Invariant Measure and the Euler Characteristic of Projectively Flat Manifolds

MANIFOLDS

KYEONGHEEJOANDHYUKKIM

Abstract.Inthispaper,weshowthattheEulercharacteristicofanevendimensionalclosedprojectivelyflatmanifoldisequaltothetotalmeasurewhichisinducedfromaprobabilityBorelmeasureonRPnin-variantundertheholonomyaction,andthendiscussitsconsequencesandapplications.Asanapplication,weshowthattheChern’sconjec-tureistrueforaclosedaffinelyflatmanifoldwhoseholonomygroupactionpermitsaninvariantprobabilityBorelmeasureonRPn;thatis,suchaclosedaffinlyflatmanifoldhasavanishingEulercharacteristic.

1.Introduction

Inthispaper,wewillshowhowtheEulercharacteristicofaprojectivelyflatmanifoldMcanbeviewedasatotalmeasureofMwherethemeasureisinducedfromaprobabilitymeasureonRPninvariantundertheholonomyaction,andthenwewillalsodiscussitsconsequencesandapplications.AprojectivelyflatmanifoldMisamanifoldwhichislocallymodelledontheprojectivespacewithitsnaturalprojectivegeometry,i.e,MadmitsacoverofcoordinatechartsintotheprojectivespaceRPnwhosecoordinatetransitionsareprojectivetransformations.Bycoordinate˜ananalyticcontinuationof

fromM

˜mapsfromitsuniversalcoveringM

,weobtainadevelopingmapintoRPnandthismapisrigidinthesensethatitisdeterminedonlybyalocaldata.ThereforethedecktransformationactiononM

˜inducestheholonomyactionviathedevelopingmapbytherigidity.(Seeforexample[4,14,15]formoredetails.)SupposethereisaprobabilityBorelmeasureλonRPnwhichisinvariantunderthisholonomyaction.ThenwewillfirstshowthataBorelmeasureµonMisinducedfromλbytheinvariancepropertyofλ,andthenshowthefollowingMainTheorem.

Theorem1.1(TheMainTheorem).LetMbeanevendimensionalclosedprojectivelyflatmanifoldandλbeaholonomyinvariantfinitelyadditiveprobabilityBorelmeasureonRPn.Then

χ(M)=µ(M),

whereµistheBorelmeasureonMinducedfromλ.

2KYEONGHEEJOANDHYUKKIM

Thisresultanditsconsequencesaresignificantlyrefinedandevolvedver-sionsofourearlierresults[8,9]initsperspective.OurinvestigationshavebeenmotivatedfromtheefforttoresolvetheChern’sconjecture(oralsoknownasSullivan’sconjecture):“Aclosedaffinelyflatmanifoldhasvanish-ingEulercharacteristic.”

An(X,G)-manifoldisamanifoldwhichislocallymodelledonXwiththegeometrydeterminedbytheLiegroupGactingonXanalytically.Forexample,projectivelyflatmanifoldisaspecialcaseof(X,G)-manifoldwithX=RPnandG=PGL(n+1,R)andsoisanaffinelyflatmanifoldwithX=En,thestandardEuclideanspaceandG=Aff(n),thegroupofaffinetransformationsonEn.Anaffinelyflatmanifoldalsocanbeviewedasaprojectivelyflatmanifoldwhoseholonomypreservesthesetofpointsat

n−1,byidentifyingEnwiththeaffinespacegivenbyxinfinity,RP∞n+1=1

inRn+1sothatRPnbecomesacompactificationofEn.SimilarlyalltheRiemannianandpseudo-Riemannianspaceformscanbeconsideredasasubclassofprojectivelyflatmanifolds,andalsoasubclassofaffinelyflatmanifoldsiftheyareflat.

