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(1)On Integrability Conditions of Equations of Contravariant Almo-t Analytic Vectors and some related ones By. Yosio MUTO DePartment of Mathematics (Received April 14, 1964). '1, ,. gl. Introduction Let us consider a real Coo manifold M of dimension n .with a (1,1) tensor. fieldgof class Coo. ･ Indices h,i,IL le,l,m=1,･･･,n are used for components qih with respect to a natural frame associated with local coordinates (xt).. A (1,1) tensor gih may be considered as a matrix g and we assume that (A) the degree of the minimal polynomial f(x) of g is constant and f(x) has only simple roots, 2,,･･･,2., that is, the Jordan canonical form of g is diagonal. Thus, if we fix a point P in M, we can take for the moment complex coordinates (xi) in some neighbourhood of P such that gt'b satisfy. (1. 1)* soih =pi5ih at P, where pi is equal to one of 2,,･･･,R., Such an equation which is valid only at a point P will be indicated by an asterisk. Let us define provisionally a symmetric affine connection 11h,:t in M. 7 is the symbol of covariant differentiation with respect to r.hii.. When g is an almost complex structure, a contravariant vector field zt is called a contravariant almost analytic vector field when za satisfies a system of partial differential equations. (1.2) zticOicqih---qikOicuh+g,iiOiuk==O, which is equivalent to ztk7kqih-gikVicuh+gicibLuk=O. ' '. ,t. or. 8g==O. u. /. This definition of contravariant almost analytic vector fields will be adopted. when g is a (1,1) tensor field and we are going to study integrability conditions of (1.2).. We consider also a system of slightly modified equations. (1.3) ukOk{;pih'-gpikOkuh'+{pkibOiuic=ak,ibuk.

(2) 2 Y. MuT6. where ajih is a (1,2) tensor of class C". In this case we assume besides (A) we 9Bs)suthheediOoOr8SovOefr tthheatMinimai poiynomiai f(x) are constant.. (C) the proper equation lg-pEl =O has no simple root. inThecase of need. ･ purpose of the present paper is to obtain necessary and sufficient conditions for (1.2) or (1.3) to be a complete system. ' By complete system we mean the following. Let. (1.4) FA(O,i,u,";zLrc;xh')=O (A==1,･･･,N) '. '. be a system of partial differential equations in m unknown functions zt', ･･･, it"".. Indices are used for the present as follows, A, B, ･･･ = 1, ･･･, N, rc,R, ･･･ =1, ･･･, m, h, i, ･･･ =1, ･･･, n.. The system (1.4) is a complete system if and only if the following conditions are fulfi11ed.. No equation in xi,･･･,x", ui,･･･,ztM only is obtained by eliminating Oiu". from (1.4). Equations obtained by differentiating (1.4) partially v times (v==1,2,･･･) with respect to x (u being considered as functions of x) will be denoted by (1.4),. Then any equation obtained by eliminating Oi..,･･･Oi,u". from.g'6`8ts,`,s,.n,oi,i,n,d,e,P,e.n,de.",t,O,`,8'i`,l',b(.",a),':'"'(i'`)v'-i' THEoREM 1. Let us assume (A). A necessary andsuX77cientcondition that (1. 2) be a cQnzPlete system is thatg be numerical, that is, M admit a covering by coordinate neighbourhoods such that the comPonents gih are constant.. THEoREM 2. Let zas assume (A), (B) and (C). Then a necessa71y and sufficient condition that (1.3) be a comPlete system is that g be numerical and aj･ih satis.lty some algebraic equations and some s.vstem of Partial dip9erential equations (see (3.14), (3.15) and (3.16) in g3).. S2. A necessary condition that (1.2) be a complete system Let us assume that (1.2) is a complete system,. From (1. 2) we get ,. zale0kqii=O.. Since there can be no equation in x' and u only, we have Oicqii==O. Hence Trace g is constant. The Nijenhuis tensor AI,,h of the tensor g is defined by IVI?･i,h':=:sp,iiOtgoi.h'--7goi,LOigoji"-(O.i{p,it-Oi,{tpl){pthJ '. ..

