Inflection points and double tangents on anti-convex curves in the real projective plane

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

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aINFLECTIONPOINTSANDDOUBLETANGENTSONANTI-CONVEXCURVESINTHEREALPROJECTIVEPLANEGUDLAUGURTHORBERGSSONANDMASAAKIUMEHARAABSTRACT.AsimpleclosedcurveγintherealprojectiveplaneP2iscalledanti-convexifforeachpointponthecurve,thereexistsalinewhichistransversaltothecurveandmeetsthecurveonlyatp.Weshallprovetherelationi(γ) 2δ(γ)=3foranti-convexcurves,wherei(γ)isthenumberofindependent(true)in ectionpointsandδ(γ)thenumberofindependentdoubletangents.Thisformulaisare nementoftheclassicalM¨obiustheorem.Weshallalsoshowthattherearethreein ectionpointsonagivenanti-convexcurvesuchthatthetangentlinesatthesethreein ectionpointscrossthecurveonlyonce.Ourapproachisaxiomaticandcanbeappliedinothersituations.Forexample,weprovesimilarresultsforcurvesofconstantwidthasacorollary.INTRODUCTIONLetP2denotetherealprojectiveplane.WeassumecurvestobeparameterizedandC1-regular.AsimpleclosedcurvesinP2issaidtobeanti-convexorsatisfyingtheBarnerconditionifforeachpointponthecurve,thereexistsalinewhichistransversaltothecurveandmeetsthecurveonlyatp.Thisconditionisthen=2caseofaconditionintroducedbyBarnerin[2]forsimpleclosedcurvesintherealprojectivespacePnforn≥2.Ananti-convexcurveisautomaticallynotcontractible.Letγ1andγ2betwoarcsinsomeaf neplaneA2 P2.Wesaythatγ1crossesγ2inaclosedarcαifαisamaximalcommonarcofγ1andγ2andthereisanopensubarcα ofγ1containingαsuchthatthetwocomponentsofα αdonotlieonthesamesideofγ2(butmightnotbedisjointfromγ2).Thearcαcanofcourseconsistofasinglepoint.Ifγ1meetsγ2transversallyinapointp,thenγ1ofcoursecrossesγ2inp.ExamplesofcrossingcurvesareshowninFigure

1.

FIGURE1.Crossingcurves

Anin ectionpointpofacurveγwillbecalledatruein ectionpointifthetangentlineofγatpcrossesγinanarccontainingp.Twoin ectionpointsarecalledindependentiftheyarenotcontainedinanarcofγconsistingoftruein ectionpoints.(Thein ectionpointsonthecurveγ2ontherightinFigure1arenotindpendent.Ontheotherhand,the

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

2GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

threein ectionpointsinFigure4areindependent.)Wewilldenotethemaximalnumberofindependenttruein ectionpointsonγbyi(γ).

AdoubletangentofacurveγisroughlyspeakingalineLthatistangenttoγattheendpointsofanontrivialarcαofγcontainedinanaf neplaneA2 P2insuchawaythatαislocallyarounditsendpointsonthesamesideofL∩A2.(Aprecisede nitionwillbegiveninSection4.)Wecallαadoubletangentarc.Asetofdoubletangentarcsα1,...αkissaidtobe

independentifanytwoofthearcsareeitherdisjointoroneisasubarcoftheother;seeFigures2and3.

FIGURE2.Twotypesofindependentdoubletangent

FIGURE3.Dependentdoubletangents

Wewilldenotethenumberofelementsinamaximalsetofindependentdoubletangentarcsbyδ(γ).ItwillfollowfromTheoremA,whichwenowstate,thatδ(γ)isindependentofthechoiceofamaximalsetofindependentdoubletangentarcsonγ.

TheoremA.LetγbeaC1-regularanti-convexcurveinP2whichisnotaline.Ifthenumberi(γ)ofindependenttruein ectionpointsonγis nite,thensoisthenumberδ(γ)ofelementsinamaximalsetofindependentdoubletangents,and

( )i(γ) 2δ(γ)=3

holds.Inparticular,thenumberδ(γ)doesnotdependonthechoiceofamaximalsetofindependentdoubletangentsifi(γ)is nite.

Formula( )isreminiscentoftheBoseformulaforsimpleclosedcurvesintheEu-clideanplanesayingthats t=2,wheresisthenumberofinscribedosculatingcirclesandtisthenumberoftripletangentinscribedcircles.Thisformulawasprovedforcon-vexcurvesbyBosein[3]andinthegeneralcasebyHauptin[7].OurmethodtoproveTheoremAwillbesimilartotheoneusedbythesecondauthortoprovetheBoseformulain[19].Theauthorsdonotknowwhetherformula( )holdsfornon-contractiblesimpleclosedcurveswhicharenotnecessarilyanti-convex.

Thereisawell-knownformulaforgenericclosedcurvesintheaf neplaneA2duetoFabricius-Bjerrerelatingthenumbersofdoublepoints,in ectionpoints,anddoubletangents;see[4].Whenthecurveshavenoin ectionpoints,Ozawa[12]gaveasharpupperboundonthenumberofdoubletangents.FormulasforrealalgebraiccurvesinP2goatleastbacktoKlein;seethepaper[20]ofWall.

Wewillalsoprovethefollowingtheorem.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS3

TheoremB.LetγbeaC2-regularanti-convexcurveinP2whichisnotaline.Thenγhasatleastthreein ectionpointswiththepropertythatthetangentlinesatthesein ectionpointscrossγonlyonce.

Thetheoremisoptimal.Anin ectionpointpiscalledcleanifthetangentlineatpmeetsthecurveinaconnectedset.Acleanin ectionpointisatypicalexampleofanin ectionpointasinTheoremB.ThenoncontractiblebranchofaregularcubicinP2hasthreecleanin ectionpoints.M¨obiusprovedthatasimpleclosednoncontractable

curvein2Phasatleastthree(true)in ectionpoints.Severalproofsthisresultareknown;see[8],

[6],and[13].Onecanshowwithexamplesthatnoneofthesehastobeacleanin ectionpoint;seeFigure4.

FIGURE4.Asimpleclosedcurvewithnocleanin ectionpoints

Asimilarresultisprovedin[17]and[18]forcleansextacticpointsonastrictlyconvexcurveintheaf neplane.Itsaysthatsuchacurvehasthreeinscribedosculatingconicsandthreecircumscribedosculatingconics.ItshouldalsoberemarkedthattheTennisBallTheorem([1]and[2]),thetheoremofSegreonspacecurvesin[14],andthere nementoftheFour-VertexTheoremin[16]canbeconsideredasgeneralizationsoftheM¨obiusTheorem;see[16].

IntheproofofTheoremBweuseanapproachthatgoesbacktoH.Kneser’sproofofthefourvertextheorem;see[9],[19],andalso[16],[15].(AfurtherdevelopmentofthisapproachisalsocrucialintheproofofTheoremA.)