TheEulercharacteristicofflatRiemannianorpseudo-Riemannianman-ifoldsvanishesbyGauss-Bonnet-CherntheoremanditisnaturaltoaskthesameforaffinelyflatmanifoldsmoregenerallyandthisisthecontentofChern’sconjecture.IfacompactaffinelyflatmanifoldMiscomplete,thentheconjectureistruebytheworkofKostantandSullivan[12],butnotethatthecompactnessdoesnotnecessarilyimplythecompletenessincontrasttotheRiemanniancase.Therehasbeenvariouspartialanswersindifferentdirectionsbuttheconjectureisnotcompletelyresolvedyet.AsoneofthecorollariesoftheMainTheorem,weshowthattheconjectureistrueiftheholonomygroupofaffinelyflatmanifoldhasaninvariantprobabilitymea-suregeneralizingtheearlierresultforamenablecaseaswellasforradiantcaseinaunifiedway.

InSect.2,wewilldefineapull-backmeasuref∗λforagivenlocalhomeo-morphismffromamanifoldMtoanothermanifoldNhavingameasureλ.

˜→X,For(X,G)-manifoldMandthecorrespondingdevelopingmapD:M

˜wheneverXhasameasureλ.IfλisinvariantD∗λiswell-definedonM

underitsholonomyaction,wecanalsodefineameasureµonMnaturally

˜→M.ThisisprovedininducedfromD∗λbycoveringprojectionp:M

Sect.3.InSect.4,wewillprovetheMainTheorem.InSect.5,wewilldiscusstheconsequencesandapplicationsoftheMainTheoremincludingtherelationbetweentheEulercharacteristicandthedevelopingmapsforprojectivelyflatmanifoldsandforementionedresultsforaffinelyflatmani-folds.

2.Pull-backmeasure

Let(N,Ω,λ)denotea(finitelyadditive,resp.)measurespacesuchthattheσ-algebra(algebra,resp.)ΩcontainsalltheopensetsofN,thatis,λisaBorelmeasureifitiscountablyadditive.InthispaperwewillalsocallsuchameasurefinitelyadditiveBorelmeasurewhenλisonlyfinitelyadditive.LetMandNbemanifoldsandf:M→Nbealocalhomeomorphism.Thenwecandefineapull-back(finitelyadditive,resp.)Borelmeasuref∗λ

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS3

onMwheneversucha(finitelyadditive,resp.)BorelmeasureλisgivenonN.

AnopencoveringB={Bi|

4KYEONGHEEJOANDHYUKKIM

additiveandIisequaltoNifλiscountablyadditive.Thenwehave

∞󰀄󰀂󰀁′

λB(∪i∈ISi)=λf((∪i∈ISi)∩Bk)

====

k=1

∞󰀄󰀄

k=1∞󰀄

󰀂󰀁′

λ∪i∈If|Bk(Si∩Bk)

λf(Si∩λf(Si∩󰀂󰀂

′

)Bk

k=1i∈I

∞󰀄󰀄

󰀁󰀁

′

)Bk

󰀄

i∈I

i∈Ik=1

λB(Si).

Noticethatthethirdequalityholdssincef|Bkisahomeomorphism.

󰀁

Lemma2.3.Let{Bi}and{Ci}beadaptedcoveringsand{ΩB,λB}and

{ΩC,λC}bethecorrespondingmeasuresystemsonM.ThenΩB=ΩCandλB=λC.

Proof.(i)ΩB=ΩC:ForeachCkandforalli,f(Ck∩Bi)isopenandthusf(Ck∩Bi)isλ-measurable.HenceCk∈ΩBforallk.SupposeA∈ΩB.ToproveA∈ΩC,itsufficestoshowf(A∩Ci)isλ-measurableforalli.

