Departmental Bulletin Paper / 紀要論文
部分ユニモジュラー構造についてのノート
(III)
Notes on Partially Unimodular Structures (III)
蟹江, 幸博
Kanie, Yukihiro
三重大学教育学部研究紀要. 自然科学. 1983, 34, p. 1-5.
http://hdl.handle.net/10076/4696
Bulletin
of
the
Faculty
of
Education,Mie
University.34肋EuTqL
Science(1983),1-5
NotesonPartia"yUnjmodularStructures(ⅠⅠⅠ)
by
Yukihiro
KANIE
β甲∬加e〝J(が肋助e〝∽血3,〟おこ加γe和砂
ThisnotesisacontinuationofPartIandⅡ(【K-5,6]),andhereweusethesamenota･
tions.Theresultsstatedhereareannouncedin[K･7].
S6.FoliationsonToruS(Ⅲ)･
Inthissection,WeStudyderivationsof2,(Tq+L)
fortheproductfo1iationJonthe(q+1)-tOruSTq+L=S主×Tq=((x,y,,…‥,yq)∈Rq+L/
r=血･
(22rZ)q+L),givenby(yi=COnStant),andthestrictlypartiallyunimodularstruCture
Then,Z=∂x,C￠(Tq+))Ly=CO(Tq),弟(Tq+))=､‰T(Tq+L)=C00(r'L)∂xand
gr(r折り=g｡,(r叶り=し鶏(r叶り⑳部(アマ)=C閣(r9)∂ズ⑳∑C00(r9)∂f,
Wherewedenote∂,iby∂i(1≦i≦q).
At丘rst,WegetthesimilarpropositionsasLemmalOandPropositionll.
Lemma.14.(i)Ifx∈.乾(Tq+L)satiiPestheTehltibns[x,∂L]=[X,SinyE∂)]=0
(J≦f≦α),班e〃ズ由eズpreJJedα∫方=C∂∬βr∫0椚eCO那ね乃Jc.
｢fり 乃ece乃ねr′〆.玖(r頼)00血血ね川血カ月∂ズ.
Proposition15.Dq伽ethelinear椚仰ingsA.B.andCq/},(Tq+L)Eoitseqas
Aし省(r叶J))=β,
A(′(γ)∂J)=(∂J′)(γ)∂ズノ
夙(し汽(T拙))=♂,
風(′(γ)∂J)=∂り′(γ)∂よJ
C(′(γ)∂ズ)=′(γ)∂ズ,
Cは(γ出))=β
わm叩γル胱瀬川/欄=㌘(γ出).乃飢A,風α〃dCαre血相Jわ那qr
gT(r叶J),鍋飢腔
叩〝か0〟肘,J如才れ助町αre仰rreα肋α抽わ′γeCわrβe肋0〃r….
肋rβ0γ〝,[A,&]=β,【臥β｣=♂,[C,A】=Aα乃d[C,&】=β`(j≦f,メ≦α).
Thenweget
¶leOreml`
｢り 上eJβ∂eα庇わ帽Jわ乃(がゑ(r叶1).7触乃,Jゐe柁α由ねαyeCわr
βと〟Ⅳ0乃r拙α〃dの乃∫ね〃ねα,∂f(J≦i≦￠)α乃dc∈Rぷ〟C力J如J
β=adⅣ+αA+冥∂一風+cC
一1-
Yukihiro
此〉reOpeち血co那加摘
α,∂`α〃d
Kanie
c∬e〟両〟eかde館〝乃加e4α乃dⅣ由加gr(r拙)
α〃d由〟両〟e椚0血わ〟柁Ce〃おrJ=月∂ズ q｢裳(r…).乃耶α柁仰湖山和J仙妙虚血Jわ那`が名(7サ=).
