Yorozu Shinsuke

Kanazawa University Repository for Academic Resources

Ann・Sci･KanazawaUniv・Vol、23,pp、15-26,1986Notesonspace-likefoliationsofcodimensiononeinLorentzmanifoldsShinsukeYoRozu(ReceivedMay6,1986)1.lntroductionThisnoteisdividedintothreeparts・Inthefirstpart,wewillgivesphericallysymmetricmaximalfoliationsofcodimensiononeinaLorentz4-manifoldwithaLorentzmetric:"2==_g(7,)"+f(")"2+72"2+72Sin2adP2.TheresultgivenbyReinhart[7]isaspecialcase.Ifg(7)=/('')=1forl'>0,thenthemetriciscalledtheLorentz-Minkowskimetric.Inthesecondpart,wewillprovethatasphericallysymmetricmaximalfoliationofcodimensiononeintheLorentz-Minkowski("+1)-spaceistotallygeodesic.ThisisessentiallyChengandYau'sTheorem[1](foliationversion).BecauseweconsideronlysphericallysymmetriCfoliations,ourmethodisdifferentfromonein[1]andiselementary.Inthethirdpart,wewilldiscussthegeometricpropertiesofSpace-likefoliationsofcodimensiononeinconnectedLiegroupswithleftinveriantLorentzmetrics,andseveralexampleswillbeshown.ThediscussioninthisnoteismotivatedbyReinhart'spaper[7]andextensionofOshikiri'sTheorem(I81)tonon-compactcase.AlltheobjectsinthisnoteareofclassCoounlessotherwisestated.2.Foliationofcodimensionone"Afoliationofcodimensiononeinamanifold"isafamilyofhypersurfacesfilling".Eachhypersurfaceiscalledaleafofthefoliation.Theexactdefinitionoffoliationofcodimensiononeisasfollows.Ifal-formのonamanifoldMsatisfiestheintegrabilitycondition,i.e.dd)=〃八の,thenのiscalledtodefineafoliationlof16 ShinsukeYOROZUcodimensiononein"([61).Apdthefoliation"iscalledtobedefinedbyの=0.Inotherwords,ifanintegablesubbundleEofrankdim〃一lofthetangentbundleT"overMannihilatesの,thenEiscalledtodefine"([8]).Here,Eisintegrableif[X,Y]Ej~'(E)foranyX,YEF(E),andEannihilatesz"if2"(X)=0foranyXEF(E)(F(E)denotesthespaceofallthesectionsofthebundleE).EachmaximalconnectedintegralmanifoldofEiscalledaleafof".Fordetails,seeReinhart[61.Afoliation､'ofcodimensiononeinaLorentzmanifold"iSspace-likeifeachleafof"isaspace-likehypersurfaceof"([4],[7]).3.SphericallysymmetricmaximalfoliationsandspeciaILorentzmetricsLetJV(")={("',x2,"3)ER31"=((%')2+(%2)2+(%3)2)'/2>a}whereα≧0.WeconsiderqaLorentzmanifold"=R×jV([z)withaLorentzmetric(1)(*2=_g(7,)"+/(7,)"2+,,2"2+"2Sin2〃や2bymeansofthesphericalcoordinate('',8,9')onJV(Cz),wheregand/arepositivevaluedfunctionsofγonanopeninterval（α,CO)．