Thetheoremswillbeprovedinlatersections.Herewewouldliketoexplainsomeofthebasicideasintheproofs.Letπ :S2→P2betheuniversalcoveringofP2.Sinceγisnotcontractible,itliftstoasimpleclosedcurveγ thatdoublecoversγ.Thereis ponS2(whichisthedoublecoverofthelinethrougheverypointponγ agreatcircleL

Lp)thatonlymeetsγ inpandtheantipodalpointT(p)= p.Theparametrizationofγ

2andtheorientationofSgiveusatangentandnormalvector eldalongγ .Wewillassume

thatthenormaldirectionpointstotheleftsideofthecurve.Wede neapositiverotationdirectionalongthecurvebyrotatingthenormalvectortowardsthetangentvector.Noticethatthepositiverotationdirectionistheclockwisedirection.Letusnowrotatethecircle paroundpasfaraspossibleinthepositivedirectionthroughcircleswhichonlymeetγL inpandT(p).WedenotethelimitinggreatcirclebyCp.Therearetwopossibilities.The rstisthatCponlymeetsγ inonecomponent.Thenpisacleanin ectionpoint.TheotherpossibilityisthatCpmeetsγ inmorethanonecomponent;seeFigure5.Inthiscasepmayormaynotbeanin ectionpoint,butitisofcoursenotacleanin ectionpoint.Wede neaclosedsubsetF(p)bysetting

(0.1)F(p)=Cp∩γ .

WeidentifyS1withtheimageofthecurveγ andintroduceonS1acyclicorderthatagreeswiththeorientationofthecurve.Wewill rstassumethatnolinemeetsγin

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

4GUDLAUGURTHORBERGSSONANDMASAAKI

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FIGURE5.Thelimitinggreatcircle

in nitelymanypointsandthendiscussthegeneralcase.IfpinS1isnotanin ectionpoint,weletδdenotethedistancefromptothenextpointq∈F(p)in(p,Tp),where(a,b)denotestheintervalfrom

a∈S1tob∈S1withrespecttothecyclicorderofS1andF(p)isde nedinequation0.1.Letp1bethemidpointoftheinterval[p,q].ThesubsetF(p1)liesintheinterval[p,q]∪[Tp,Tq].Ifp1isnotacleanin ectionpointweletδ1denotethedistancetothepointq1closesttop1inF(p1)∩(p1,Tp1).Noticethatδ1≤δ/2.Iteratingthisprocess,weeitherarriveatapointpnwhichisacleanin ectionpoint,orwegetasequence(pn)thatconvergestoacleanin ectionpoint.AswewillseeinSection2,thisapproachleadstotheexistenceofatleastthreein ectionpoints.IntheproofofTheoremBweonlyuseafewaxiomaticpropertiesofthefamily{F(p)}p∈S1ofclosedsubsetsinS1.Itcanthereforebeappliedtodifferentsituations.

InSection5,weapplythemethodtoconvexcurvesofconstantwidth.

FIGURE6.Thesupportingfunction

LetγbeastrictlyconvexcurveinR2.Foreacht∈[0,2π),thereisauniquetangentlineL(t)ofthecurvewhichmakesangletwiththex-axis.Leth(t)bethedistancebetweena xedpointointheopendomainboundedbyγandthelineL(t);seeFigure6.Notethattgivesaparametrizationofthestrictlyconvexcurveγwhichwewillusefromnowon.Thefunctionhiscalledthesupportingfunctionofthecurveγwithrespecttoo.Astrictlyconvexcurvehasconstantwidthdifandonlyifh(t)+h(t+π)=dholds.

Wenow xacurveγofconstantwidthd.Foreachpointponthecurve,thereexistsauniquecircleΓpofwidthdsuchthatΓpistangenttoγatp,thatisΓpandγmeetatpwithmultiplicitytwo.SinceΓpisthebestapproximationofγatpamongthecirclesofwidthd,wecallΓptheosculatingd-circleatp.Generically,theosculatingd-circleofγatpdoesnotcrossγatp.

WewillprovethefollowingtheoreminSection5.

TheoremC.LetγbeaC3-regularstrictlyconvexcurveofconstantwidthd.Thenthereexistatleastthreeosculatingd-circleswhichcrossγexactlytwice,bothtimestangentially.Moreover,thesethreecirclescoincidewiththeosculatingcircles(intheusualsense)ateachoftheircrossingpointsonγ.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS5

Theabovetheoremisare nementofthefactthattherearesixdistinctpointsonγwhoseosculatingcircleshaveradiusd/2.(Basicpropertiesofcurvesofconstantwidthcanbefoundin[21].)InFigure7weindicatethethreeosculatingcirclesofdiameterdofthecurveofconstantwidthwhosesupportingfunctionis(d/2)+sin3t.

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FIGURE7.Thethreeosculatingcircles

WewillalsoproveaformulaanalogoustotheoneinTheoremAforcurvesofconstantwidthinSection5.

1.INTRINSICLINESYSTEMS

Inthissection,weshallderivesomebasicpropertiesofthefamilyofclosedsubsets{F(p)}p∈S1de nedinequation(0.1)intheintroduction.Weshallthenusetheseproper-tiestode newhatwewillcallan‘intrinsiclinesystem’.

Letγ:P1→P2beaC1-regularanti-convexcurveinP2,whereP1isaclosedcircleconsideredasaprojectiveline.WeassumethattheimageofγisnotalineinP2.Letπ :S2→P2andπ:S1→P1bethecanonicalcoveringprojections.Thenthereexistsasimpleclosedcurveγ :S1→S2suchthat

π γ =γ π.

ponS2suchthatπ p)=Moreover,foreachpointponγ ,thereexistsagreatcircleL (L pintheclockwisedirectionthroughgreatcirclesthatonlymeetγLπ(p).ByrotatingL inp

andtheantipodalpointTp,wearriveatthelimitinggreatcircleCpasintheintroduction.

psuchthatitpassesintoDγLetDγ .WeorientL bethedomainonthelefthandsideofγ aftergoingthroughp.TheorientationofthegreatcircleLpinducesanorientationonthe

limitinggreatcircleCp.

IfCisanorientedgreatcircle,wedenotebyH+(C)(resp.H (C))theclosedhemi-sphereontheleft(resp.right)handsideofC.

ByapplyingasuitablediffeomorphismtoS2,wecanmapγ ontotheequatorandDγ ontheupperhemisphere.Ifwecomposethiswiththestereographicprojectionintothe p)lookasinFigure8.plane,γ andH+(L

Thoughγ maynotbestar-shapedingeneral,weshallfrequentlyusethiskindofsketchesofγ tosimplifythe gures.

Thefollowingassertionisobvious.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

6GUDLAUGURTHORBERGSSONANDMASAAKI

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FIGURE8.

p)Proposition1.1.Thearcofγ :S1→S2fromptoTp(resp.fromTptop)liesinH (L p)).(resp.H+(L

Proposition1.2.ThelimitinggreatcircleCphasthefollowingproperties.

(a)Thearcofγ fromptoTp(resp.fromTptop)liesinH (Cp)(resp.H+(Cp)).(b)ThesetF(p)hasatleastthreeconnectedcomponents,ifCpisnotthetangentline

ofγ atp

.