󰀂󰀁

f(A∩Ci)=f(A∩Ci)∩(∪∞B)k=1k

󰀁󰀂∞

=f∪k=1(A∩Ci∩Bk)

=∪∞k=1f(A∩Ci∩Bk)

Thelastunionisinfactafiniteunionsince

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS5

(ii)λB=λC:ForeachA∈ΩB=ΩC,wehave

λB(A)=

󰀄∞λk=1󰀂f(A∩B′

k)󰀁=λf((A∩Bk)∩(∪∞j=1Cj)k󰀄

∞=1󰀂′′

󰀁=λk󰀄

∞=1󰀂∪∞j=1f(A∩B′k∩Cj

′)󰀁=λk󰀄

∞=1j󰀄∞=1

󰀂f(A∩B′

k∩Cj′)󰀁=j󰀄

∞=1󰀂

λ(f(A∩Cj′∩B′k

))k󰀄∞=1󰀁=

󰀄∞λj=1

󰀂f(A∩Cj′

)

󰀁=λC(A),

wherethefourth,fifthsincethesummation󰀇and∞thelastholdevenifλisfinitelyadditivej=1and󰀇equalities∞

k=1areinfactfinitesums.󰀁FromLemma2.2and2.3,wehavethefollowingtheorem.

Theorem2.1.Letf:M→Nbealocalhomeomorphismandλa(finitelyadditive,resp.)BorelmeasureonN.Thenthereexista(finitelyadditive,resp.)measureµonMsuchthat

(i)everyopensubsetofMismeasurable,(ii)forµ-measurablesubsetAofM(with

Acompact,resp.)ofM.

Proof.TheexistenceofsuchameasureisprovedbyLemma2.2,2.3andthedefinitionofµrespectively.Sotheonlythinglefttoproveistheuniqueness.LetAbeaopensubsetofMand{Band󰀇Biareµ′-measurableandµ(A)=kµ′(A∩B′

k)=µ′(∪k(A∩B′k))=µ′󰀇i}beanadaptedkµ(A∩B′(A).Noticethat

k)=󰀇covering.Thenkλ(f(A∩B′Ak))=sumif

󰀇kisinfactafinite6KYEONGHEEJOANDHYUKKIM

Remark2.1.Iffisahomeomorphism,thenf∗Ω=f−1Ωandf∗λ(A)=λ(f(A))forallA∈f∗Ωwhenλiscountablyadditive.Butthisdoesnotholdifλisfinitelyadditive.Infact,id∗λ=λifλisfinitelyadditiveandMisnotcompact.Forexample,ifM=RandλisanyfinitelyadditivetranslationinvariantprobabilitymeasureofR(theamenabilityofRensurestheexistenceofsuchameasure),thenanyboundedmeasurablesubsetofRhasameasure0andthusid∗λ≡0bythedefinitionofid∗λ.Thisstrangephenomenonarisessinceourmeasureid∗λispulledbackonlylocallyandthenisgivenasthesumoftheselocalmeasuresnotreflectingtheglobalnatureoftheoriginalmeasureλ.

Theorem2.2.SupposetopologicalgroupsGandHactcontinuouslyonMandNrespectively.Let(φ,f):(G,M)→(H,N)beanequivariantpairwhereφ:G→Hisahomomorphismandf:M→Nisa∗localhomeomorphism,andλbeanH−invariantBorelmeasure.ThenfλisG−invariant.

Proof.LetAbeameasurablesubsetofMand{Bi}beanadaptedcoveringofM.Thenforeachfixedg∈G,{gBi}isalsoanadaptedf∗

λ(gA)==󰀄󰀄λ󰀂f(gA∩(gBi)′)

λ󰀂f(gA∩gB′󰀁coveringofM.

=󰀄λ󰀂i)f(g(A∩B′󰀁==󰀄󰀁󰀄λλ󰀂i))φ(g)f(A∩B′i)󰀂f(A∩B′i)

=f∗λ(A)

󰀁󰀁NoticethefifthequalityholdssinceλisH−invariant.

