トップページ - 横浜国立大学学術情報リポジトリ

全文

(1)On some (1,1) Tensor Field and Connections By. Yosio MUTO (Received April 30, 1963). gl. Introduction. We consider a Cco manifold･M of dimensiQn n with a (1, 1) tensor field g of class Coo. Indices h,i,i,･･･ =1, ･･･,n are used for components gi･h with respect. to a natural frame associated with local coordinates (xi) and indices a, B, r, ･･･. =1,･･･,n are used for components gBcr with respect to a more general frame. An even dimensional manifold which admits a (1, 1) tensor field FEh satisfying FsiFi･h=:-6?- is an almost complex manifold. An almost Hermitian space, an almost Kahler space, and a Kahler space are also almost complex spaces. If a manifold admits a (1, 1) tensor field Fih satisfying ,PIJiFEh ==6ig･, it is called. an almoSt product manifold. These are very often studied by many mathematicians.. A (1,1) tensor gi･h may be considered as a matrix q. Then the tensor giigih is the matrix g2 and so on. Some properties of a manifold admitting a (1,1) tensor field q such that gP=±L where P is' some'integer and l is the. unit matrix, were studied by S. TAcHiBANA [5] and S. IsmHARA [2]. In this case the.minimal polynomial f(x) of q is xP Fl or a divisor of it, hence f(x) has only simple roots and the Jordan canonical form of g is diagonal.. Matrices g considered in the present paper are also assumed to have diagonal Jordan canonical forms. Some years ago M. KuRiTA studied'parallelism of such tensor field [3]. The NiJENHuis tensor Al<g) of a tensor field gi･h is defined by. (1. 1) ･ Al> ･h ==qyo,gih-gila,gbh-(ajgEi-a,gbt)gih. If N(qr) and gr are denoted by N andrr g respectively, we get (1.2) rliiVl>1.h.=[[Ei,',=.),(r￠S`cT￠i,bqdigh+r￠;･.cripg.b9g"Zh)+,S.,ebc&i.g2rg':hq]lvbba.. ' ' - .; ' Hence IV(gr) vanishes whenever 2V<g) vaniShes. Geometrical properties of a manifold admitting a (1,1) tensor field with vanishing NiJENHuis tensor were studied by A. NuENHuis [4], by A. FRdLIcHER and A. NuENHuis [1], and also by. M. KuRiTA [3].i) In the present paper some properties of g are studied in relation to inte1) These papers contain more general results..

(2) 2 Y. MuT6. grability, g-connections, and parallelism. q-connections are obtained explicitly.. A (1, 1) tensor field g is said to be numerical in M when for each point P of M a neighbourhood Ui) and local coordinates xi valid in Ul) exist such that the components gih with respect to the natural frame F(Ub;xi) belonging to this coordinate system (x'`) are constant in Ul). Then a set {(Ua);x2z),･･･,. xeb);2EA} of coordinates and neighbourhoods which cover M can be chosen in such a way that the constants sD(z)ih do not depend on R. This is proved as. follows. First assume that U(.)AUk.+i) is not vacuous and that q(.)ih and g(.+i)i･h that are components of g with respect to F(U(.);x&'.)) and F(U(.+i);. xe.+)) respectively are constant. Then there are matrices A and B such that･ BiMAthh=:6,h･ and g(.+,)i･h==BiMg(.),hiAih. If we use new local coordinates fc?.+i) =Bihxea+i) in U(a+i), we get ip(a+i)ih=Ai･Mq(a+i)thiBih =g(.)ih. Such a process can. be carried out to any U(.)AUt.+i) repeatedly and also to any chain U(i),･･･, U(.) of neighbourhoods where U(.)AUk.+i) (a==1, ･･･,m-1) are non vacuous, and finally we get a desired set {F(U(z);xez))} of natural frames.. Let M be an aMnely connected manifold. If the covariant derivative 7g of q with respect to this connection vanishes in M the connection is said to be a g-connection. As we assume that the minimal polynomial f(x) of q has no multiple root, there is, at each point P of M] a complex frame EYp such that the components g)Bev of g with respect to Srp satisfy. SPAex=,op(5S. But, in general, there is no natural frame F(U;xi) such that gi･h satisfy. 9ih =piS,h･ in U even if complex coordinates are admitted. Such a natural complex frame F(U;xi) exists for some neighbourhood U of every point if g is numerical, or, if the covariant eigenvectors are all X},-,-forming, or, if, for every point. PEM and some neighbourhood U), there are fields of n covariant X},-i-forming eigenvectors which are linearly independent throughout the domain Ul). In the last case we can take in Ul, n gradients Oif(i), ･･･,6if(") such that. gJ-iOif(h)= p(,)a,.f(h). and det (aif(h))tO. Then we can take f('),･･･,f(n) as new local coordinates xi',･･･,xn' for some neighbourhood U such that PG I7cUl), and obtain adxj'g))h,t=:g)bi6ixht=ph,adxht,. hence gP'=:pi,62･t'. It is also evident that, if we have gi･h==pi6,h･ in some neighbourhood U and for. some coordinate system, then we can take n linearly independent cQvariant eigenvector fields in U which are X},-i-forming. Since pi are functions, Al<g).