FIGURE9. p,thepropertyin(a)followsfromPropositionProof.SinceCpisthelimitofcircleslikeL

1.1.Toprove(b),wesupposethatCpisnotatangentlineofγ atp∈S1.ThenCpmeetsγtransversallyatpandTp.HenceifCponlymeetsγ inthesetwopoints,onecanrotateitslightlyinpositivedirectionthroughcurvesthataretransversaltoγ inpandTpandonlymeetγ inthesetwopoints.Thiscontradictsthede nitionofCp.ThusthereexitsapointqinF(p)=Cp∩γ whichisdistinctfrombothpandTp.Sinceγ isnotagreatcircle,pandTpbelongtodifferentconnectedcomponentsofF(p).SincebothCpandγ aresymmetricwithrespecttoT,itfollowsthatCpisneitheratangentlineatpnoratTp.IfqisinthesameconnectedcomponentofF(p)asp(orTp),Cpcontainsthesegmentofγ betweenpandq(orTpandq),whichimpliesthatCpmustbethetangentlineatp(resp.Tp),acontradiction.

Conversely,wehavethefollowing

Proposition1.3.IfagreatcircleCthroughpandTpsatis esthefollowingtwoproperties,thenCcoincideswithCp.

(a)Thearcofγ fromptoTp(resp.fromTptop)liesinH (C)(resp.H+(C)).

(b)Cistangenttoγ atallpointsinC∩γ differentfrompandTpandifCisnot

tangenttoγ atpandTp,thenC∩γ containsapointdifferentfrompandTp.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS7

Proof.SinceCistangenttoγ atallpointsinC∩γ differentfrompandTp,wecanrotateCslightlyinnegativedirectionintoagreatcirclewhichmeetsγ transversallyinpandTpanddoesnothaveanyfurtherpointswithitincommon.Itnowfollowsfromthede nitionofCpthatC=Cp.

WewilldenotebyF0(p)theconnectedcomponentofF(p)=Cp∩γ containingpfor1eachpointponS.

Proposition1.4.Supposethatγ:P1→P2isananti-convexcurvewhichisnotalineandmeetsalineinP2inatmost nitelymanyconnectedcomponents.Thenthecorrespondingfamily{F(p)}p∈S1ofsubsetsofS1satis esthefollowingproperties:

(L1)p∈F(p).

(L2)F(p)isaclosedpropersubsetofS1andhas nitelymanyconnectedcomponents.(L3)Ifq∈F(p),thenTq∈F(p)whereT:S1→S1istherestrictionoftheantipodalmaponS2toγ .

(L4)Supposep′∈F(p)andq′∈F(q)satisfy

p≤q≤p′≤q′(≤Tp)

p≥q≥p′≥q′(≥Tp),

where≥and≤arethecyclicorderofS1.ThenF(p)=F(q).

(L5)Ifπ(F(p))=π(F0(p)),thenπ(F(Tp))=π(F0(Tp))whereπ:S1→P1denotesthecanonicalprojection.

(L6)q∈F0(p)ifandonlyifF(p)=F(q).

(L7)Let(pk)beasequenceinS1thatconvergestoanelementpinS1,andlet(sk)beanothersequenceinS1suchthatsk∈F(pk)andlimsk=s.Thens∈F(p).

Proof.(L1)isobvious.(L2)isatrivialconsequenceoftheassumptionthatγandalinemeetinatmost nitelymanyconnectedcomponents.(L3)followsfromthefactthatγ and parebothsymmetricwithrespecttotheantipodalmapT.L

Wenowprove(L4).IfCpandCqaregreatcircleswhichmeetintwopointswhicharenotantipodal,thenCpmustbeequaltoCq.Supposep′∈F(p)andq′∈F(q)andp≤q≤p′≤q′(≤Tp)orp≥q≥p′≥q′(≥Tp)holds.ThenthesubarcofCqbetweenqandq′must

meetCptwice.Oneisbetweenpandp′,andtheotherisbetweenp′andTponCp.(SeeFigure10forthecasep≤q≤p′≤q′.)ThusCp=Cqholds.or

FIGURE10.

Nowweprove(L5).Ifπ(F(p))=π(F0(p)),thenF(p)consistsoftwoconnectedcomponents.ByProposition1.2(b),Cpisatangentlineatp.ThegreatcircleCTpcoincideswiththegreatcirclewhichwegetbyrotatingCpinnegativedirectionthroughgreatcirclesmeetingγ onlyinpandTpuntilithitsγ .ThegreatcirclesCpandCTpcannotcoincidesinceγisnotaline.ItfollowsthatCTpisnottangenttoγ atpandhencealsonotatTp.ByProposition1.2(b),F(Tp)containsatleastthreecomponents,twoof

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

8GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

whichconsistofpandTpsincetheintersectionbetweenCTpandγ istransversalinthesepoints.Henceπ(F(Tp))isnotconnectedandweseethatπ(F(Tp))=π(F0(Tp)).

Wenowprove(L6).Supposeq∈F0(p).Wemayassumethatq=p.ThenF0(p)isaclosedintervalandCpmustbethetangent

linebothatpandq.ItfollowsthatCpmustbeequaltothegreatcircleCqbyProposition1.3.ThisimpliesF(p)=F(q).NowweassumethatF(p)=F(q).WeletAdenotethesetofpointsrinF(p)=F(q)suchthatthetangentgreatcircleofγ inrcontainsF(p)=F(q)andrisnotatruein ectionpoint.LetBdenotethecomplementofAinF(p)=F(q).ByProposition1.2thesetBcoincideswithF0(p)∪T(F0(p))=F0(q)∪T(F0(q)).NownotethatasetT(F0(r))cannotcoincideswithasetF0(s)foranyrandsinS1sincethecurveγ crossesCrfromrighttoleftinF0(r)andCsfromlefttorightinF0(s);seeFigure11.

Finallyweprove(L7).WemayassumethatsisneitherpnorTp.Afterreplacing(pk)byasubsequenceifnecessary,wemayalsoassumethatCpkconvergestoagreatcircle

C.SinceCpksatis esproperties(a)and(b)inProposition1.3forallk,sodoesC,andit

followsthatC=Cpholds.Hences∈F(p). Remark1.5.Wewillcallafamily{F(p)}p∈S1ofclosedsubsetsofS1anintrinsiclinesystemifitsatis esproperties(L1)–(L7)inProposition1.4.Thisisananalogueofthesomewhatsimplerintrinsiccirclesystems,see[19]and[16],efulinprovingtheexistenceoftwoinscribed(resp.circumscribed)osculatingcirclesofagivensimpleclosedC2-regularcurveintheEuclideanplane.

FIGURE11.Negatvein ectionandpositivein ection.

Aninfectionpointofacurveγ iniscalledpositiveifthetangentgreatcirclecrossesγ fromrighttoleft,andnegativeifthetangentgreatcirclecrossesγ fromlefttoright.De nition1.6.Let{F(p)}p∈S1beanintrinsiclinesystem.Apointp∈S1satisfying

π(F(p))=π(F0(p))(resp.π(F(Tp))=π(F0(Tp))),

iscalledapositivecleaninfectionpoint(resp.anegativecleanin ectionpoint).

Apositive(resp.negative)cleaninfectionpointofγ isapositive(negative)in ectionpointbyde nition.Sincethesignofacleanin ectionpointisreversedbytheantipodalmap,thenotionismeaningfulforγ butnotforγ.