｢可
乃e伽Jco如椚0わ紗J㍗=J㌢(裳(T州);名(r叶り)由げ肋乃e〃血乃中2,
α〃dαぶα上feα如ム相計加ざヴゼ邸乃e和ね柑A,β`α乃dCw肋作加わ那【A,&】=♂,
【&,昆】=0,【C,A]=Aand[C,Bi]=Bi.
bea
L>oqf:Sincethestatement(i)implies(ii),WePrOVeherethe丘rststatement.LetD
derivationof2T(Tq+)).
Let
D'bethelinearmappingof9t(Tq)toitselfwhichassignstoX∈9t(Tq)the
9t(T.)-COmPOnent
OfD(X)･Then,SimiladyasintheproofofTheorem12,WegetaVeCtOr
丘eld取∈9t(Tq)suchthat
X∈9t(Tq).
D,(X)=【町J,X]fbrany
Let
DL=D-adWL,then
DLisaderivationofし弟(Tq+))suchthat
D)(9t(Tq))
⊂し汽(γ出).
De血ethefunctions
hi(y)∈C∞(Tq)asD)(ai)=hi(y)∂x(1≦i≦q).Thenweget
that∂Ehj=∂jhi.Infact,aPPly
DL
tOtheequality[∂t,∂j]=0.Put
bi(y)=(27r)
L
L2㌔i(y)dyi,thenweget
∂J∂刷=(2けイ2て∂勅))れ=(2けイ2て∂勅))れ=(2が[ゐ刷
hencethesefunctions
biareCOnStantS,SOWemayPut
∂f=(2がJ2も`(β･t･‥伽,♂･…･･のれ
柚)=真上ツ紘(♂,…･♂,抽+1,…･･釣十-れ)d′･
Now,We
de丘ne
vector丘eldI佐
asl偽=-H(y)∂xin
D2=DL
L7t(Tq+L),andlet
-ad取-∑biBi.Then,WegetthatD2isadeTivationofし7;(Tq+))suchthat
D2(9((Tq))
⊂し先(γ出)α乃dβ2(∂`)=β(J≦才≦わ.Inねct,
∂酌)=鼠1上方∂`招･…･･柚仙…･,γq肋+払肌…伽,γ
=引篭メ紬…･裾γ出…･,γ｡)d什…,…･,♂,γ跡目
=諦ゐf(β,…･,抽げ‥,γ9)】ごご+れ(♂,…･,紬,γ
=巌(γ㍉…･,γ｡)-∂`,
鋸∂f)=鋸∂fト【眺,∂f]一兵岨(∂f)
=ゐf∂∬-(広一占`)∂∬-∂f∂ズ=仇
-2-
NoTES
Apply
that
ON
PARTIALLY
UNIMODULAR
SrRUCTURES(皿)
D2tOtheequalities[∂x,∂i]=[∂x,Sinyi∂)]=0,thenbyLemma14weget
forsomeconstant
D2(∂x)=C∂x
Let
D3=D2-CC,
c∈R.
then
yr(Tq+L)suchthat
D3isaderivationof
D3(9((Tq))
⊂し苑(T軒J)andβ3(∂ズ)=β∫(∂i)=♂.
men,Weget
b皿m17.7加虎〟帽Jわ乃βぷ由z〝00〃伽肋Jし汽(r巾｣).
L>0(がDe丘nethefunctionsん,fAij,ghiandgAij∈C￠(Tq)as
D3(sinkyi∂x)=fhi∂x+∑f頼∂jandD3(coskyL∂x)=ghi∂x+∑g柏∂j.
D3tOthe
Then,WegCtthatthesefunctionsdependonlyonthevariableyi･Infact,aPPly
qpalities8i,ksinkyi∂,=koskyi∂x,∂,]and8ijkcoskyL∂x=[∂j,Sinkyi∂x]･Moreover,by
thesimnarargumentsasintheproofofTheorem14,Wegetthatthesefunctionsvmish,and
thestatementofthelemma.