󰀁

3.Holonomyinvariantmeasure

LetonM

󰀆p:M

󰀆→MbearegularcoveringmapandλbeaBorelmeasure.Assumeλisinvariantundertheactionofthedecktransformationgroup.ThenwewilldefineaBorelmeasureµonMsuchthatp∗µ=λ.AnopencoveringB={Bi}i∈NofMwillbecalledacoveringadaptedtocoveringmapp,ifBiisevenlycoveredand

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS7

Definition3.2.LetB={Bi}beanadaptedcoveringofM.Defineafunction

µB:ΩB→{x∈R:x≥0}∪{+∞}byµB(A)=󰀄∞λ󰀂󰀂p|B󰀆󰀁−1

(A∩B′

k)󰀁,

k=1

k

whereB′k=Bk\\∪ki=1−1Bi

.ItiseasytoproveB′k∈ΩB

bysimilarargumentasinSect.2.Hence,µBiswelldefinedindependentlyofthechoiceoftheliftingofthedecktransformationgroup.

󰀆Bisinceλis

invariantundertheactionLemma3.2.LetB={Bi}beanadaptedcoveringofM.Then(M,ΩB,µB)isa(finitelyadditive)Borelmeasurespace.

Lemma3.3.Letboth{Bi}and{Ci}beadaptedcoveringsofMand{ΩB,µB}and{ΩC,µC}bethecorrespondingmeasuresystemsonM.ThenΩB=ΩCandµB=µC.

FromtheaboveLemmas,wehavethefollowingtheorem.

Theorem3.1.Letp:M

additive,resp.)Borelmeasuredecktransformation󰀆→Mbearegularcoveringandλbea(finitelyonM

󰀆.Assumethatλisinvariantundertheactionofthegroup.Thenthereexistsa(finitelyadditive,resp.)BorelmeasureµonMsuchthatp∗µ=λ.Furthermoreµisuniqueinthefollowingsense:′ifµ′isanother(finitelyadditive,resp.)Borelmeasuresuchthatp∗µ=λ,thenµ(A)=µ′(A)foreachopensubsetA(with

A

iscompactandthusthefirstandfifthequalitiesalsoholdinthefinitelyadditivecase.Alsothesecondandthirdequalitiesholdsince

8KYEONGHEEJOANDHYUKKIM

Remark3.1.LetAbeasubsetofMsothatAiscontainedinsomeevenlycoveredgeometricchart.Assumethat

2

󰀅

Sn

fi1···fin−rdλ

Clearly,

λ(sn)=󰀅

fn+1dλ

Sn

1···fλ(

snistheantipodalimageofsn.Thenλ(sn)=λ(

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS9

andholonomygrouprespectively.LetλbeanH-invariantfinitelyadditiveprobabilitymeasureonRPn.Thenforeachfaceσr<σntheangleatσrinσn,denotedbyα˜(σr,σn),isdefinedas

n

α˜(σr,σn)=α(sr0,s0),

nrn˜where(sr0,s0)isadevelopingimageofaliftingofapair(σ,σ)inM.

nNotethatα˜(σr,σn)iswelldefinedbythefollowingreason.α(sr0,s0)is

actuallythemeasureofsomesubsetofRPnandforanyotherchoiceofa

nliftinganditsdevelopingimages1thereexistsh∈Hsothatsn1=h(s0)rnnrrnrandsr1=h(s0).Thereforewegetα(s0,s0)=α(h(s0),h(s0))=α(s1,s1)

sinceλisH-invariant.Buttheangledependsonthedevelopingmap.Fromnowon,we’llsimplydenotetheangleα˜byαbyabusingnotationLet

󰀄

S(σi)=α(σi,σn),

k(σ)=

andforavertexνinK,let

d(ν)=

n

σi<σn

n󰀄r=0

(−1)

r

σr<σn

󰀄

α(σr,σn),

n󰀄(−1)rr=0

Then|Si,j|λ([V∩W])=0ifVandWbelongtoSi,jandV=W.ThereforeSi=

n−1

∪∞j=1Si,jiscountableandsoisS=∪i=0Si.Thereforewehavethefollowingproperties:

(i)|S|iscountable.

(ii)IfX󰀇Rn+1andXistransversaltoeachelementofS,thenλ([X])=

0.