2.CLEANINFLECTIONPOINTS

InthissectionweproveTheoremBintheintroduction.Thecrucialpointisthatweonlyuseproperties(L1)–(L6)ofintrinsiclinesystemstoprovethetheoremundertheassumptionthatγmeetsalineinatmost nitelymanycomponents.ItisonlyinthelaststepwhereweremovethisassumptionthatweusespecialpropertiesofcurvesinP2.

Lemma2.1.Letp∈S1.Supposethatq∈F(p)∩(p,Tp).Letrbeapointin(p,q).SupposethatrisnotcontainedinF0(p).Then

π(F(r)) π((p,q))

holds.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS9

Proof.Supposeπ(F(r))containsanelementa∈π((p,q)).Let{a +,a }bethepreimageofaunderπ.Withoutlossofgenerality,wemayassumethata +∈(p,Tp].Sincea∈π((p,q)),wehavea +∈(q,Tp].Hencewehavetheinequality

p<r<q≤a +≤Tp.

By(L4),wehaveF(p)=F(r).Inparticularr∈F0(p)by(L6),whichisacontradiction.

Withsimilarargumentswecanprovethefollowinglemma.

Lemma2.2.Letp∈S1.Supposethatq∈F(p)∩(Tp,p).Letrbeapointin(q,p).SupposethatrisnotcontainedinF0(p).Then

π(F(r)) π((q,p))

holds.

Nextweprovethefollowinglemma.

Lemma2.3.Letp∈S1.Supposethatq∈F(p)∩(p,Tp)and(p,q)∩F0(p)= .Letrbethemidpointof(p,q).Thenatleastoneofthefollowingthreecasesoccurs:

(i)risapositivecleanin ectionpoint.

(ii)Thereexistp1,q1∈F(r)∩(r,q)suchthatp1∈F0(r)and(p1,q1)∩F0(r)= .(iii)Thereexistp1,q1∈F(r)∩(p,r)suchthatp1∈F0(r)and(q1,p1)∩F0(r)= .

Proof.Assumethatrisnotapositivecleanin ectionpoint.Thenthereexistsapointb∈π(F(r)),suchthatb∈π(F0(r)).Let{q1,Tq1}bethepointssuchthatπ(q1)=b.Since(p,q)∩F0(p)= ,wehaver∈F0(p).ThusbyLemma2.1,wehaveb∈π(F(r)) π((p,q)).Sowemayassumethatq1∈(p,q)withoutlossofgenerality.Sinceb∈π(F0(r)),wehaveq1∈F0(r).Therearetwopossibilities,onebeingq1∈(r,q)andtheotherbeingq1∈(p,r).

First,weconsiderthecaseq1∈(r,q).SinceF0(r)isapropersubsetofS1,itisalinearlyorderedsetwithrespecttotherestrictionofthecyclicorderofS1andonecande neitssupremumandin mum.Weset

p1:=sup(F0(r)).

SinceF0(r) (p,q)andr∈F0(r),itholdsthatp1∈[r,q].Ontheotherhand,sinceq1∈F0(r)andq1∈(r,q),wehave

r≤p1<q1<q.

Thisiscase(ii).

Next,weconsiderthecaseq1∈(p,r).Weset

p1:=inf(F0(r)).

SinceF0(r) (p,q)andr∈F0(r),itholdsthatp1∈[p,r].Ontheotherhand,sinceq1∈F0(r)andq1∈(p,r),wehave

r≥p1>q1>p.

Thisiscase(iii).

Similarlywegetthefollowinglemma.

Lemma2.4.Letp∈S1.Supposethatq∈F(p)∩(Tp,p)and(q,p)∩F0(p)= .Letrbethemidpointof(q,p).Thenatleastoneofthefollowingthreecasesoccurs:

(i)risapositivecleanin ectionpoint.

(ii)Thereexistp1,q1∈F(r)∩(r,p)suchthatp1∈F0(r)and(p1,q1)∩F0(r)= .(iii)Thereexistp1,q1∈F(r)∩(q,r)suchthatp1∈F0(r)and(q1,p1)∩F0(r)= .

WewilluseLemma2.3andLemma2.4toprovethefollowingproposition.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

10GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

Proposition2.5.Letp∈S1.Supposethatq∈F(p)∩(p,Tp)and(p,q)∩F0(p)= .Thenthereexistsapositivecleanin ectionpointsin(p,q)suchthatπ(F(s)) π((p,q)).Proof.Supposethattherearenopositivecleanin ectionpointsin(p,q).Letδbethelengthoftheinterval(p,q).Letrdenotethemidpointoftheinterval(p,q).ByLemma

2.3orLemma2.4,therearetwopointsp1,q1∈(p,q)satisfyingthefollowingproperties:

(1)q1∈F(r)andp1∈F0(r).

(2)(p1,q1)∩F0(r)= ifq1>p1and(q1,p1)∩F0(r)= ifq1<p1.

(3)Thelengthoftheintervalbetweenthetwopointsp1andq1islessthanorequaltoδ/2.

Sincep1∈F0(r),wehaveF(r)=F(p1)by(L6).Sowehave′(1)q1∈F(p1).

(2′)(p1,q1)∩F0(p1)= ifq1>p1and(q1,p1)∩F0(p1)= ifq1<p1.

Wecanrepeatthisargumentreplacing{p,q}by{p1,q1}.ApplyingLemma2.3andLemma2.4inductively,we ndsequences(pn)and(qn)satisfyingthefollowingproper-ties:

(a)pnliesintheintervalbeteenpn 1andqn 1,andqn∈F(pn).

(b)(pn,qn)∩F0(pn)= ifqn>pnand(qn,pn)∩F0(pn)= ifqn<pn.

(c)Thelengthoftheintervalbetweenthetwopointspnandqnislessthanorequaltoδ/2n.

ItfollowsfromLemma2.1andLemma2.2that

π(F(pn)) π(pn 1,qn 1).

Inparticular,thelengthofπ(F(pn))islessthanδ/2n 1.Weset

y=limpn=limqn.

Thelimityliesbetweenpnandqnforalln.

Wewillnowprovethatπ(F(y))={π(y)}.Supposethatπ(F(y))doesnotonlyconsistofπ(y).Thenthereisapointz∈F(y)suchthatTy>z>y.Forsuf cientlylargen,weeitherhave

Tpn>z>qn>y>pn

or

Ty>Tqn>z>pn>y.

Inbothcases(L4)impliesthatF(y)=F(pn).Inparticulary∈F0(pn),whichcontradicts(qn,pn)∩F0(pn)= .Thuswecanconcludethatπ(F(y))={π(y)},whichimpliesthatyisapositivecleanin ectionpoint.Thisisacontradiction.Hencethereisapositivecleanin ectionpointsin(p,q).ByLemma2.1,wehaveπ(F(s)) π((p,q)).