(3) On lntegrability Conditions of Equations 3 or AI,iih=:sojt7tsoi.it-{tpitVLgo/e-(J7jspit-l7igj`)goe'".. Iif we put ,. Lih=8{plblr'uicVicsplb-spikl7teuib+goichVittic, N. Lj.iib==￡IiSg, '. u.. ttt. '. u. where ￡/ denotes the Lie derivative with respect to the vector field u, then the Lie der2 vative of AT),h can be written in the form i'`. llitAl17･ih==9{spjicVicgDih-sptk7ksc)/b-(7j{piic-7igojk)sp,ib}. =Lj･ic7icgiii-Liic7icg/b-(Vjgik-Vig,･ic)Licib. +9.iic(7kLih-Licii9ti`+Liclhgit)-gik(VicLjh-LicJ･igeib+L,thqj･t) -(7jLiic-7iLl･k-L,･iispele+Lij`{ptic+Ljzicgpii-LiLicsDjt)gDic'b. for we have ￡/(Vjgii")=7jLiib'Ljiigtib+Ljehgit･. 'tt. Lj,i'e being symmetric in ]' and i, we obtain (2. 1) ￡/ Al17･iib=LjleVicgiie-Lik7ksolb-(Vjgt)ik-Vig.iA;)Licib. u. +9,･ic7kLiic-g,icV,L/V-(J71iLiic-VtLjk)g,ib,. which relation holds good for any vector field u.. If u is a contravariant almost analytic vector, we have Lih==O. Hence (2. 1) becomes 2 Al>ih== O, or. u. (2.2) uicO,2Vl)･,h-.ZV17･,leO,ztib+IVk,ibb,-u,+IVI?･,hO,uic==O.. We can regard (2.2) as linear partial differential equations in functions zt. and independeht variables x. Then, since we assume completeness of (1.2), every equation of (2.2) must be a linear combination of equations of (1.2), Hence equations of the form (2.3) zticO,IVIi,h-IVIi,icO,uib+IV),,ibOduk+IVI?･,it:O,ztic ==Aj'ilb･,nl(ZticOicSt)L"b-SPeicOicZtM+{PkMOeUk). '. must be fu1fi11ed identically when the coefficients A,,.'L.,t are chosen suitably.. Comparing coeMcients of Oicui and of uic in both sides of (2.3) we get tollowing equations, (2･4) mlVIir,ic5tic+IVZiib6jk+:ZVIith6,k=-4,ieeMs;o.ic+Aj,ib.A'speM,. (2, 5) O,A71i,h =Aj,hiMO,giM. Putting l=le in (2.4) and summing for k we get. IVI)･ih =O.. ' g is 'numerical (see From this and Trace q= const we can conclude that.

(4) 4 Y. Muhr6 g4 of [2],)D. Thus we get the LEMMA 2.1. 'VVe assume (A). IIIr a system (1.2) of Partiat dijSferential equations, zvhich defines coiztravar'ict7Lt ctlmost anctlytic vecte't" .fielcls, is cotiz-. Plete, g is cL numerical tensor,. S3. A necessary condition that (1.3) be a complete system Let us assume (A) and (B) and that (1,3) is a complete system. Since T?i is symmetric in 1' and i we can write (2. 1) in the form (3.1) ukO,2VI),ib-2Vli,icD,uh+2Vk,ib6,･uic+IVI),hD,uic. ' ･ =L,･kO,g)iii---LiicO,g,･h-(O,･g)iic-O,gpjic)L,iL. '. +gjicO,L,h-g,icO,L,･h-(O,･Lik-OiL,.ic)g,h. This relation holds good for every (1, 1) tensor g and every vector field u.. Substituting ' Lih == uicaicih. into (3.1) we get (3.2) ukOicN17-iib-A7IiFikOkui"+MiibOj･uk+A7IfichOiz{ic. -ut{atj･kOkgplb-aeikOicgDlb-(Oj･qik-Oigpjic)atich +gpjicOica"h-spikOkat,-h-(Oj･aiiic-Oiaij･k)gpicit} -spjkatihOicui+gikaLj･haicui+atiicspichOj･ui-atj･lesi)ichOiui=O.. Then every equation of (3.2) must be a linear combination of equations of (1. 3), hence (-A71?-iic5tit+.ZVLiibSjk+IVIiehSiic-stpjkaLiib+{Dikaij'b. +atiMsp.h5j･ic-aij･Msp.hSiic)Oleui+{OleAIIiiit-aifOispiiV+aleteOtip/b. +(Odspie-Oispji)akLh-{DjL6iaicih+giiOtabjib. +(O,-akei-Oiak,･`){Dih}uic = A,fihLM(UkOic9oml-gomibOkut+ 9oicIOmuh-zahaicn,l).. ' and of uic we get relations of the form Comparing coethcients ofJ aicuL. (3.3) -IVIiiicbth+IVLi'v(5jk+IVI7zhSik-sDjicaLi'i+spikaL/b '. ' +a"Msp.ibSjic-aLjMsp.hSik=-AjiibeMgp.ic+Ajiib,.ksDe"e, (3.-4) Ok2Vlii'b-aic,･`Oigpih+akitOtgpjh+(O,･{;pi`-Oi{DX)aicLh -g,･LOLakih+giiOtaic,･h+(Oj･aiciL-OtakJ･i)gih. ==AjihtM(Ohg.t-aic.i). 1) On this occasionIwish to remark that in order to prove that, if the NiJENHuis tensor of g vanishes and every eigenvalue of g is constant, then g is integrable we. have to refer' to NEwLANDER and NiRENBERG [3] or NiJENHuis and WooLF [4]..