(3) On some (1,1) Tensor Field and Connections 3 =O is not satisfied in general.2). Before closing the introduction some theorems will be mentioned.3) THEoREM 1.1. if a (1,1) tensor .fiela g.is numerical, then its Alijenhuis tensor AI(g) vanishes and M admits a symmetric g-connection. PRooF. The first half is evident because we can take local coordinates such that the components gi･h are all constant. Let {F(Ukz);xea));REA} be a set of natural frames such that V2U(A) covers Mand q(z)i･h are constant. A set {fka,;REA} of Coo real valued functions is taken such thatz;Afk2)=1 and such that fa)(P)=O if PG- UkA). Let J"(z) be an afline connection such that the components T(z),h･i of r(A) with respect to F(Ukz);xez)) al! vanish. The compo-. nents of T(z) with respect to F(Uk.);x?pt)) are denoted by T(2,di･i. Then z.2Afkz)Ta,di･i are components of an affine connection T, which may be denoted. by 2fo)r(a). We show that AcliA. (1.3) rtz)T(z,pt),h･ig(,)i･i-fl[A)T(z,pt)S･isb(p)Zh==O in Uk.). Since -ICkz) vanishes in Ut.) if Ukz)AUk.) is vacuous, and since ra,z),h･i. =ra)?･i==O, we need only to show. (1.4) Ta,pt)3ig(p)EimT(z,pt)S･ig(pt)ih==O for the points of (7(z)AUt.) where R#pt. As we have T(2,a)?･i==O, we get. axe.) 62xez). T(2,ptB'i=oxe'2, 6xz'.,oxe,,,,' On the other hand spa)i･h and gp(nih are' constant. Hence, differentiating parti-. ally the equalities. .iOxez) Ox?lpt})m .h 9(p)m oxe.T ox?iJm-9a)i we get (1.4) immediately. Thus (1.3) is proved and T isag-connection. As 'T is obviously a symmetric connection, we have obtained a symmetric g-connectlon.. THEoREM 1.2. [f M admits a symmetric g-connection, then the Aiijenhuis tensor N(g) vanishes.. This is proved immediately by substituting 6jqih=-T?･igi･`+1-'S･igih into (1.1).. g2. Existence of a g-connection.. We assume that the minimal polynomial. (2. 1) f(x) =- xP+a,xPdi+ ･･･ +ap-x+a.`) of the matrix q under consideration has only simple roots Ri, ･･･, 2p, and search. 2) See also A. NiJENHuis [4], A. FRdLicHER and A, NiJENHuis [1]. 3) See also M. KuRiTA [3]･. 4)PisassumedtobeconstantthroughoutM. L.