ByreversingtheorientationofS1,Proposition2.5impliesthefollowing

Proposition2.6.Letp∈S1.Supposethatq∈F(p)∩(Tp,p)and(q,p)∩F0(p)= .Thenthereexistsapositivecleanin ectionpointsin(q,p)suchthatπ(F(s)) π((q,p)).Corollary2.7.Letp∈S1.Supposethatq∈F(p)∩(p,Tp)andq∈F0(p).Thenthereexistsapositivecleanin ectionpointsin(p,q)suchthatπ(F(s)) π((p,q))andF(s)∩F0(p)= .

p′=supF0(p).

Sinceq∈F0(p)andF0(p′)=F0(p),wehave

q>p′≥p,(p′,q)∩F0(p′)= .

ApplyingProposition2.5tothepair(p′,q),we ndapositivecleanin ectionpointsin(p′,q) (p,q).WehaveF(s)∩F0(p)= sinceπ(F(s)) π((p′,q)).

Similarlywegetthefollowingcorollary.Proof.Weset

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS11

Corollary2.8.Letp∈S1.Supposethatq∈F(p)∩(Tp,p)andq∈F0(x).Thenthereexistsapositivecleanin ectionpointsin(q,p)suchthatπ(F(s)) π((q,p))andF(s)∩F0(p)= .

ApplyingCorollary2.7andCorollary2.8,wegetthefollowing:

Corollary2.9.Supposethatq∈F(p)satis esq=Tpandq∈F0(p).LetJbetheopenintervalboundedbypandq.Thenthereexistsapositivecleanin ectionpointsinJsuchthatπ(F(s)) π(J)andF(s)∩F0(p)= .

TheoremBintheintroductionisaconsequenceofthefollowingtheoremifthecurveγmeetsalineinatmost nitelymanycomponents.

Theorem2.10.Let{F(p)}p∈S1beanintrinsiclinesystem.Thenthereexistthreeposi-tivecleanin ectionpointss1,s2,s3inS1suchthats2∈(s1,Ts1)ands3∈(Ts1,s1).Moreover,thesetsF(s1),F(s2),F(s3)aremutuallydisjoint.

Proof.Takeapointpwhichisnotacleanin ectionpoint.Thenthereexistsapointq∈F(p)suchthatq∈F0(p).ByCorollary2.9,thereisacleanin ectionpoints1betweenpandq.By(L5),wehaveπ(F(Ts1))=π(F0(Ts1)).Thenthereexistsapointu∈(s1,Ts1)suchthatu∈F(Ts1)butu∈F0(Ts1).ThenbyCorollary2.9,we ndacleanin ectionpoints2on(u,Ts1) (s1,Ts1).NoticethatTu∈F(Ts1)andTu∈F0(Ts1).Hencewe ndanotherpositivecleanin ectionpoints3on(Ts1,Tu) (Ts1,s1)byCorollary2.9.ThesetsF(s3)andF(s2)aredisjointsinceF(s2) (u,Ts1)andF(s3) (Ts1,Tu).

SupposethatF(s2)∩F(s1)= .SinceF(s2)=F0(s2)andF(s1)=F0(s1),wehaveF(s2)=F(s1)by(L6).ThenTs1∈F(s2)contradictingF(s2) (u,Ts1).ThusF(s2)∩F(s1)= .SimilarlyweshowF(s3)∩F(s1)= .

Untilnow,wehaveassumedthatγmeetsalineinatmost nitelymanycomponents.WenowproveTheoremBinthegeneralcaseusingthatsuchcurvesaregenericinthesetofanti-convexcurves.IntheproofwewillneedthatthecurveγisC2.SofarweonlyusedthatitisC1.

ProofofTheoremB.Letγbeanarbitraryanti-convexcurveonP2thatweassumetobeπ-periodic,thatisγ(t)=γ(t+π)fort∈R.Apointp∈R3\{0}determinesapoint[p]inP2,where[p]denotesthelineinR3spannedbyp.Thereisanπ-antiperiodicC2-regularmapF:R→R3suchthat

γ(t)=[F(t)]∈P2

whereamapF(t)iscalledπ-antiperiodicifitsatis esF(t+π)= F(t)forallt∈R.ThemapFhastheFourierseriesexpansion

F(t)=a0+ ∞ ancos(2n+1)t+bnsin(2n+1)t ,

n=1

wherea0,a1,b1,...areallvectorsinR3andthisseriesconvergesuniformlytoF(t).Weset

FN(t)=a0+ N ancos(2n+1)t+bnsin(2n+1)t .

n=1

OnecaneasilyshowthatγN(t)=[FN(t)]isalsoanti-convexregularcurveforsuf cientlylargeNsinceγisC2.Weset

γ (t)=FN(t)N

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

12GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

Bytakingasubsequence,wemayassumethatsj(N)convergestosjforj=1,2,3.Sinceγ isnotagreatcircle,cleanpositivein ectionpointsdonotaccumulatetocleannegativein ectionpoints.Thuswehave

0≤s1<s2 π<s3<s1+π<s2<s3+π<2π.

Thesesixpointsmaynotbeclean exes.However,thetangentgreatcirclesatthesesixpointstopologicallycrossγ exactlytwice.Hencethecorrespondingtangentlinesofγonlycrossγonce.

3.FURTHERPROPERTIESOFINTRINSICLINESYSTEMS

Inthissectionwederivesomepropertiesofintrinsiclinesystems,whichwillbeusedinthenextsectiontoproveTheoremAintheintroduction.Throughoutthissectionwewillassumethatanintrinsiclinesystem{Fp}p∈S1isgiven.

Forapointp∈S1,weset

Y(p):=F(p)\(F0(p)∪TF0(p)),

Y+(p):=Y(p)∩[p,Tp],

F+(p):=Y+(p)∪F0(p),Y (p):=Y(p)∩[Tp,p],F (p):=Y (p)∪T(F0(p)).

Forexample,inthecaseof

Figure12,wehave

F0(p)={p},Y+(p)={q1,q2,q3},Y(p)={q1,q2,q3,Tq1,Tq2,Tq3}.

FIGURE12.De nitionofY(p).

De nition3.1.Anopeninterval(a,b)issaidtobeadmissibleifb∈(a,Ta)andtherearenopositivecleanin ectionpointsin(a,b).

Let(a,b)beanadmissibleinterval.ThenY+(p)isnon-emptyforallp∈(a,b).Soweset(SeeFigure13)

µ (p):=infY+(p),(p,Tp)µ+(p):=supY+(p)

(p,Tp)

forp∈(a,b).Forexample,

µ (p)=q1,µ+(p)=q3

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS

13

FIGURE13.De nitionofµ±(p).

holdsinthecaseof

Figure12.Moreover,weset inf[a,Ta]Y+(a)ifaisnotapositivecleanin ectionpoint,µ (a):=inf[a,Ta]TF0(a)ifaisapositivecleanin ectionpoint, sup[b,Tb]Y+(b)ifbisnotapositivecleanin ectionpoint,µ+(b):=sup[b,Tb]F0(b)ifbisapositivecleanin ectionpoint.

Figure14explainsthede nitionsofµ (a)andµ+(b)whenaandbarecleanin ectionpointsandneitherF0(a)norF0(b)reducesapoint.