Therefore(i)impliesthatwecanchooseageometrictriangulationKonMbyasmallperturbationsuchthateveryhyperplaneinRPncontainingadevelopingimageofsome(n−1)-dimensionalgeometricsimplexinKistransversaltoSandsoithasameasurezeroby(ii)

j}.

10KYEONGHEEJOANDHYUKKIM

WenowprovetheMainTheorem:S(σi)=1foranygeometricsimplexσiinKbytheaboveconsiderationandthusd(ν)=0forallvertexν∈K.

nnnLetsn0beanydevelopingimageofσ.Thenk(σ)=k(s0)bydefinitionof

nnα(σr,σn)andthuswegetk(σn)=k(sn0)=λ(s0).Wemayassumeσis

nevenlycoveredandliesinsomegeometricchart,λ(sn0)=µ(σ)byRemark

inSect.3.Thereforek(σn)=µ(σn)foralln-simplexσn∈K.NowbythepolyhedralGauss-BonnetTheorem,

󰀄

χ(M)=µ(σn)

σn∈K

Butwehavechosenatriangulationsothatthefacesofσnhavemeasure

zeroandhence󰀄

µ(σn)=µ(M).

σn∈K

Thiscompletestheproof.

5.ConsequencesandApplications

󰀁

Therighthandsideµ(M)oftheformulaintheMainTheoremissupposedtodependontheholonomyinvariantmeasurechosenandonthedevelopingmap,namelytheprojectivelyflatstructureofM.Butthetheoremsaysthatinfactitdoesnot,andisalwaysequaltotheEulercharateristicofM,atopologicalinvariant.Futhermore,thereisnoreason,apriori,thatthetotalmeasureofM,µ(M)shouldbeaninteger.Thetopology,geometryandthemeasurerelatedtoMareinterlockedbytheformulaandwecanexpectinterestingapplicationsfromtheseobservations.Wewillseesomeoftheimmediateconsequencesandapplicationsinthissection.

LetMbeaclosedprojectivelyflatmanifoldwithamenableholonomygroupHandmaninvariantmeanonB(H),thespaceofallboundedfunctionsonH,see[5]fordefinitionsofamenablegroupandinvariantmean.WemayassumethatmisrightinvariantsinceHisagroup,thatis,m(fs)=m(f)foralls∈H,wherefsisaboundedfunctiononHgivenbyfs(t)=f(ts).ThenwecandefineanH-invariantfinitelyadditiveprob-abilitymeasureonRPnasfollows.Chooseanyprobabilitymeasureλ0onRPn.Thenforanyλ0-measurablesubsetEofRPnwecandefineaboundedfunctionfE:H→[0,1]by(5.1)

fE(h)=λ0(h(E))λ(E)=m(fE)

forallλ0-measurablesubsetE⊂RPn.Then,bythepropertyofinvariantmean,λisafinitelyadditiveH−invariantprobabilitymeasureonRPn.Moreprecisely,

λ(hE)=m(fhE)=m((fE)h)=m(fE)=λ(E)

since

fhE(h′)=λ0(h′(hE))=λ0(h′h)E)=(fE)(h′h)=(fE)h(h′),

andthepropertym(1)=1impliesthatλisafinitelyadditiveprobabilitymeasureonRPn.

forallh∈H.NowdefineanewmeasureλonRPnby

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS11

Theorem5.1.LetMbeanevendimensionalclosedprojectivelyflatman-ifoldwithholonomygroupH.SupposethereexistanH-invariantfinitelyadditiveprobabilityBorelmeasureλonRPn.Thenwehavethefollowing.

(i)χ(M)isnonnegative.

(ii)Ifthedevelopingmapisinjective,thenχ(M)=0.

(iii)Ifχ(M)=0,thenthedevelopingmapisnotsurjective.

(iv)Iftheinvariantmeasureλiscountablyadditive,thenχ(M)=0if

andonlyifλ(Ω)=0andχ(M)>0ifandonlyifλ(Ω)=1,whereΩisthedevelopingimage.Proof.Toprovethistheorem,itsufficestoprove(ii),(iii)and(iv)because(i)istheimmediateconsequenceoftheMainTheorem.