(5) On rntegrability Conditions of Equations 5 Let us co.nsider (3.3) at a point P, Then, using complex coordinates satisfying (1.1)* at P, we get. ' (3.5)* -.ZVIti,'"6e'b-i-bX{AZI,'e-i-(pt,-p,)at,ib} -5,iA{IVL/b-e(p/,-p,ti)acJ･'t'}. =(Pl-pic)A7･ilblic.. Forland le such that pL4pk we can regard (3.5)* as equations determining Ajihiic.. On the other hand we can eliminate Adihtk from (3.5)* by taking l and k for which pi==pic. For the present we use indices ajb, c, d, e, ･･･,P,q, r, ･･･, x, y,･･･ as follows,. tt. '. Pa =Pb =Pc =Pd== Pe== ''' Jt Pp== Pq =Pr = '''. #Px==Py= '''･ Putting le=d, l=e in (3.5)* we get (3.6)* -2V17･id6,'b+S,･a{2(Vl,ih+(to,,-ioj)a,tiV} -5iel{IVbjib+(toh-toi)a,/b}=O. Putting 7'=c, i=b, h= a, in (3. 6)* we get. Alb,a==O. Putting 1'=q, i=P, h==a :d==e we get. AIQ,a==O., and putting 7'=x, i=P, h==a= d=e we get. Alle.a=o, These results show that the components AThiii must vanish except those for which the numbers 1' ,i, h satisfy. ･ pj=ph7!pi or pi=phit!pj. Putting 7'=c, i=4, h=P in (3.6)" we get. AlbqP+(Pp-Pa)aeqi'=O, hence. (3.7)* A7i,b"==(p,-p.)a,b". Now let us consider the minimal polynomial. f(x)==xP+c,xP-i+ ･･･ +cp-,x+cp of 9･. From f(g)==O we get.

(6) 6 Y. Mu'r6. g2)-i2 g+g2)-2(8 g)g+ ････ +(8 g)g2)-i. 2k U U. +ci(sDi'-'2il gp+ ･･t +(8ill so)sp2'L`2). -]. ------}-----. +c,-,2 sD=O u for ci, c2, ･･･, cp are constant by virtue of assumption (B).. Since we have. an6a.i)*,weget ¥gzh==aleihu'` , ' {toiP-･i+toiP--2ioh+ ･･･ +tohP-i+c,(toiP-t2+ ･･･ +tohP--2) + ･･･ +cp-,}ajiibuj-TO. As (1.3) is assumed to be complete, this must be identically satisfied, hence {toipu-'+ioiP"-2toh+ ･･･ +ioi,a'-'i+c,(ioi2'-'2+ ･･･ +toi,P'2). + '･･ +cpk,}adiib=O. For i and h such that pi#ph this is equivalent to {(toiP-io,P)+c,(ioiP-i-to,,2)-i)+･･･+cp-,(ioi-to,)}ctjiib=O. which is a trivial identity.. Foriandhsuchthatp,=phweobtain ' {PpiP-"+<P-1)c,piP-2+ ･･･ +c,-.,}aji'b=O. from which we can deduce aj･,i,je==O. by virtue of the assumption (A). Hence we have. (3. 8)* ajb"=aj･qi'=ajy'"== ･･･ ==O, From this result and (3.7)* we get. (3. 9) IVI,･,iV =O .' suchTthieant' SinCe We aSSUMe (B), we can take iocal compiex coordinates (xi). ' gtpeh=toiO,h'. Hence asterisks oan be removed hereafter, Thus, substituting (3.9) into (3.5) and putting k= a, l=e, we get. (3.10) , S,･d(ph-p,･)a,ih-Sia(ph-pi)a,,･h=O. Putting 7'=c, i=b, h==P in (3.10) we get. 6,da,bP-SbdaecP==O, hence ' '.