FIGURE14.De nitionsThesede nitionshaveanaloguesinthetheoryofintrinsiccirclesystem;seep.190in

[19]byUmehara.TheresultsinthissectioncorrespondtoLemma1.3,Theorem1.4andTheorem1.6in[19].TheleftandtherightofFigurecorrespondtothede nitionofµ (a)andµ+(b)whena,bispositivecleanin ectionpoints,respectively.

1Remark3.2.LetSrevbethe1-dimesionalsphereS1withthereversedorientation.Then

1{Fp}p∈Srevgivesanotherintrinsiclinesystem.Anadmissibleinterval(a,b)of{Fp}p∈S1

1correspondstotheadmissibleinterval(b,a)of{Fp}p∈Srev,andµ (p)(p∈(a,b))with

1respectto{Fp}p∈S1coincideswithµ+(p)withrespectto{Fp}p∈Srev.

Lemma3.3.Let(a,b)beanadmissibleinterval.Thenwehavetheinequalities

b≤µ+(p)<Ta

forallp∈(a,b]and

b<µ (p)≤Ta

forallp∈[a,b).

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

14GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

Proof.We rstassumethatp∈(a,b).Thenpisnotapositiveclean

in ectionpointandY+(p)isnonempty.We xq∈Y+(p)arbitrarily.ThenbyCorollary2.9,thereisapositivecleanin ectionpointron(p,q).Since(a,b)isanadmissiblearc,wehaveq>r>b.Supposethat

(Tb>Tp)>q≥Ta.

Thenwehave

b>p>Tq≥a.

SinceTq∈Y (p),thereisapositivecleanin ectionpointon(Tq,p) (a,b)byCorol-lary2.9,whichcontradictsthefactthat(a,b)isanadmissiblearc.ThuswehaveTa>q,whichimpliesq∈(b,Ta).Sinceqisarbitrary,wehave

b<µ (p)≤µ+(p)<Ta

forallp∈(a,b).

FIGURE15.Thecaseµ+(b)≥Ta.

Next,weconsiderthecaseq=b.Ifbisnotapositivecleanin ectionpoint,µ+(b)∈Y(b)andtheaboveargumentsyieldb<µ+(b)<Ta.Soweassumebisapositivecleanin ectionpoint.Thenb≤µ+(b)holdsbyde nition.Supposenowthatµ+(b)≥Ta.(SeeFigure15.)ThenT(µ+(b))∈F0(b)andµ+(b)=Tb.Thereisthereforeapositivecleanin ectionpointbetween(T(µ+(b)),b)byCorollary2.9,whichisacontradictionsinceT(µ+(b))∈(a,b)and(a,b)isadmissible.Thuswehaveµ+(b)<Ta.

Finally,weconsiderthecaseq=a.Ifaisnotapositivecleanin ectionpoint,µ (a)∈+Y(a)andtheaboveargumentsyieldb<µ (a)<Ta.Soweassumeaisapositivecleanin ectionpoint.Thenµ (a)≤Taholdsbyde nition.Supposenowthatµ (a)≤b.(SeeFigure16.)+

FIGURE16.Thecaseµ (a)≤b.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS15

Thenµ (a)=Ta.Sinceµ (a)∈F0(a),thereisapositivecleanin ectionpointbetween(a,µ (a))byCorollary2.9,whichisacontradictionsinceT(µ (a))∈(a,b)and(a,b)isadmissible.Thuswehaveb<µ+(b). Proposition3.4.Let(a,b)beanadmissibleinterval.Thenwehavetheinequalities

(b≤)µ+(b)≤µ+(p),

µ (p)≤µ (a)(≤Ta)

forallp∈(a,b).

Proof.Inthepreviouslemma,wealreadyprovedthat

b<µ+(p)

forallp∈(a,b).Supposenowthatµ+(p)∈(b,µ+(b)).ApplyingLemma3.3to(p,b),wegetb≤µ+(b)<Tp.Thus

p<b<µ+(p)<µ+(b)(<Tp)

holds.Sincep,µ+(p)∈F+(p),wehaveF(b)=F(p)by(L4).Thusbislikepnotapositivecleanin ectionand

µ+(b)=µ+(p),

contradictingthetheassumptionµ+(p)<µ+(b).Sowehaveµ+(p)≥µ+(b).

ByLemma3.3,wehaveµ (p)<Ta.Nowwesuppose

µ (a)<µ (p)<Ta.

ApplyingLemma3.3to(a,p),wegetp<µ (a).Thus

p<µ (a)<µ (p)<Ta(<Tp)

holds.Sincep,µ (p)∈F+(p),wehaveF(a)=F(p)by(L4).Thenaislikepnotapos-itivecleanin ectionpoint.Thuswehaveµ (a)=µ (p),contradictingtheassumptionµ (a)<µ (p).Sowehaveµ (p)≤µ (a). Corollary3.5.(MonotonicityLemma)Let(a,b)beanadmissiblearcandp,q∈(a,b).Supposethatp<q.Thenwehave

µ (p)≥µ (q),µ+(p)≥µ+(q).

Moreoverµ (p)>µ+(q)holdswhenF(p)=F(q)andµ (a)>µ+(b)iftherearepointspandqin(a,b)suchthatF(p)=F(q).

Proof.The rsttwoinequalitiesfollowdirectlyfromProposition3.4.

Wenowprovethatµ (p)>µ+(q)whenF(p)=F(q).AssumethatF(p)=F(q)andµ (p)≤µ+(q).ByProposition3.4wehave

(a<)p<q<µ (p)≤µ+(q)<Ta.,

whichimpliesby(L4)thatF(p)=F(q),whichisacontradiction.Henceµ (p)>µ+(q).

Finallyweprovetheinequlityµ (a)>µ+(b)undertheassumptionthattherearepointsp,q∈(a,b)suchthatp<qandF(p)=F(q).FromProposition3.4andtheinequalitywehavejustprovedfollowsthat

µ+(b)≤µ+(q)<µ (p)≤µ (a)

whichprovestheclaim.

Proposition3.6.(Semi-continuity)Let(a,b)beanadmissiblearc.Then

x→a+0 limµ (x)=µ (a),x→b 0limµ+(x)=µ+(b).

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

16GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

Proof.Weshallprovethe rstformula.Thesecondformulacanbeprovedsimilarly.(SeeRemark3.2.)Whenthereisapointp∈(a,b)suchthatp∈F0(a),theassertionisobvious.Sowemayassumethat(a,b)∩F0(a)= .Let(rn)beastrictlydecreasingsequencein(a,b)convergingtoa.Therearepointspnandqnintheinterval(a,rn)suchthatF(pn)=F(qn)sinceotherwisetheclosedsetF(q)wouldcontaintheinterval[a,rn]forallq∈(a,rn)anditwouldfollowthat[a,rn] F0(a).HencebyProposition3.4andCorollary3.5wehavethat

µ+(r1)<µ (rn)<µ (rn+1)<µ (a)

holds.Sothesequenceµ (rn)hasalimits.Sinceµ (rn)∈F(rn),(L7)impliesthat

s∈F(a).

Sinceµ+(p1)≤qn≤µ (a),wehaveµ+(p1)≤s≤µ (a).Since(a,b)∩F0(a)= ,wehavethat(a,µ (a))isdisjointfromthesetF(a).Thuswehaves=µ (a)sinces∈F(a). Theorem3.7.Let(a,b)beanadmissiblearc.Thenforanyq∈(µ+(b),µ (a)),thereexistsapointp∈(a,b)suchthat

µ (p)≤q≤µ+(p).