(4) 4 Y. MuTO. for a necessary and suthcient condition that M admit a q-connection. We. also obtain/a g-connection. or. oo be the coefficients of an arbitrary affine connection. Covariant Let T?･i. * and, differentiation with respect to this aMne connection is denoted by V, when X is any contravariant vector with components Xh, the matrix with. ･. ' Let T?･i be the coefficients of another afiine elements Xiigi･h is denoted by qi.'' connection. If we define a matrix ZIi by. * (2.2) Tir=(X)Tyih), Ts,:h=I"?･,-r?,, , then a necessary and suflicient condition that T,h･i define a q-connection is that. 71isatisfy ･. ' (2.3) 71x9-9Tx=9x for every vector field X. , Let EYp be a complex frame at P such that the components gAcr of the tensor g with respect to Srp satisfy. (2. 4) gpAa'=top6Bat ,. and F(U;xi) be a natural frame where I7 is a neighbourhood of P. Then there are matrices (Ai･a') and (BAib) at P such that. ' ' .BAig･hAhcr AB==l :pp6B". If matrices obtained by transforming g, gx, and Z'Ly' by B and A are denoted by gf, gx' and Tir' respectively, (2.3) becomes TJi-'g/-g/Ti'==gx/. Since (2.4) are the elements of g/, we get. '. (2. 5) (p.-pp) 712rbat=g2ect･ From (2. 5) 712Aa are determined completely for a and B such that p.lpB, but, for a and B such that p.==pp, these are left undetermined. On the other hand glBcr must satisfy g2ecr=O if p.==pp, and this is a necessary and sufficient condition that M admit a g-connection. Differentiating covariantly the tensor equation. (2. 6) qP+a,gPd'+ ･･･ +a.-,q+a,l==O with respect to *, T?･i, we get. (2. 7) gxgP-i+qqxgP-2+ ･･･ +gPHigx +ai(9zg"'2+ggxgP'3+ ･･･ +gP-2qx). + -･･--･-･. +ap-i9x +(JZiai)gP'i+ ･･･ +(Jlrap.i)g+(7Lra.)l. ==o. Using the frame SXp we get. `. :. is.

(5) On some (1,1) Tensor Field and Connections 5 {PR.Pdi+ lvdl)aiRrP"2+ '" +apu-i}92Bcr=O. for the numbers at and B such that a4B and p.=pp=R.. Hence we find f'(R.)szpti}Bcr=O. But, since a. is a simple root and do not satisfy .f'(2.) =O, we. get g2ecr=O for such a and B. Thus, the condition that g2sa vanish for the numbers at and B such that p.=pB is always satisfied as long as atB. We also obtain from (2.7) , f'(p.)g2dict+(JLra,)p.P-i+･･･+(Iia.-,)p.+J7Ltia.==O,. where we do not sum for a. Then putting ndicr=O we get (JZxai)p."di+ ･･･ +(JLrapMi)p.+VLrap=O,. hence (JZzai)2.Pri+･･･+(JZza,:i)2.+JLra.=O (r=1,･･･,P). vAaSnisikj'W' Rep gaerte diStinCt' JLrait'･', JLrap Must vanish･ Conversely, if lz.a,, ･･., Jz.ap. ' f/(Pcr)9Idiev==O. ･･ .Sincef/(p.)#Owegetgi2Bct=Ofora=f9. '. '. tt. '. '. Thus we obtain the following theorem which was already obtained by M. KuRITA [3]. THEoREM 2.1. Let zas assume that the minimal Polynomial f(x) of g has only simPle roots. Then a necessan7 and sufiZcient condition that M qdmit a g-connection is that the coev[7Zcients of f(x) be constant, that is, lhe roots be. constant. , '' -. g3. An explicit form of a. t- t g-connection.. '. '. .. We assume that the coefficients ai,'･･･, a. in the minimal polynomal f(x) of. q are constant. A g-connection is obtained if we put. '. (3.i) z12}Bct=',,.Ii,,pq.i}Rct ･" '. for a and B 'such that p.tpp. But this is not a tensor expression.. We put '. '. ' p-1 (3. 2) 7Lr= ]2] C.,g)rsz)ig)S -r,s==O. tt t. and try to determine C., is such a way that Tz may satisfy (2. 3). At first we prove that a matrix 7-lz of the form (3. 2) exists which satisfies. <2.3) for every vector X. . ,, ''. If we put for example p.=Ri, pp=Z, in (3.1), we get. Asthed'iscriminant ' 7'lieBct.==R,ELi,sz'2B". '. D= (Ri-R2)2 "' (Rp-i-2p)2 , ' can be expressed as a polynomial P(ai, ･･･,ap) bf ai,･･･,api we have.