(7) On lntegrability Conditions of Equations 7. (3. 11) a,bP=O. if p. is a multiple root of the proper equation lg-pEl==O. Putting 1'=c, i=x, h=P in (3.10) we get. aexP =:O . ' Thus we find that the components aji'b vanish except those for which ph.=pjlpi and those for whichi=i#h. the latter vanishing if p.i is not a simple root of l{D-pE[=O. Let us assume (C). '. Then a.i,;ib vanish except those for which ph=-=--to,i'-1'tO･i･. We get from (3,3) (K)t-toic)Aj,,ibtic==(pj,----toj)6j･leat,ib-(toh-pi,>6,A'ai/L. =(toJ,-tok)(Sj･icaiiib-5,iicai1'J),. which is equivalent to. (3.12) A,･ihLle=Sj･leaeih-5iicai,-'b (toi"t!toic) because atih and ai" vanish if pt iEph.. From (3.4) we get (ph-pj)Ojaicih-(ph-pi)Oiabjh+Ajihtma,.t==o.. Since ak.i vanishes if p.=pt, we need in the last term of the left member only to sum forland m such that p.tpi. Hence we get by virtue of (3.12). ' (3.13) (ph-p,･)O,-aki'L-(p,,-pi)Oiaicj･h+aic,･tatih-akiiaij･'b=O.･ ' Let us express the obtained conditions in general coordinates., We, find immediately that. (3. 14) ajitspth-g,-iaeih =O, (3. 15). P{pMiimaj.h+Pgoi ,2･maj.Lg)Lh+ ･･･ +qjitPi5;h. '. p-2 +ci( giMapmib+ ･･･ +ajiL p-2 gLh)+ ･･･ +cp.-iajih=O, r where g=gr, are equivalent to the condition that the components ajiib vanish. except those for which ph=pjtpi. ,. Now, as we assume (A) and (B)t.and have obtained (3.9), there exists in. M a symmetric g-connection (see Theorem 9 of M. Kurita [1] or Theorem 1.1 of [2]). gih being constantin our special coordinates, the coeMcients r,h･,. of this connection must satisfy. (p,,-p,)T?, = O, hence T,h･, vanish except those for which ph==pd=pi･. Then taking (3.14) into account we can write (3.13) in the form (3.16) -{pdiVtaicih+soiL7taic,･'V+(Pl-aki`-7iabjL)sDeh +akji'a"ib-aicitaifh==O,.

(8) 8 ･ Y.MuTo. (3.14), (3.15) and (3.16) are validin any coordinates fortheir left members. aretensorexpresslons. ･ Thus we get the following lemmas. LEMMA 3.1. Assume (A) and (B). A necessa?ly condition that the system of Partial dijfZerential equations (1.3) be comPlete is that the ATij'enhuis tensor. satis.ICIy (3.15). ･ of g vanish and ,･,ib a LEMMA 3.2. Assume (A), (B) ana (C). A necessary condition that (1.3) be comPlete is that g be numerical and a,･i7b satistv (3,14), (3,15) and (3.16). where 7 denotes covariant diLfferentiation with resPect to a symmetric g-connection.. g4.,Proof of Theorems1and2 ,. Assume that g is a numerical tensor and take complex coordinates such that,. (4.1) gi'b=piSi'b (piareconstant). Then (1.2) can be written in the form. (4. 2) (p,-p,)O,uh==O, '. hence '. Oiza h =. o. when ph￠p,. But (4,2) is complete, admitting solutions such that uil= ui'1(xl, ..., xnl) s ui2 = ui2(xnl+1, ..., xn2) ) e-----e----------------uzp = uzp(xnp-1+1, ..., xn). where. ii =1, "', ni, pi=m=pni =2i, i2tni+1,''',n2, pni+i='''==Pn2==22, -'-'t--------e---'" ----ti-}-t-t'"-i----'i''-. ip=np-i+1,'･',n, pnp.i÷i=''･=Pn=:2p. Hence (4.1) is a sufficient condition that (1.2) be complete. Next, let us assume besides (4. 1) that ajih vanish except those for which. Ph=pd#pi･ Then (1.3) takes the form (ph-pi)Oiuh,==aicihuic and the system is separated into P parts. Fof example we have for zti, ･･･, ztni. Oi2uibi=(pi-p2)-"iakii2hiuki, --･-----------t--i}----t------E)i,puh'i=(toi-K)p)-]ak･ii,7,i"izt･ic],.

(9) On lntegrability Conditions of Equations 9 Regarding xi,･･･,x"i to be fixed we find immediately that this system of equations is completely integrable if (3.13) ,is satisfied. Consequently (1.3) is. a complete system when g is numerical and (3,14), (3.15) and (3.16) are satisfied by aj･ih.. Thus we have proved Theorems 1 and 2. References [1]. KuRiTA, M.: Tensor fields and their parallelism. Nagoya Math. J., 18 (1961),. [2]. MuTO, Y.: On some (1, 1) tensor field and connections. Science Reports of the Yokohama National University, Sec. I, No. 10 (1963), 1-11.. [3]. NEwLANDER, A. and NiRENBERG, L.: Complex analytic coordinates in almost. [4]. NiJENHuis, A. and WooLF, W. B.: Some integration problemsin almost-complex. 133-151.. complex manifolds, Ann, of Math. 65 (1957), 391-404, and complex manifolds, Ann. of Math., 77 (l963), 424-489..

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