=C00(ぶ;),即r2)=乳r(T2)=C∞(5;)亀andどr(r2)=ど｡r(T2)=町r2)⑳鴎) WemayCOnSidertheproductfo1iationFonT2=宝×S;=i(x,y)∈R2/(2nZ)21given Structures(II)

Bulletinof theFacultyof Education,MieUniversity･33mturalScience(1982),ト6頁

Notes on PartiaJJy UnimoduJar Structures(II)

by Yukihiro Kanie

かepαγ加e氾才イ〟α山肌α〜∫cぶ,肪e U乃れ,er吉和

This noteis a continuation of PartI(〔K‑6〕),and ^here _we｡Sethe ^same

§5･F'0[iationsonTorus(ⅠⅠ).Inthis section,WeSt｡dyderivati｡nS｡f iT(T2)forthelinearfoliation苫Onthe2‑dimensionaltorus T2witharational

Slopelandthe corresponding s･p･u.StruCture Tl･Duetothe remarksi｡§4,

WemayCOnSidertheproductfo1iationFonT2=宝×S;=i(x,y)∈R2/(2nZ)21given

byly=COnStant【,and the s･p･u･StruCture

T=dx.Then,Z=∂∬,C脚(T2)j

=C00(ぶ;),即r2)=乳r(T2)=C∞(5;)亀andどr(r2)=ど｡r(T2)=町r2)⑳鴎)

=C伽(5;)も⑳C∞(5;)も･

At first,We _get

easily the following

Lemma10･(∫)〝■ズ∈ゴて(r2)ざα棚e‥んere′α涼乃β〔考朝=〔ろsi叫朝

=0,沌e乃gよぎe叩γe5ざedα5 g=α屯 カγざOme

(よりrんe ce雨eγ∂扉どr(r2)co加古de…油R∂ズ･

Proposition

H･DdinelhelinearmappingsA,BandCqfろ(Thoilseぴ

A(乳(rう)=0, A(≠(γ)屯)=(鋤)(#)∂ズ;

月(乳(T2))=0ラ 郎拍)屯)=拍)∂∬;

C(≠(ッ)色)=頼)色, Cは(S三))=0

力汀̀川〟≠∈Cの(5こ)･rんe乃,A,月α乃dCαγedeγ∫γα〜go乃ぶ扉どr(r2),｡几d輌｡γe

pγOperJy

肋γeO靴γ,〔A,β〕=0,〔c.A〕=A α乃d〔c,β〕=且

Proq[Since〔≠Zl躍〕=0,〔≠∂y,躍〕=棉湧Z

and鳩,吋=1≠(㌔功一

4･(Py4)ia｡forany≠,4･∈C∞(S:),WeCaneasilycheckthederivati｡np,｡Perty｡f

A,BandC･And _also bysimp)ecalculations,We _can showtheircommutationrelations.

Now,We

Showthat ^A,Band Care

properly

‑1‑

Yukihiro

Kanie

At first,aSSume that

field XonT2gives the derivation Aas

A=adX･Since〔x,qr]=〔x,∂y〕=〔x,Siny∂x〕=0,We

by the similar

arguments ^as LemmalO that ^Xis written ^as X=a∂x(a∈R).

Apply Ato the

field siny∂n,then

cosy∂y=A(sinyPy)=〔apx,Sinyも〕=0･

Thisisimpossible,thatis,Ais

properly

Secondly,aSSume _that X∈別(T2)gives ^{the derivati｡n B as B=adX.}

Since[x,qr]=0,the coefficient ^functions of Xareindependent of ^x,SO

Xis written ^as X=≠(y)∂x+￠(y)∂y(≠,4･∈C∞(S:))･

On the other hand,the function≠has the constant derivative‑1.In ^fact,

∂ズ=月(∂〟)=〔g,∂〟〕=‑(∂〟≠)∂ズー(㌔胡∂y･

This contradicts that periodic functions have

non‑ZerO

derivatives.

Finally,aSSume that X∈せl(T2)gives ^{derivation C as C=adX.}

Since[x,別(S;)]=0,it ^{follows easily that Xis written as X=f(x)∂,}

for some f∈C∞(S:)･

ApplyC to the

field∂x,then

∂ズ=C(∂ズ)=〔g,生〕=‑(屯J)(∬)生･

hence㌔J=‑1･This contradicts the periodicity off･ Q.E.D.

Theorem12･(f)エe古かゐeαdeγ血〜fo陀げろ(r2)･rんeれ,加γe ^P∬f5f5α

γeCわγイブeJdⅣ0乃r2α乃d coれ5如才5α,占α乃d c∈R5"｡ん〜ん｡〜

β =adⅣ+αA+ 占月+

cC.

〟0γPOぴeγ,亡んe

α,占α氾d ^{c α∫re} 混和f叩eJy de才eγmよ几ed,

α乃d〜β 祝乃gq腔 mO血Jo

√∫=R∂ガイろ(T2)･

(わ)rんeγeαγ̲e mO乃α加αJ仙ねγdeγ血〜fo几ざイゴr(T2)･

β1=β1ぱr(T2);ゴ丁(r2))g50Jd加乃ざioれ3,αれヴα5αエie

ge乃Prα拍rざA,βαれd Cぴi〜んγe′αffo耶[A,β〕=0,〔c,月〕

=

β.