˜.LetD:M˜→RPn(ii)LetFbeanopenfundamentaldomainofMinM

bethecorrespondingdevelopingmapandφ:π1(M)→Hbetheho-lonomyrepresentation.Supposeφ(ξ)=idforsomeξ∈π1(M).Then

˜.SinceDisinjective,ξx=xforD(ξx)=φ(ξ)D(x)=D(x)forallx∈M

˜,i.e,ξ=id.Thereforeφbecomesanisomorphism.NotethatHallx∈M

isnon-trivial:Ifitweretrivial,MissimplyconnectedandDbecomesahomeomorphismsinceMiscompactandDisaninjectivelocalhomeomor-phism.Butthisisimpossiblesinceπ1(RPn)=Z/2forn≥2.Leth∈Hbeanon-identityelement.ThenbytheinjectivityofD,D(F)∩h(D(F))=∅.Sinceλ(D(F))=λ(h(D(F)))andλ(D(F))+λ(h(D(F)))≤1,theEulercharacteristic,beinganon-negativeinteger,hastobezero.

(iii)Letµbetheinducedmeasure.Ifχ(M)=0,thentheMainTheorem

˜)thereexistsanopenimpliesthatµ(M)=0andthusforeachx∈D(M

neighborhoodUxofxsuchthatλ(Ux)=0bythedefinitionoftheinduced

˜)hasmeasurezero.Sup-measure.ThereforeanycompactsubsetEinD(M

˜).ButRPniscompactandthusposeDissurjective.ThenRPn=D(M

λ(RPn)=0.Thiscontradictsthatλisaprobabilitymeasure.

(iv)LetµbetheinducedmeasureonM.χ(M)=0impliesµ(M)=0bytheMainTheoremandthusagainforeachx∈ΩthereexistsanopenneighborhoodUxofxsuchthatλ(Ux)=0bydefinitionoftheinducedmeasure.ThereforeanycompactsubsetEinΩhasmeasurezeroandthusλ(Ω)=0bythecountableadditivityofλ.Theconverseisclear.Thereforeχ(M)>0impliesλ(Ω)=0.Supposeλ(Ωc)=α(α>0).Byconsidering

˜=(1/α)λ|Ωcsupportedonthecomplementofanotherinvariantmeasureλ

Ω,wegetχ(M)=0.Thisisacontradiction.Soλ(Ωc)=0󰀁Theorem5.1(ii)saysforinstancethattheholonomygroupofevendi-mensionalhyperbolicmanifoldcannothaveafinitelyadditiveinvariantprobabilitymeasure.(Butitdoeshavecomplexinvariantprobabilitymea-sure.)Amuchbroaderclassofconvexprojectivelyflatmanifoldsshouldhavethesameproperty.Andinthiscasetheholonomygroupofsuchmanifoldscannotbeamenablesinceamenabilityenablesonetoconstructaninvari-antprobabilitymeasurestartingfromanyprobabilitymeasurebyaveragingprocess.Butingeneraltheholonomygroupofprojectivelyflatmanifoldisfarfrombeingamenableevenwhenithasaninvariantprobabilitymeasure.ThecaseofamenableholonomygroupisaninterestingspecialcaseandwecanobtainasharperresultasthefollowingTheorem5.2shows.

12KYEONGHEEJOANDHYUKKIM

Theorem5.2.LetMbeanevendimensionalclosedprojectivelyflatman-ifoldwithamenableholonomygroup.Thenthefollowingsareequivalent.

(i)Thedevelopingmapisnotonto.(ii)χ(M)=0.

(iii)ThereexistsfinitelyadditiveinvariantprobabilityBorelmeasureλon

RPnsuchthatλ(K)=0foranycompactsubsetKofthedevelopingimage.