D3isaderivationof裳(Tq'L)
Returningtotheproofofthetheorem,Weknowthat
and
suchtbatヱ)J(乳(Tq))⊂し苑(r拙),β3(∂f)=β
fij and
De丘nethefunctions
gtj(1≦i,j≦q)inC∞(Tq)as
D3(sinyi∂j)=fij(y)∂x
D3(cosyt∂j)=gtj(y)∂x.
and
Thenwegetthatthesefunctionsfij
apply
上)よし汽(r叶J))=β.
and
depend
gij
only
onthe
variabley`･Infact,
D3tOtheequalities
8LhSinyi∂j=kosyi∂j,∂.]
8i鳥COSyi∂j=[∂h,Sinyi∂j].
and
Moreover,bythesimi1arargumentsasinthepr00fofTheorem14,Wegettheconstants
のり≦i≦仔)sucb血at
D3(sinyi∂j)=8ijaiCOSyi∂x
D3(cosyL∂j)=-8ijaiSinyi∂x.
and
乃瑠eeO那ぬ乃ねのの加c克おw肋朗C力0助仇
bmma18.
Hereweshowthat
伽qf
aL=a2.Puty=y)and
z=y2･De丘nethefunctions
′andg∈C00(r9)as
D3(sin(y+z)∂))=f∂x
D3(cos(y+z)∂L)=g∂x
and
(notethat∂,=∂,,.).Simnarlyasabove,Wegetthatthefunctionsfandgdependonlyon
thevariable
yL
Apply
and
y2.
D3tOtheeqpalities
kos(y+勿∂L,∂i]=Sin(y+可∂Land[∂L,Sinh7十Z)∂L]=
cos付+z)∂,(i=1,2),thenwegctthatthesefunctionsfandgsatisfytheconditionofthe
followinglemma.
-3-
Yukihiro
bmmal,･上eりα〃d
Kanie
gα柁伽α加∫げル仰W由抽ぷγ∫α乃dγヱ,∬上砂加g伽
蜘乃J血Jeq〟αJわ耶
∂f′=g
α乃d
∂fg=一′
(オ=j,2).乃e〃班即∬eCO耶ぬ乃ねαα乃dβ乳化力∫血〟
f=aSin(yL+y2)+β.cos(yL+y2)and
g=-βsin(y,+y2)+acos(y,+y2).
WecanprovethislemmabyuslngLemma13twice.
Here,this
constantαPrOVeS
tObezero･Infact,aPPlyD3tOtheequality∂)=
kos(γ+之)∂J,Sin(γ+g)∂止
ApplyD3tOtheequalitysinz∂]=[sin(y+z)∂L,Siny∂L],thenweget
O=hgcos(y+z)∂x,Siny∂L]+[sin(y+z)∂,,a)COSy∂x]
=βsinysin(y+z)∂x-aLSin(y+z)siny∂L,
hencewegetthatβ=aL,thatis,
β3(sin(γ+z)∂ヱ)=αJCOS(γ+z)∂ズ.
Simi1arlywegetthat
β∂(sin(γ+z)∂2)=α2COS(γ+z)∂ズ.
Apply
D3tOtheequality
【sinz∂L,Siny∂2]+kosy∂2,COSZ∂L]=Sin(y+z)∂2-Sin(y+z)∂,,
thenweget
♂=α2COS(γ+z)∂ズーα∫COS(γ+z)∂ズ,
hence,a)equalsto
a2･Thus,Lemma18isproved.
Returnmgtotheproofofthetheorem･Put
a=ai,andlet
D4=D3-aA,thenD4
isaderivationof9Pr(Tq+L)suchthat
D4(5i(Tq+)))=0,D4(9I(Tq))⊂Lr;(Tr(T…),
β一(∂f)=β4(sinγf∂J)=β｡(cosγf∂J)=β(J≦f,ノ≦わ.
Moreover,bythesimilarargumentSaSintheproofofTheorem14,WeCanShowthat
D4(f∂L)=0foranyf∈C00(Tq),hence
D4(裳(Tq+L))=0,thatis,
β=ad(帆+取)十αA+∑れ&+cC.