α乃dⅣ〜5f･乃ごて(T2)

TんeJfγ5f coゐomoJog〟

α′geゐγαガ1んα‥んγeP.

=Aα托d[C,β〕

Proq[Let ^Dbea derivationofゴT(T2)･

Let D'bethelinear mappingof P((S;)toitself ^{which assigns}

X∈ヅ((ギ)

the別(S3)‑part ^{Of D(X)･Then D'gives a derivation of} や【(i),because

gT(T2)is anideal･Hence by F･Takens〔T‑1〕,there ^{exists a unique}

field Wl副(Sこ)suchthat D'(X)=Lwl,X〕foranyX∈Y((S:)･

Let Dl=D‑adwl,thenDlisaderivationoffT(T2)suchthat Dl(PL(i))

⊂汀(r2).

NoTES

ONPARTIALLYUNIMODULARSTRUCTURES(II)

Definethefunction〆0(y)∈C∞(S;)as Dl(も)=≠｡(y)∂x･Put

占=(2わー1如(y)dy ^α乃d ￠(〟)=1#胸トムid弘

then thefunction￠(y)is well‑definedon S;･

Let W;=‑￠(y)∂x∈gT(T2)and D2=Pl‑adw2‑bB,thenD2isaderiva‑

tionofiT(T2)suchthat P2(やt(Sこ))⊂gT(T2)and 巧(∂y)=0･hfact, 巧(屯)β1(亀)+〔呪,∂g〕‑d月(∂タ)=j≠｡(y)‑(∂y￠)(y)‑いも=0.

Apply P2tO ^the equalities〔∂x,Py〕=〔∂x,Sinyも〕=0,thenby ^LemmalO

We get that 上㌔(∂x)=c∂x ^{for some}

^c∈R.

Let D3=D2.CC,thenD3isaderivationof2(T2)suchthatlWt(S:))

⊂乳(門 ^and q(∂ズ)=巧(免)=0･

Definethefunctions≠k,≠;,警 ^and 粍′∈C∞(S:)as P3(sinky㌔)=≠kPr+軌 andIucosky㌔)=範∂ズ+吃∂y･

ApplyD3tOtheequalities ksinky∂x=[cosky∂x,㌔〕and kcoskyqr=〔∂y,Sinky㌔〕

(k∈Z),then we get

枕∂ズ+ん≠去∂y=〔叱∂ズ+仇,∂ダ〕=‑(も亀)∂ズー(∂〟吃)屯, 片帆+た帆=〔も,≠々㌔十≠こ㌔〕=(兢)∂ズ+(も船∂y,

hencei≠々,顧 andl≠;,万 ^{satisfythe conditions of the} followinglemma.

Lemma13･⊥e才≠肌d￠ゐビル乃C王〜0れ55α‡よぎ血乃g偏d抑eγeれ〜ねJeq祝αfよ0托ざ

も≠=烏￠ and も￠=一点≠

Joγ50me∫乃〜egeγ ん,rんeれ,〜んeγeαγe

CO柁β如才ざααれdβ∈Rβ〟｡ん血才

≠(y)=asinky+βcosky _and ￠(y)=‑βsinky+acosky.

Proqf Thefunctions≠and￠satisfytheequations

∂訝=‑k2≠and∂:￠

=‑K2￠ ^So,aSis wellknown,≠and4･arelinearcombinationsof ^sinkyand COSky.Write≠and ￠as

≠(y)=aSinky+βcosky and 4･(y)=γsinky+8cosky

forsomeconstants

a,β,γand81Put･themintotheequation∂y≠=kめthen kacosky‑kβsinky=kγSinky+k5cosky,

hence

a=5andβ=‑yl Q.E.D.

Thus,there are

ak,Pk,a々 ^{and 且'such that}

≠々(y)=a々Sinky十島cosky, 亀(y)=一色sinky+akCOSky,

‑

‑

YⅦkiIliro Eanie

≠;(y)=宅Sinky+色′cosky ^and 吃(y)=一銭Sinky+宛COSky･

Apply上もtothe equality kも=〔sinkyPy,‑COSkyqr〕+〔cosk咤,SinkyPr〕,then O=〔sinkyも,(βkSinky‑akCOSky)px+(&'sinhy‑a;cosky)も〕

+〔cosky曳,(a烏Sinky+鶴cosky)∂x+(∂;sinky+色′cosky)も〕

=klβ烏Sinkycosky+akSi㌔ky十a々COS2ky‑βkCOSky‑βkCOSkysinkyi∂x 十kla;sin2ky+αこCOS2ky+a;cos2ky+a;sin2kyi∂｡