Let'beaspace-likefoliationofcodimensiononein"definedbyの=0.Thefoliation､ダissphericallysymmetricifのcanbeexpressedby(2)の＝鋭十"(γ)〃where/iisafunctionofl､)n(zz,oo)([71).Thefoliation."･ismaximalifthetraceofthesecondfundamentalform(bymeansoftheLevi-CivitaconnectionVwithrespecttothemetric(1))ofeachleafof"iszero([7],[8]).Wewillseekaclosedl-formの＝伽十九(γ)"on"suchthatthespace-likefoliationldefinedbyの=0ismaximal.TheunitnormalvectorfieldTtotheleavesof､ダisgivenbyT=(g(")-!一ﾙ)-"(")')-'"(-9(,"｣:+/(,)-!"(");)Wenoticethatg(7')-'-/(7')-1"(r)2>0because"isspace-like.AnorthonormalOframefield{Xi,XZ,Xb}tangenttotheleavesof"isgivenbyXi=(g("'-f(7)-'"("汁'"(-ル)-'/2g(7")-!曜吻(γ):+/(,)-'2g(")-'f)脇=γ-‘諸ルバsin'"!*Thenwehavespace-likefoliations亜=,＜▽x‘Xj，T＞=(g("'-/(7)-'"(7)2)-劃い{-'ﾙ)-Ig(7)-'"(")＋2/(γ)-2"(γ)3－が(7)-'g(7)-!",(γ）‐が(7)-'g(")-2g'(")"(")+告げ(7)-2/'(7)g(7)-'"(7)＋告げ(")-2g(,"|g'(")'("%}"17where<,>denotestheinnerproductwithrespecttothemetric(1),and'denotesthederivativewithrespecttoγ，Thetraceofthesecondfundamentalformvanishesifandonlyif(3)〃(γ)=(券(γ)‘(γ)-g(")-'g'(")-2,-|)'(")+(#(,)-!g'(,)+'ﾙ)-!g(,)"-|)"("'Thedifferentialequation(3)iseasilysolvedbymeansofanauxiliaryvariable:"(")=g(")''2/(7)-!ﾉ2"(γ)．Then(3)canberewrittenintheform(4)"(")-(g(""｣g'(")+2"-!)(一偽(7)+"('')Then,theeqUation(4)hasthesolutions(々γ)＝±(1+CE(")74)-]ﾉ2,Or:==01whereCisaconstantThuswehaveTHEoREM1.Lgｵ〃=R×Ⅳ(")6eczZ,07'ig"た4-"""加〃”鋤aLo7'g"iz"@g"iC(1),α”〃'6e"sp"g"""sy加沈e鯛c”αce-"g/b肋加〃"co""@g"sわ〃o"e伽〃18ShinsukeYOROZU《域""hyの=0,""27'gの=di+"(")成別g〃夕2s”z‘z〃"α／がα"do"なが●凸"(γ)＝±g('')-!ﾉ2f(',｡)｣′2(1+CE(,")7"4"1/2,07,=00〃(cz,oo)/b"sowzeco,zs〃"ォCThecaseg(")=/(")=1for"E(0,oo),thatis,thecaseoftheLorentz-Minkowskispace,isdiscussedinthenextsection.Thecaseofg('')=1-2"'/"and/(")=g(""]for7/E(211@,oo)wasdiscuSsedbyReinhart[7],andthen(",ds2)istheSchwarzschildexteriorspace-time([41).WemustremarkthatReinharthasfoundallsphericallysymmetricmaximalfoliationsinextendedSchwarzschildspace([7]).Nowwemayconsiderthecaseg('')=1-4"z2/7'2and/f(")=7-2g(")-]forl"E(2wz,oo).Thenwehave(5)"2=_(1_4"z2/'/2)"+7"-2(1_41"2/"2)-1"2+7/2"2+"2sin28〃Ifweset(6) ？＝r＋log(γ2-4沈2)1ﾉ2thenthemetric(5)Canberewrittenintheform(7)曲2＝－(1-41"2〃2)〃＋2γ-1‘〃,＋γ2"2+7"2Sin28〃．Themetric(7)hasbeenintroducedbyOtsuki[5],whohasdiscussedthe@@blackholes"bymeansofsmooth"generalconnection"onthespace-timewiththismetric(7).Ifweset(8)zノー(γ2-4”2)'ﾉ2sinhr〃＝(γ2-4〃)'ﾉ2coshjthenthemetric(5)canbealsorewrittenintheform(9)曲2二=γ-2(－"2+CZZZ2)+7/2"2+7'2SinZadp2ThuswemayhaveanalogousdiscussiononthespaceR×Ⅳ(O)w肋themetric(5)asReinhart'sdiscussion([71)onextendedSchwarzschildspace.