(iv)ForanyfinitelyadditiveinvariantprobabilityBorelmeasureλon

RPn,λ(K)=0wheneverKisacompactsubsetofthedevelopingimage.

(v)ThereexistsacountablyadditiveinvariantprobabilityBorelmeasure

λonRPnsuchthatλ(Ω)=0,whereΩisthedevelopingimageof˜.M

(vi)ForanycountablyadditiveinvariantprobabilityBorelmeasureµ,

µ(Ω)=0.Proof.(i)⇒(ii);Letλ0betheDiracmeasureconcentratedatapointx0outsidethedevelopingimage.LetmbeaninvariantmeanonB(H).ThenwecandefineameasureλonRPnbyλ(E)=m(fE)foreachsubsetE⊂RPn,wherefEisdefinedastheequation(5.1).ThenλisaninvariantfinitelyadditiveprobabilitymeasureandforeachsubsetEcontainedinthedevelopingimage,λ(E)=0.Thereforeχ(M)=0bytheMainTheorem.(ii)⇒(i)hasalreadybeenshowninTheorem4.1(iii).

Sincetheholonomygroupisamenable,thereexistsafinitelyadditivein-variantprobabilityBorelmeasurebyaveragingandfurthermoretherealsoexistsacountablyadditiveinvariantprobabilityBorelmeasurebycompact-nessofRPn.

(ii)⇒(iv),(vi);Supposethatλ1isafinitelyadditiveinvariantprobabilityBorelmeasureonRPnandλ2isacountablyadditiveinvariantprobabilityBorelmeasure.Letµ1andµ2bethecorrespondinginducedmeasureonMrespectively.Thenµ1(M)=µ2(M)=0sinceχ(M)=0.Bythedefinitionoftheinducedmeasure,foreachxinthedevelopingimage,thereexistanopenneighborhoodUxsuchthatλ1(Ux)=λ2(Ux)=0.Thereforeλi(K)=0(i=1,2)foranycompactsubsetKofthedevelopingimageandfuthermoreλ2(Ω)=0sinceλ2iscountablyadditive.

(iv)⇒(iii)and(vi)⇒(v)aretruesincetheholonomygroupisamenable.(iii)and(v)eachimply(ii)bytheMainTheorem.󰀁AnotherinterestingspecialcaseinwhichtheexistenceofH-invariantprobabilitymeasureisguaranteediswheretheholonomygroupHhasafixedpointormoregenerallyhasafiniteorbit.Inthiscase,wehavethefollowingtheorem.

Theorem5.3.LetMbeanevendimensionalclosedprojectivelyflatman-ifoldwithholonomygroupH.SupposeHhasafiniteinvariantsetI.Thenwehavethefollowings:

˜)ifandonlyifχ(M)>0.(i)I⊂D(M

˜)=∅ifandonlyifχ(M)=0.(ii)I∩D(M

Inparticular,ifHhasafixedpointoutsidethedevelopingimage,thentheEulercharacteristicofMmustvanish.

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS13

Proof.DefineaninvariantprobabilityBorelmeasureλonRPnbyλ(E)=󰀇1

a∈E∩I

λ(I1)λ|I1

andλ2=

1

Hiscompactor(ii)thereisapropersubspaceV0suchthatλ[V0]>0andV0isinvariantbyasubgroup

14KYEONGHEEJOANDHYUKKIM

ofHwithfiniteindex.Anaffinelyflatmanifoldalsocanbeviewedasa(Sn,P+GL(n+1,R))manifold,whereP+GL(n+1,R)∼=GL(n+1,R)/R+.LetSL±(n+1,R)={A∈GL(n+1,R)|detA=±1}.ThenP+GL(n+1,R)∼=±SL(n+1,R).NoticethatPGL(n+1,R)∼=SL(n+1,R)ifniseven.Let