FortheunlqueneSSOftheexpressionofD,itissumcienttoshowthat
andW∈,,ifthe
derivation
a=bE=C=O
adW+aA+∑biBL+cC(W∈夙(T…))iszeroon
2r(Tq+))･Infact,aPPlythisderivationtothevectornelds∂x,∂Landsinytaj.
-4-
Q.E.D.
N)TES
ON
UNIMODULAR
PARTIALLY
this
§7.FoliationsonCylinder(Ⅰ).In
SrRUCTURES(Ⅲ)
section,We
Study
derivations
of
裳(R]×Tq)fortheproductfbliationontheproductmanifo1d点LxTq,givenby2r-L(t),
t∈Tq,andconsiderthes･♪･u･StruCtureT=dx･Thenwehave
名(Rlxrq)=夙(Rヱ×丁年)⑳部(rq)=C00(Tq)∂ズ+∑C虚(r冨)∂ズ･
C
A,Bi(1≦i≦q)and
Wecande丘nethederivation
inProposition150f§6.ThederivationsAandBt
of.玖(R]×Tq)similarlyas
areproperlyouter,butthelatestCcan
C=-ad(x∂x)･
berealizedbythevector丘eldas
Thenwegetthefo1lowingtheoremsimilarlyasTheorem16･
職∞托m20.｢り
裳(RJxrq).乃e乃,伽reα血α
上eJβ如α庇わ相加〝〆
γeCわrβe揖Ⅳ0乃R】×㍗α舶00脚加仙川崩れり≦f≦α)∈月∫〟C力血J
β=adⅣ+αA+茅∂f昆･
肋柁卯叫伽co那ぬ畑αα乃d∂f〝e以呵〟ゆ虎ねm加d,α乃dⅣ由加島(R∫×r9)
+R∬∂ズα乃d如殉以e椚α加わ助ece乃托r′=R∂ズOrg;(Rlxrq).
仲ノ 乃eβざJco力0椚0わ紗餅=餅(名(RJxT9);g右肘×r9))ね〆成me那加針2,
4凰(j≦よ≦留)and
仇緑川烏山廟如上鮮血臥西行抑矧加ゞ
【A,BL]=[Bt,Bi]=0,[C,A]=A
C
w肋柁血ぬ耶
and[C,Bi]=Bi.
RefereれCeS
【K-1】Y.Kanie:α力omo物ね∫げ上fgα加ゎ招∫げγeCわ′βe肋w肋の亜おぉ旭加α郎)如
Univ･11(1975),
representa(ions:Cbsedchzssical卯e,Publ･RJMS･,Kyoto
213-245.
【K-2】-:αゐ0仰わ如∫げ上ねα鹿ム和∫げyecわ′舟肋w肋co敵お咄加α尋わ如
rq?reSentatibns:Eblbtedase,Publ.RIMS.,Kyoto
Univ･,14(1978),487-501･
【K-3】-:助椚e上おα如加∫げγeCわr舟肋0乃βぬねd〝昭〃紳助α乃d伽か血ガ用血〃
abbns,Proc.JapanAcad.,55,Ser.A(1979),409-411･
【K-4】-:助meエfeα如加Ⅶ∫げyecわr押通α乃d鋤か血血血那ご
c加∫血J加〉e,Nagoya
αぶeげp∬ぬ砂
Math.J.,82(1981)･
【K-5]-:Nbtesontwtb砂unimodLhzrstructures似Bul1･Ofthe
Mie Umiv.,32NaturalScience,(1981),17-26･
Fac･OfEducation,
【E-6】-:肋細0〃卯r血砂〟〃血0血お∫助血昭働鋸d･,33(1981),ト6･
【K-7】-:0囁α珊〟明り〃㌦0如甲脇ばC呵)′仰卯αズ〟〝0∼M∽OZ〟那α必中此
c8幻afLfLbLrCfLLLN14,tOapPearin
Russ.Math.Surv･(inRussian)･
【T-1]F.Takens:DerivatibnsdvectorJkhis,Comp･Math･,26(1973),15l-158･
ー5-

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