＝一spacelikefoliations 194．SphericailysymmetricmaximalfoliationsintheLorentz-Minkowskispace.Let"betheLorentz-Minkowski("+1)-spacewiththeglobalcoordinate(ｵ,"1,･6･,x")andtheLorentzmetricⅢ好=-"2+("')2+…+(伽羅)2Let7'=(("1)2+…+("")2)｣ﾉ2,andwesetthat"+={(/,"',…,"")EM|">0}.Then"+isanopenLorentzsubmanifoldof"([4]).Bymeansofthesphericalcoordinate('",e,,･･･,am_,),theinducedmetricon"+from(10canberewrittenintheform01)"2=－〃2+"2+"2(dB,)2+刃濁γ2Sin2&…Sin2a,_,(daj)2Weconsideral-formのon"definedby0》の="+2W=｣灸倣‘where/iarefunctionson",andwesupposethatのdefinesafoliation､'ofcodimensiononeinM，Thenthefoliation"issphericallysymmetricifthel一formのiswrittenasfollowson"+;側の＝鋭十"(γ)〃where"isafunctionof">0.Undertheassumptionthatthefoliation"in"isspace-like,"istotallygeodesiCifthesecondfundamentalform(bymeansoftheLevi-CivitaconnectionwithrespecttothemetricOO)ofeachleafof'vanishesidentically([4],[81).Hereafterwesupposethatthefoliation,'ofcodimensiononein"definedbyの=0issphericallysymmetricandspace-like.Let"|"+betherestrictedfoliationof"in〃七.Sincell"+isspace-like,wehavethatl－"(7)2>0for">0.Bymeansofthecoordinate(#,",&,…,8"_,)in"+,theunitnormalvectorfieldTtotheleavesof"|"+isgivenbyT=('-"('(-:+"(");)andanorthonormalframefieldtangenttotheleavesof"|"+isgivenby20ShinsUkeYOROZU???‐????? ????、???‐??????．????、????、?? ｜?? ）????? ????庁＝3，4，…，〃ThenwehaveZILI<Vx,X,T>=-(1－"(7)2)｣ﾉ2{"'(7)+("－1)"-'"(")(1－"(")2)}ThusthetraceOfthesecondfundamentalformofeachleafof､ダ|"+iszeroifandonlyif側〃'(7)=-("-1)"-1"(")(1－"(")2)(">0)ThedifferentialequationMhasthesolutions(15a)(15b)"(γ)＝±(1+C72("-'))-]ﾉ2(">0)"(γ)＝0（γ＞0）whereCisapositiveconstantREMARK.Afunction/('')=(1+C"2("~'))~'/2iswell-definedon[0,oo)forariypositiveconstantC.As"goestoirifinity,/(7)approachesO・Andwehavethat/(0)＝1．ThenwehavethefollowingtheoremTHEoREM2.Z,g＃〃6g"eLo""iz-MMMoz"s"("+1)-Wzcg"""ダα妙"〃加吻SWzWZg師bα""S""-雌g允肋肋〃"CO成加g"SZO"O"g伽〃〃；"g"hyの=0,""27'gの●="+ZL]灸血＃α"’たαγE""c加泥so”肱〃'""""""e""ダおわ〃"gEoc"siC.WewillgiveaproofofTheorem2for"=3.Byolirassumption,thefunction"巳inO3isgivenby(15a)or(15b).Wemayset???????????????〆21space-likefoliations%3=7'SinalSin&ByO3for"=3and(13,wehavejicosa,+/Zsin6,cos&+たsin8,sin&="(7,･)-/isin8,+/Zcos8,cos&+/hcosBIsin&=0－たsin8,sin&+たsin&cos&=0.