Proof.Weset

Bq:={x∈(a,b);µ+(x)≤q}.

ByProposition3.6wehavethatx→limb 0µ+(x)=µ+(b)+0.Thusapointx∈(a,b)suf cientlyclosetobbelongstoBq.SinceBqisnon-empty,wecanset

p:=[infa,b](Bq).

Sinceµ (a)>q,wehavep∈(a,b).Bythede nitionofp,thereexistsasequence(rn)inBqsuchthatnlim→∞rn=p+0.Byde nitionofBq,wehave

µ (rn)≤µ+(rn)≤q.

Sincenlim→∞µ (rn)=µ (p)byProposition3.6,wehave

µ (p)≤q.

Ontheotherhand,let(sn)beasequencesuchthatlimsn=p 0.Byde nitionofBq,wehaveq<µn→∞

+(sn).Sincenlim→∞µ+(sn)=µ+(p),wehaveq≤µ+(p).

4.DOUBLETANGENTS

Wewewillassumethroughoutthissectionthatγ:P1→P2isananti-convexC1-regularcurvewhosenumberi(γ)oftruein ectionpointsis nite.ItfollowsfromthelastassumptionthatalineinP2meetsthecurveγinatmost nitelymanycomponents.

Lemma4.1.Letγ:P1→P2beananti-convexcurve.SupposethatγmeetsalineLinγ(a)andγ(b)anddenoteoneoftheclosedintervalsonP1boundedbyaandbby[a,b].ThenoneofthetwoclosedlinesegmentsL1andL2onLboundedbyγ(a)andγ(b),sayL1,hasthepropertythatγ([a,b])∪L1liesinanaf neplaneandγ([a,b])∪L2isnothomotopictoapoint.Thecurveγ([a,b])∪L1boundsacontractibledomainhavingacuteinterioranglesatγ(a)andγ(b)ifitisfreeofself-intersections.

WecallL1thechordwithrespecttotheinterval[a,b]anddenoteitby

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS17

Proof.Wechooseapointc∈[a,b].ThenthereisalineLcwhichmeetsγonlyinγ(c).ThenLcmeetsLinonepointwhichweassumetobeonthelinesegmentsonLboundedbyγ(a)andγ(b)thatwedenotebyL2.Thenγ([a,b])∪L1liesinanaf neplane.

SinceLisnotnull-homotopic,eitherγ([a,b])∪L1orγ([a,b])∪L2isnotnull-homotopic.Soγ([a,b])∪L2isnothomotopictoapoint.

Assumeγ([a,b])∪L1isfreeofself-intersectionandletDdenotethecontractibledomainintheaf neplaneboundedbyγ([a,b])∪L1.Ifitsinteriorangleatγ(a)orγ(b)isnotacute,anylinepassingthroughthepointmeetsγ,whichcontradicstheanti-convexityofγ.

Thefollowingassertionisoneofthefundamentalpropertiesofanti-convexcurves.

Proposition4.2.Letγ:P1→P2beananti-convexcurve.Let[a,b]beaclosedintervalonP1andsupposeγ([a,b])meetsalineLinA2at

a=t1<t2<···<tn=b.

Then

γ(t1),γ(t2),...,γ(tn)

lieon

γ(a)γ(ti 1).Thenanylinepassingthroughγ(ti)mustmeetγ((t1,ti)),whichcontradictstheanti-convexityofγ.

ByLemma

4.1,γ([a,b])andthechord

fort∈[a,b],

fort∈(a,b)),γ(t)b a

whichisthecurveonegetsbyreplacingγ([a,b])by

γ(a)γ(b)istangenttoγatγ(a)andγ(b).

(2)thereisapointinγ([a,b])whichisnotcontainedin

γ(a)γ(b).

Remark4.5.If(a,b)isadoubletangentinterval,thenthesamecannotbetruefor(b,a)=P1\[a,b].Infact,thereductionγ2ofγwithrespecttotheinterval[b,a]has[b,a]asanin ectionintervalwhichviolatesproperty(3)inDe nition4.3.ThisphenomenonisexplainedinFigure17wherethetwosketchesindicatethesamecurveγindifferentaf neplanes.

De nition4.6.Letγ:P1→P2beananti-convexcurve.Twodoubletangentintervals(a1,b1)and(a2,b2)arecalledindependentiftheyaredisjointoriftheclosureofoneiscontainedintheother.

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

18GUDLAUGURTHORBERGSSONANDMASAAKI

UMEHARA

FIGURE17.γinthedifferentaf neplanes

WenowbegintheproofofTheoremAinIntroduction.

ProofofTheoremA.Toproveformula( ),wewillstartwithadoubletangentinterval(a,b)andintroducethefollowingreductionsofγ.Weletγ1bethereductionofγwithrespecttothedoubletangentinterval[a,b]andweletγ2bethereductionofγwithrespecttotheinterval[b,a];seeFigure

18.

FIGURE18.γ1andγ2

Wenowbringacoupleoflemmasandpropositionsthatwillbeneededto nishtheproofofTheoremA.

Lemma4.7.Thecurvesγ1andγ2arebothwithoutself-intersections.

Proof.Wewillprovetheclaimforγ1.Supposeγ(P1\[a,b])meetsthechord

γ(a)γ(c)

inthisordersincea<b<c.Thisisacontradiction.Itfollowsthatγ1doesnothaveself-intersections.Onecansimilarlyprovethatγ2doesnothaveself-intersections.

Thefollowingisakeytoproveformula( ).

Proposition4.8.Thecurvesγ1,γ2arebothanti-convexandtheidentity

(4.1)i(γ)=i(γ1)+i(γ2) 1

holds.

Proof.We rstshowthatγ1isanti-convex.Wemayassumethatγ([a,b])liesinanaf neplaneA2.Forapointx∈P2,thepenciloflinespassingthroughxisaaprojectivelineinthedualspaceofP2thatwedenotebyP1(x).Forapointt∈P1,wede neasubsetBγ(t)ofP1(γ(t))suchthateachlineLinBγ(t)meetsγonlyatpandListransversaltothetangentlineatp.Sinceγ(t)isananti-convexcurve,Bγ(t)isnon-emptyforallt∈P1.OnecaneasilyprovethatBγ(t)isanopenintervalinP1(x).WewillcallBγ(t)theBarnersetofγ.

WehavethatBγ(t)iscontainedintheBarnersetBγ1(t)ofγ1foreveryt∈[a,b],sincenolineL∈Bγ(t)canmeetthechord

γ(a)γ(b)at

a=t1<t2<···<tn=b.

ByProposition4.2,

γ(t1),γ(t2),...,γ(tn)

lieon

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS19

Supposenowthatthereexistsapointx∈

γ(ti)γ(ti+1)andx=γ(ti),γ(ti+1)

.

FIGURE19.

Wenowset

I:=[ti,ti+1].

InthefollowingargumentweworkinA2thatweequipwiththeorientationsuchthat

γ(t)x,

(3)β(t)generatesalineinBγ(t).