(6) 6 ･ Y.MuT6. 1 R,-R, D. 1i-R2 = P(ai, ･･･, a,) ' (Ri-R,)2' e. On the other hand (A,-R,)-2D is a sYmmetric polynQmial of Ri, R, with coeffi-. cients which are symmetric polynomials of Z,,･･･,R,. Hence (2i-Z,)-2D is a polynomial of fundamental symmetric functions of Ri, R, and fundamental. L. t. symmetric functions of R,,･･･,2p. Since any symmetriG, polynomial of Z3,･･･,Ap. is expressible as a polynomial of fundamental symmetric functions of Zi,R2 and fundamental symmetric functions of Z,,･･･,2,, we find that (R,-R,)-2D is. apolynomialofab･･･,ap,ZiZ2. Hence ' ' -' 1'. ･ ,Z,.-R,. can be expressed as a polynomial P,,(R,, R,) of Ri, R, where the coefficients are. rational functions of a,,･･･,ap. Since 2, and Z, are roots of f(x), the degree Of Pi2(Ri,22) in Zi and R2 respectively may be considered to be less than p. If this polynomial is expressed in the form. p-1. Pi2(Ri, R2) = ￡ CrsR2rRiS. r,s =O. and the coeMcients C., in this expression are used as the coethcients C.s in (3.2), then the matrix 7HIi obtained satisfies (3.1). Of course (3.2) is not the. unique solution of (2.3). , Since a.･･･,a, are constant, we get from (2.7). (3. 3) gxgP-i+ ･･･ +gP-igx +a,(9x9Pd2+ ･･･ +9P--29.). + ･--･---･ +ap-igx==O･ N. When (a.,) is a matrix of degree p,5) the expression. '. p-1 (3. 4) Za.,g'PxqS r,s= O. v. will be denoted by gz(a.,). Theri we get from (3.3). (3. 5). 9x. ap-1 ap-2. ････- ai 1. ap-2 ap-3. ･-･- 1 O. ----------------- ----i----- ------. =o.. ------i---t------ ---{------------. ai 1. 10. ･-- b o ･-- o o. :. If this equality is multiplied by q`, from the right for example, and the equality (2. 6) is used, we get 5) a., is the element in the (r+1)-th row and in the (s+1)-th column..

(7) ' On some (1,1) Tensor Field and Connections. 7. (3.6), spx(g` O.z?,-,)=O. . oo -･-･o O O ･･････-ap-a -ap-i '. p. i.. -----e--------------- --t------------------ --e--t------. s, =. --t------------------------te-------------------------. O -ap ･･････ -ap.t÷3 -ap-t+2 -ap -ap-1 ･･･････-ap-t+2 -ap-t+1 and aq.-i aq-2 ･････. ai 1 aa.-2 aq-3 ･･･'･･ 1 O --------------------------t------. Rq= ･. ------t--------------------------. a, 1 ･･････O O. 1 O ･-･-OO We have P linearly independent relations (3. 6), (t=O, 1, ･･･,p-1).. Because of these relations we can reduce the number of terms and put' p-1 p-2. (3.7) Tx=2 Z] C.,grqxgS･ - r=O s=O. Substituting (3.7) into 71rg-g ZHIr we get. p-1 p-2 Tig-g71x== Z ]E) C.,(grg.gS+i-gr+ig.gs). r=O s=O '. = ]2] 2. p-l p-1 p-lZ p-2 C,･,--ig)TspxspSZ C.-i,grg)xspS r=O s==1 r=l s=O. -p-2 ･. + 2 Cp-i s(aigP'i+ ･･･ +ap-ig+a.)gxqS. s=o. p-l p-1 p-l2 p-2 = Z 2 C.,.-igrgxgSZ C.-i,qrgxgS r=O s=1 r=l s=O. .. p-1 p-2 +22ap-.Cp-i,grgxgS, r=O s=O '. sothat(2.3)becomes ･ ' '' ' 9x(brs)=O ' '. where. boo==apCp-io-1, bos= apCp-i s+ Co s-i,. bop-i=Cop-2, ･ bro=ap-rCp-io-Cr-io, brs == ap-rCp-i s- Cr-is+ Cr s-i,. in (3.2).