=如α鳥∂ズ+2αヱも),

hence

a烏=屯=0･Apply ^{D3tOthe equality} sin2y∂x=2〔siny∂y,SinyPx〕,then

J%cos2y屯+攫cos2yay=2〔sinyh,AcosyPr+β;cosy∂y〕

=‑2且sin2ypx‑2(AIsin2y+A′cos2y)py=‑2β1Sin2y∂x‑2A′Py,

hence 色cos2y=β1(cos2y‑1)and 考COS2y=‑2β;･Theseimply ^that

β1=A′=0,thatis,巧(siny生)=彗(cosyPx)=0･

Let≠∈C叫(S:)･Thenweget ^that D3(穐)=0,bytheformula 4ax=〔≠siny∂｡,‑COSy∂x〕十〔≠cosyP,,SinyPx〕･

Hence巧(どr(r2))=0･

Definethefunctions≠"and V∈C∞(S;)as

Lt(sinyPy)=≠′′(y)Px ^and A(cosyも)=少'(y)∂x･

Apply D3tOthe equalities siny屯=〔cosy▼㌔,∂y〕and cosy∂y=〔㌔,Sinyay〕,then

We get Similarly ^as above that Py≠"=g'and P9￠'′=‑≠"･Then ^{by Lemma} 13,there are constants ar'andβ"such ^that

≠'′(y)=a''siny+β′′cosy and g'(y)=‑β'′siny+a''cosy･

Apply P,tO ^the equality Py=[cosy∂,SinyPy〕,then ^{we get}

0=〔(a"cosy‑β''siny)qr,SinyP,〕+〔cosy曳,(a"siny十β"cosy)∂x〕

=α′旦,

Put a=β"andlet P.=巧‑aA,then ^D.is aderivationof 2T(T2)

suchthat 上乙(9t(S;))⊂gT(T2)and P,(Px)=旦(Py)=q(siny∂y)=q(cosya｡)

Let fEC00(S:)･Define thefunctions fi∈C∞(S;)(i=0,1,2)as

q(J∂y)=f｡∂x,D.(fsinyも)=f∂x ^and 彗(fcosy∂｡)=ム∂x･

Apply ^D tothe equalities

2fち=Ucosyay,Siny屯〕‑LFsinyPy,COSyち〕,

NoTES ONPARTIALLYUNIMODULARSTRUCTURES(ⅠⅠ)

2fsinyも=LFcosyP,,∂y〕‑Lfも,COSyも〕

2fcosyPy=LF∂めSinyP｡〕‑LFsinyPy,q,〕,

〔も,f∂y〕+LFsiny屯,SinyPy〕+Ucos鴫,COSyPy〕=0,

then we get

2ム∂ズ=叛∂ズ,Sinyも〕一レ1屯,COSy∂〟〕=明美)cos#‑(揖)si叩lち,

21㌔=抗生,も〕一払∂ズ,COSy∂y〕==屯ム)cosダー(もム)l㌔, 2L∂x=LhPx,Siny∂｡〕‑u∂x,も〕=i(PyL)‑(PyL)sinyi∂,,

〔亀,兢〕+扶∂∬,Si噂も〕十抜毛,COSy∂タ〕

=‡(亀j;)‑(亀f)siny‑(もh)cosylpx=0.

benee,

2jこ=(Pyf)cosy‑(∂｡ム)siny･ 2fl=(∂｡ム)cosy‑∂yj;,

2f2=Py美.(∂yf｡)siny ^and Py亮=(∂y美)siny+(∂y美)cosy.

In particular,

2L=(Pyf)(siJy+cos2y)‑(∂yム)siny

=(∂｡fo)siny‑(∂yム)sinycosy+2jLcosy+(u)sinycosy‑(∂｡差)siny

=2J｡COSy,

thus j;=jLcosy,Similarly f=j;siny･Hence 2亮(㌔工)cosダー(も美)sin封

=鳩ム)siny+Lcosyicosy J(㌔ム)cosy‑Lsinyisiny

=J｡(cos2y+sin2y)=ん

Hence _weget that jL=0,thatis,q(f∂｡)=0･Consequently weget that

q(fT(T2))=0,thatis,D=ad(Wl+W2)+aA+bB+cC･

For _{the uniqueness}

of ^the expression of D,itis sufficient

showthat

a=b=c=O and′W∈3,if thederivation adW十aA+bB+cC(W∈3r(T2))

iszeroon2r(T2)･Infact,applythisderivationtothevectorfields Px,∂y

and sinyPy･ Q.E.D.

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〔E‑2〕‑:Coんomo毎よeざ扉エfeα′geムγα5げむeC加/ieJd=βよ〜んco卿cfe乃〜ざ

in adjoinlreprentalions:Foliated Case,Publ･RIMS･,Kyoto Univ.,

14(1978),487‑501.

〔E‑3〕‑:Someエゴeα′geゐrαざげ即eC加Jfe′dβ0犯♪Jfα‡edmα項Jdβα兜d

‑

Yukihiro

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〔S‑1〕N.Steenrod:The ^Topology _of ^Fibre Bundles,Princeton Univ.Press, Princeton,NewJersy(1951).

〔S‑2〕S.Sternberg:Lectures ^on DiffrentialGeometry,Prentice‑Hall,(1964).

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