⑩LetNbethevectorfielddualtoの,thatis,(''!v=-3+2跡器Since"'isspace-like,JVisatme-likevectorfieldon"([4]).Foranyfixedr*ER,weconsideracurvec:R－－'"givenbyc(S)=(#*,2~1ﾉ2S,2~is,2~1s).ByOO,wehave,fors≠O,21ﾉ2/i(c(s))+/i(c(s))+/h(c(s))=2"(IsI)-2'ﾉ2/i(c(s))+/Z(c(s))+た(c(s))=0一九(c(s))+/i(c(s))=0.(IWWesupposethat"(,,)=±(1+C7/4)-]ﾉ2(i.e.(15a)for"=3).As7'approachesO,h(7)approaches±1.Thus,assapproachesO,03impliesthat,?????????， ?? 、???????????Bymlwehave（2-"-2‐＋2砦'券 ）＋2-1糸"‘}’3肌(0)＝一所十 敗Ⅱc(0） c(0） c(0）whichisalight-likevector([41).ThiscontradictsthefactthatIVisatime-likevectorfieldon".Therefore,wehavethat/Z(")=0(''>0).Then,by(10,wehave/i=0j=1,2,322 ShinsukeYOROZUon".Thuswehavethatの=dion",whichmeansthat､ダistotallygeodesicBythesamewayasabOve,wecanproveTheorem2for">3.5．Space-likefoliationsinLiegroups.LetGbean"+1dimensionalconnectedLiegroupand9betheassociatedLiealgebraofallvectorfieldsonGthatareinvariantunderlefttranslations.IfwetakeaLiesubalgebraDof9,thenwehaveafoliation.'(6)inG([8]).Infact,foreachpoint%EG,asubmanifoldLx(H)ofGistheleafthrough"ofthefoliation"ダ(6)inG,whereLxdenotesthelefttranslationofGbyxG,andHisaconnectedsUbgroupofGwhoseLiealgebraisb.Let{eo,e,,…,e")beabasisof9.ThenwedenotebyCIKthestructureconstantsof9withrespectto{eo,e,,･･･,e"}.Hereandhereafter,unlessotherwisestated,I,J,K=0,1,…,"andi,/==1,…，〃・Inthissection,weconsiderGwhoseLiealgebra9isgivenby(19 9={eo}+6wheref)isanidealofcodimensiononeof9,and{e,,…,e"}isabasisoff)..NowweノcantakeaLorentzinnerproduct<,>on9suchthat<eo,eo>=-1<eo,e2>=0<ei,Gj>=6".ThenwehavealeftinvariantLorentzmetric<,>onGinducedfromtheLorentzinnerproduct<,>on9([2],[3]).LetVbetheLevi-CivitaconnectiononGwithrespecttotheleftinvariantLorentzmetric<,>([21,[31).Thuswehaveaspace-likefoliation"(6)inGwiththeleftinvariantLorentzmetric<,>.Weset"lj=<Veiej,e｡>=(CKj+CWi)/2Thenwehavethefollowingdefinitions([8]):〆))ismaximal(resp・totallygeodesic)ifZ匙,脇＝0 (resp."=0foralli,ノ).And夕(0）isofconstantmeancurvatureifZ筵,脇isaconstant・Next,thecurvaturetensorRofVdefinedbyspace-likefOliations 23R(e,,eJ)eK=V[g"鋤】eK-▽e,Ve,eK+▽動▽e,ek,andtheRiccicurvatureRjC(eo)inthedirectionofeoisdefinedbyRic(e｡)=国是,<R(eo,ei)eo,ei>([2],[41).Bythedirectcalculation,wehaveRic(e｡)=-邸4(")'-z],"(cK,+〃ThuswehavethefollowingProposition:PRoPosITIoN3.Leオ､ダ(b)69α妙αcg-"g/b加伽〃Qfco""e"sわ〃0"g加娩gα加2ﾉeLiegm"G"肋伽α加zﾉg賊〃”γi""Z,071g"た〃ze"c.ZWe犯娩gRiccic"γ沈z如彩Rib(e｡)加地g或形c肋〃Qfeois"o"-加si""9．〃り形o"gγRjb(eo)=0がα"αo〃なが"(b)たわ"/"ggod@sic.""7伽""必ず幼g賊加”""オLo""た，"g鯛c"〃オ肋g〃‘ダ(I)）たわ〃〃配0"sic.EXAMPLE1.I刺"…|■H伽…ⅢG=|"%｡"