Weset

IL:={t∈I;α(t),β(t)isapositiveframe},

IR:={t∈I;α(t),β(t)isanegativeframe},

thatisIL(resp.IR)consistsofthosetwiththepropertythattheBarnerdirectionβ(t)isontheleftof(resp.rightof)of

γ(a)γ(b).Henceitfollowsthatti∈ILandti+1∈IR,and

thusthatneitherILnorIRisempty.ThisisacontradictionandwecanconcludethattheBarnersetofγ1atapointx∈

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

20GUDLAUGURTHORBERGSSONANDMASAAKIUMEHARA

By(4.2)and(4.3),wehencehavei(γ1)+i(γ2)=I1+I2+1=i(γ)+1,

whichproves(4.1).

Corollary4.9.Ifi(γ)=3,thentherearenodoubletangentintervalsonγ.

Proof.Supposethatthereisadoubletangentinterval.Thenwecanconsidertheanti-convexcurvesγ1andγ2asinProposition4.8.Sincebothi(γ1)andi(γ2)areatleast3byTheorem2.10,wehave

i(γ)=i(γ1)+i(γ2) 1≥3+3 1=5

whichcontradictsi(γ)=3.

Weareassuminginthissectionthatthenumberi(γ)is nite.Thishasaconsequencefornumberofelementsinasetconsistingofindependentdoubletangentintervalsasthenextcorollaryshows.

Corollary4.10.Thenumberofelementsinasetofindependentdoubletangentintervalsis nite.

Proof.Weassumethatthisnumberisin nite.Letnbeanarbitrarypositiveinteger.Thenwecan ndindependentdoubletangentintervals(a1,b1),(a2,b2),...,(an,bn).Weordertheintervalssuchthat(ai,bi)doesnotcontain(aj,bj)fori<j.Wecanassociateto

(1)(1)(a1,b1)twoanti-convexcurvesγ1andγ2aswasdonebeforeLemma4.7.Thenwe

(1)(2)usethesameconstructiontoassociateto(a2,b2)andγ1twonewanti-convexcurvesγ1(2)(k)andγ2.Inthiswaywecangeta nitesequenceofpairsofanti-convexcurvesγ1and

(k)γ2fork=1,...,n.ByProposition4.8wehave

i(γ)=

(k)(k)(n)i(γ1) n+k=1n i(γ2)(k)Sincei(γ1),i(γ2)≥3,wehavei(γ)≥3 n+3n=3+2n.Sincenisarbitrary,this

contradictsthefactthati(γ)is nite.

TheproofofthenextpropositionreliesontheresultsofSection3.

Proposition4.11.Iftherearenodoubletangentintervalsonγ,theni(γ)=3holds.

Letγ :S1→S2betheliftofγtoaclosedcurveonS2.Wewillneedthefollowinglemmaintheproofofthepoposition.

Lemma4.12.Let(a,b)beanadmissibleintervalonS1inthesenseofDe nition3.1.Supposethattherearenodoubletangentintervalsonγ.Thentherearenotruein ectionpointson(µ+(b),µ (a)).

Proof.Let{F(p)}p∈S1betheintrinsiclinesystemassociatedtotheliftγ .Supposethatthereisatruein ectionpointc∈(µ+(b),µ (a)).ByTheorem3.7,thereexistsapointp∈(a,b),suchthat

µ (p)≤c≤µ+(p).

Sincecisatruein ectionpoint,thelimitinggreatcircleCpcannotpassthroughγ (c).Thisimpliesthatthereisadoubletangentintervalonγ.Thiscontradictionprovestheclaim. ProofofProposition4.11.ByTheorem2.10,thereareatleastthreepositivecleanin ec-tionintervals[a1,a2],[b1,b2]and[c1,c2]onS1someofwhichmayofcoursereducetopoints.Weassumethat

a1≤a2<b1≤b2<c1≤c2

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delt

INFLECTIONPOINTS21

andthattherearenopositivecleanin ectionpointson(a2,b1)and(b2,c1).

ByLemma4.12,therearenoin ectionpointson(c2,Tb1)since(b2,c1)isanadmis-siblearcandµ+(c1)=c2,µ (b2)=Tb1.Sinceπ((c2,Tb1))=π((Tc2,b1)),therearenoin ectionpointson

(4.4)A:=(c2,Tb1)∪(Tc2,b1).

Therearealsonopositivecleanin ectionpointson[a2,b1].ApplyingLemma4.12totheinterval(a2,b1),weconcludethattherearenoin ectionpointson

(4.5)C:=(b2,Ta1)∪(Tb2,a1).

(c2,a1)=(c2,Tb1)∪(Tb1,Tb2)∪(Tb2,a1).

ApplyingLemma4.12totheinterval(c2,a1),weconcludethattherearenoin ectionpointson

(4.6)B:=(a2,Tc1)∪(Ta2,c1).Inparticular,therearenopositivecleanin ectionpointson

Nowitfollowsfrom(4.4),(4.5),(4.6)thattherearenoin ectionpointson

S1\([a1,a2]∪[Tc1,Tc2]∪[b1,b2]∪[Ta1,Ta2]∪[c1,c2]∪[Tb1,Tb2])=A∪B∪C,andhencethati(γ)=3.

Wecannow nishtheproofofTheoremA.Wewillletδ(γ)denotethenumberofelementsinamaximalsetofindependentdoubletangentintervals.Thenumberδ(γ)is nitebyCorollary4.10.Itwillfollowfromtheproofthatδ(γ)doesnotdependonthemaximalsetthatwasusedtode neit.

Weshallproveformula( )byinductionoveri(γ).Wheni(γ)=3,then( )holdssinceδ(γ)=0byCorollary4.9.Soweassume( )holdswheni(γ)≤n 1andn≥4andproveitfori(γ)=n.Sincei(γ)≥4,thereexistsatleastonedoubletangentintervalI=(a,b)byProposition4.11.Thereexistnon-negativeintegersiandjsuchthat

(1)I,I1,,...,Ii,J1,,...,Jjisamaximalfamilyofindependentdoubletangentin-

tervals.

(2)I1,...,IiaresubetsofI,

(3)J1,...,Jjlieon P1\(a,b).

Thenwegettwoanti-convexcurvesγ1,γ2withrespecttoI=[a,b].Bytheinductionassumptionδ(γ1)andδ(γ2)donotdependonthechoiceofthesetofindependentdoubletangentintervals.SinceI1,...,IiandJ1,...,Jjaremaximalsetsofindependentdoubletangentintervalsonγ1andγ2respectively,wehave

i+j+1=δ(γ2)+δ(γ2)+1.

By(4.1),wehave i(γ) 2(i+j+1)=i(γ1) 2δ(γ1)+i(γ2) 2δ(γ2) 3.

i(γ1) 2δ(γ1)=i(γ2) 2δ(γ2)=3.

Thuswehave

i(γ) 2(i+j+1)=3,

whichimpliesthatthenumberi+j+1oftheindependentdoubletangentintervalsisindependentofthechoiceofI,I1,...,Ii,J1,...,Jj.Thuswehaveδ(γ)=i+j+1.This nishestheproof. Bytheinductionassumption,

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