±q:SL(n+1,R)→PGL(n+1,R)bethecoveringhomomorphismand

˜→RPnbethep:Sn→RPnbetheusualcoveringmap.LetD:M

˜:M˜→SnbeitsliftingsothatD˜◦p=D.LetH˜⊂developingmapandD

˜sothatitistheP+GL(n+1,R)betheholonomygroupcorrespondingtoD

liftingofH.IfH)iscompactinGL(n+1,R)sinceSL±(n+1,R)isclosed.Thereforethereexistsq−1(

H′iscompactorthereexistaproper

subspaceWofV0suchthatλ[W]>0and[W]isinvariantbyasubgroupofH′withfiniteindex.ButbyminimalityofV0,

H′)iscompactinGL(m,R)wherem=dimV0.Sothereexists

aq−1(

INV.MEASUREANDTHEEULERCHAR.OFPROJECTIVEMANIFOLDS15

PropositionA.1.LetµfbeafinitelyadditiveprobabilityBorelmeasureonacompactHausdorffspaceX.ThenthereexistsacountablyadditiveprobabilityBorelmeasureµconX,whichcorrespondstothemeasureµf.Proof.LetB(X,Σ)betheBanachspaceconsistingofalluniformlimitsoffinitelinearcombinationofcharacteristicfunctionsofsetsinBorelalgebraΣ.ThenthedualspaceofB(X,Σ)isisometricallyisomorphictotheBanachspaceba(X,Σ)consistingofallboundedfinitelyadditivemeasuresonΣ(SeeTheoremIV.5.1in[1]).Inthiscorrespondence,aprobabilitymeasureµfinba(X,Σ)correspondstoapositivelinearfunctionalΛµfonB(X,Σ)andΛµf(χX)=1forthecharacteristicfunctionχX∈B(X,Σ).SinceΣistheBorelalgebraonXandB(X,Σ)iscompletewithrespecttothesupremumnorm,theBanachspaceC(X)consistingofallcontinuousfunctionsoncom-pactspaceXisaBanachsubspaceofB(X,Σ).SotherestrictionΛµf|C(X)ofΛµfisapositivelinearfunctionalonC(X)withΛµf|C(X)(χX)=1sinceχX∈C(X).Consequently,wehaveacountablyadditiveprobabilityBorelmeasureµconXcorrespondingtoΛµf|C(X)bytheRieszRepresentationTheorem.Thiscompletestheproof.󰀁RemarkA.1.Thiscorrespondencedoesnotimplythatµf(E)=µc(E)forallsubsetEofXwhichiscontainedintheBorelalgebra.Forexample,considerafinitelyadditivetranslationinvariantprobabilityBorelmeasureµf.Infact,µfcanberegardedasafinitelyadditiveprobabilityBorelmeasureontheclosedinterval[−∞,+∞],thecompactificationofR1,suchthatµf({−∞})=µf({+∞})=0.Butforthecorrespondingcountablyadditiveprobabilitymeasureµc,µc({−∞,+∞})=1.Infactµf(I)=0foranyboundedintervalI⊂Randthisimpliesthatµc(R)=0usingtheMonotoneConvergenceTheorem.

PropositionA.2.LetthegroupGactonacompactmetricspaceXandµfbeaG-invariantfinitelyadditiveBorelmeasureonX.ThenthecountablyadditiveprobabilitymeasureµcwhichcorrespondstoµfisalsoG-invariant.Proof.G-invarianceofµfimpliesthatµf(E)󰀃=µf(gE)forallmeasurable󰀃

Eandg∈Ganditfollowsthatfdµf=g·fdµfforany󰀃󰀃f∈C(X)−1where(g·f)(x)=f(gx).Sinceµf=µconC(X),fdµc=g·fdµcforanyf∈C(X),whichinturnimpliesthat

󰀅󰀅󰀅

µc(E)=χEdµc=g−1·χEdµc=χgEdµc=µc(gE)forallmeasurableEandg∈GbytheMonotoneConvergenceTheorem.󰀁

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E-mailaddress:khjo@math.snu.ac.krE-mailaddress:hyukkim@math.snu.ac.kr