R|9＝＝i剛|←|網|-WIlileO＝[eO,el]=e2[eO,e2]=0[el,e2]=0I)={e,,e2}<,>:aleftinvariantLorentzmetriconGsuchthat<eo,eo>==-1<eo,ei>=0<ej,aj>=6"(i,/=1,2)RiC(eo)=-1/2Thenthespace-likefoliation､ダ(b)ofcodimensiononeinGismaximalandnottotally24ShinsukeYOROZUgeodesic.ExAMPLE2.｜職|">α＆■-"|G=｜淵"…’9＝＝｜淵l-l鮒|"1州leO＝＝[e｡,e,l=e![e｡,e2]=e2[e,,e2]=06={e,,e2}<,>:aleftinvariantLorentzmetriconGsuchthat<eo,eO>=-1<eO,ei>=0<ei,e,>=8"(',/=1,2)Rib(eo)=-2ThentheLorentzmanifold(G,<,>)isofconstantcurvature-1,andthespace-likefoliation､ダ(6)ofcodimensiononeinGisofconstantmeancurvature(notmaximal).ExAMPLE3|惇専制Ⅲ"州|鰯鯛裟伽ⅧG=|;1｡1"%｡R|9二＝|索‘ルーI蝿lo-I州｜動＝＝[eo,e,]=e2【eO,e2]=-4[e,,e2]=06={e,,e2}<,>:aleftinvariantLorentzmetriconGsuchthat25space-likefoliations<eo,ei>=0●(i,ノ=1,2)????ー?????????????Ric(eO)=0Frhen(G,<,>)isaflatLorentzmanifold,andthespaCe-likefoliation"(fi)of|CodimensiononeinGistotallygeodesic.ExAMPLE4｜|刺">α‘…｜G=｜|嫌|"%｡…’9＝｜榊l-l剛←|繍|･-蝋IIeO＝二[eo,ei]=0(j=1,2,3)[el,e21=0[el,e3]=e2[e2,e31=0f)={e,,e2,e3}<,>:aleftinvariantLorentzmetriconGsuchthat<eo,eO>=-1<eO,ei>=0<ei,ej>=6"(j,/=1,2,3)RjC(eo)=ORjC(e,)=-1/2Thenthespace-likefoliation""(I))ofCodimensiononeinGistotallygeodesicReferencesS.-Y・ChengandS.-T.Yau,MaximalspacelikehypersurfacesintheLorentz-Minkowskispaces,Ann.ofMath.104(1976),407-419.J.Milnor,CurvaturesofleftinvariantmetricsonLiegroups,AdvanceSinMath.21(1976),293-329.K.Nomizu,LeftinvariantLorentzmetricsonLiegroups,OsakaMath.J.16(1979),143-150.B.O'Neill,Semi-RiemannianGeometrywithApplicationstoRelativity,AcademicPress,NewYork,1983.T.Otsuki,Acertainspace-timemetricandsmoothgeneralconnections,KodaiMath.J.8(1985),307-316．B.L.Reinhart,Foliatedmanifoldswithbundle-likemetrics,Ann.ofMath.69(1959),119-132.B.L.Reinhart,MaximalfoliationsofextendedSchwarzschild5pace,J.MathPhys､14(1973),719.R.TakagiandS.Yorozu,MinimalfoliationsonLiegroups,TohokuMath.J.36(1984),541-554､[1］[2］{3］【4］[5］[6］[7］[8］:訓.91黍:126 Shinsuke;YOROZU！I‐DepartmentlofMathematicSICollegeofLiberalArtslKapaZaWaIIniVersity'Kanazawa,920Japan《－一ソI’’?

Notes on space-like foliations of codimension one in Lorentz manifolds

https://kanazawa-u.repo.nii.ac.jp/?action=repository_action_common_download&item_id=40762&item_no=1&attribute_id=26&file_no=1

Notes on space-like foliations of codimension one in Lorentz manifolds

Abstract

Similar works

Full text

Available Versions

Kanazawa University Repository for Academic Resources