ely, w e ha v e13 2W ATA N AB E 5CoR O L LA RY 2. 3 . Zf M iB a e O mP乙ete Ein Biein manifoZdwi士h n o npo 8i士ive 8 eaZap e u pvatu r7e, 士he n 七he prlOduL3七 mc mi-foLd EmxM admits e O mPZete Ein8士ein m eねie 8 Wi七h nega士ive8Ca乙a r3 e u r7Va士u r7e .Mo r e o v e r sup po s e that M is a n E in stein m a nifol dwithn egativ e c u r v atu r e. The n fr o mLe m m a1. 1 a nd く央央1, w e cans e e that ExfM is of n egativ e c u r v atu r e. T hu s we ha v eTH E O R EM 2. 4. Zf M i8 a e O mPZeie Ein stein ma nifo乙dwith n ega士ive c um atu p e3 士he n 士he w a rped pp odu c舌 ExfM8 a士isjTyin.g く叫 iB a C O mP乙e士e Hin 8士ein ma nifo苦d wi七hn ega士iv牢 C u P 7J a士u 21e .Mak ing u s e of The o r e m2. 4 indu ctiv ely, w e hav eCo RO L L A RY 2. 5. Zf M i8 a e O mPZe士e Ein 8iein manifo乙dwi士h n egative c u TVatu r7e, the n the pr30du e七 m a nifoLd EmxMa 血it8 C OmPZe士e Ein stein m etr3ie8 Wi士h 7 Wga七ive e urvatu p e.ExAM P L ES. Let M be a n ir r edu cib le R ie m an nia n globallysyTrm etric spa c e of n o n c o mpa ct ty pe . T he n, sin c e M is a nEin stein m a ni fold of n egativ e c u r v atu r e, w e f ind fr o mT he o-re m 2. 4 亡hat the pr odu ct m a ni fold EX M adTnits c o mpleteEin stein m etrics with n egativ e c u r v atu r e, a nd fu rther m o r efr o m co r ol la ry 2. 5 that the pr odu ct m anifold EmXM admitsc o mplete Ein s亡eln m etrlc s wl亡b n egativ e c u rv atu r e.No w we c a n c o mpute d ir e ctly the le ngth of the cur v atu r ete n s o r sof the w a rped pr odu ct EXfCHnand EXfQHns u chthat f s atisf ie s く叫 , whe r e C Hnis the c o mple x n- d im -en sio n al hy pe rbol ic spa c e a nd 耶nis the qu ate r nio n hy pe r-13 36 WATANABIbolic spa c e9 bec a u s ew e ha ve alr e ady kn o w ntheir e xplicite xpr e s sio n of the c u r v atu r e te n S O r S. Then w ef ind v ario uscomplete n o nho m oge n e o u sEin stein n etric s of n egativ e c u rv -atu r e, be c a u s e w e c a n cho o s epo sitiv e c o n sta nt cx a nd Bs u ch 亡hat the s c ala r fu n ctlo n sく of the le ngth of c u r v atur etenso rl a r e c o n sta nt .REFERENC E Srlコ R . i . Bishop and B . OYNei l 1,Ma nifol ds of n egativ ec u r v atu r e,Tr a m s. Amer . Hath. Soc . , 145く1 9 6 9l, ト 斗S.r2コ N . Ejiri, A n egativ e a n s w e rto a c onje ctu r e of c o n-fo r m al tr a n sfo r m ation s of Rie m a n nia n ma nifolds,J . Hath. So c. Japa n, 3 3く1 9 8 1l, 26 ト 2 6 6.t31 S. Ish i ha r a, Note s o nqu ate r nio n K ah le ria n m a ni folds,J. Dif fe r e ntial Ge o m etry , 9く1 9 7 4l, 48 3- 5 0 0.r41 Y. Wata n abe , On the cha r a cte ristic fu n ctio n s of quate r-nio n Ka h le ria n.spa ce s of c o n sta nt Q- s e ctio n al c u r v-atu r e,Ko-dai Hath. Se m. Rep . , 2 8く1 9 7 7,, 2 8 4- 2 9 9.r5J K. Ya n o, D i f fe r e ntial Ge o m etry o n co mple x a nd Alm o stCo mple x Spa c e s, Pe rga m o nPre s s, く1 9 6 5J.Depa rtm e nt of Mathe m atic s くRec eiv ed Apri l 2 4, 19 8 1Toya m aUniv er sityToya m a9 Japa n1 3 4

A construction of Einstein metrics by warped product

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A construction of Einstein metrics by warped product

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