(8) 8 Y. MuTO '. '. , br p-i=Cr p-2. /. (r=1,･･･,p-1;. s=1,･･･,p-2).. .. Thus, a solution of (2.3) is obtained if we find C., such that the matriX (b.s) is a linear combination of the matrices s. (gt ft.-,). ExAMpLE 1. When P==2 we get. (t=O, 1, ･･･,P-1).. t" t /t. . (b,s)=:( ZiSl::},, S2: )･. . (gO %,)-(fi6),(gi '%,)-(,-a22. )･. HenceCoo,Cioareobtainedfrom'' ' ..d ., fi.d ( Z?CCIOO:}oo S:Oo )-COO( fi 6)-Cio( o-a2 Ol ). =o,. tt Cio=4i,2-a,2'. ' ' . ,. Coo==4a,a-i]zil72,'. Th,. 7k. is. obtained in the fOrTM. .,. .,4qis-2.g,gi,. orinamoresymmetricform, .-,. .t ' ,,i.. Taj==q4q.x,--q.f2q, '. .. * by virtue of ggx+qzg+aigx=O. Thus, T2･i,;r2･i+Tsih, where. ** .h- gi･`J71iqih-(L･gii)qih. . Z'i- 4a,-ai2 ' ' ls ExAMpLE a g-connectlon. 1 2. If ai=･･･==apki=O'land ap=ewhere. and Rq in (3.6)t become .. '. OO ･- O1 OO'･- 1O ' St=-e '-"'"''''''''''" , -------!-"-----------. ., o 1. ･･,.,o o. ･ .. 1O -･ OO. b-. E2=1,. '. the matrices Se.

(9) On some (1, 1) Tensor Field and Connections 9. OO -･ O1 OO -･ 1-O. Rq= ･ O1 ･- OO 1O ･- OO Then boo=eCp-io-1, bos==eCpds+Cos-i, bop-i=Cop-2, bre=-Cr-io, brs=-Cr-is+Crs-i, brp-i==Crp--2, where r=1, ･･･,p-1 and s=1, ･･･,P-2, must satisfy. boo==-ebi p-i=-eb2 pA2== ･･･ =-Ebp-n, boi=bio== -eb2 p-i== -eb3 p-2= ･" == -ebpu-i 2, "."....."."･-･･･-.･..･"･".･.-.. bo p-2=bi p-3== '" =bp-2o= -Ebp-i p-i, bo p-i=bi p-2== ''' =bp-i o･. FromtheseequationsandC.p.i=Oweget . , 'F, Cip-,==-S--, c,p-.,= 2pe, L....., cp-,,.. (P-pl)e,. /. all other C.,=O.. Hence ' 71i= S-{gnqP-2+2g2gzqP-3+ ･･･ +lv-1)gP-iq.} is a solution of (2.3), and I"k･i==iEi?･i+Ts'ih, where 71}l･h are given by T):,ib = i- f):.li rcai･m(ill,･gthi)P -a7･S,. ls a g-connectlon.. g4. Integrability of 9･ It was mentioned in gl that, if q is numerical, then the eigenvalues of g are constant and N(q) vanishes. Conversely, if the eigenvalues are constant and N(q) vanishes, then q is numerical. This is proved in the following way. Let the distinct eigenvalues of g be Z,, ･･･, R. and put. p,==R, if i=1,･･･,m,,. pi=R, if i=m,+1,･･･,m,+m,, -t-----------------------l-------------------. ' pi=Rp if i=m,+･･･+mp-,+1,･:･,m,+･･･+mp, where m. is the multiplicity of the eigenvalue R... ' Asystemofpartialdifferentialequations. ttt tt.

(10) 10 Y. MuT6. (4.1)r ･(9i'h-Zr6ij')aar.h ' =O .. with an unknown variable f is a complete system when each one of the. equations , (4･2)r {(9PS`-Rr6S')Oi(9'i'h-Rr(Sij')-(9'i'`-Rr6g')Oi(9'bh-,]lrtB?)}oar.h=O. '. is a linear combination of (4.1).. This condition is fulfi11ed if R. is constant. and N(q)==O. Since for any fixed r (1$r$P) the system of equations (gi･h-Z.6ij･)uh=O has m. Iinearly independent solutions uh, we have m, independent solutions. f=:f` (i=Mi+'"+Mr-i+1,''･,Mi+'''+Mr) Of (4･1)r･. Mor'eover, we can take these solutions of (4.1). (r =1, ･･･,P) in such a way. thatfi,･･･,fn form local coordinates. Then the components of the tensor q with respect to the natural frame belonging to this coordinate system (fi, ･･･,. fn) are constant. Thus we obtain the THEoREM 4.1. Assume that the minimal polynomial of g has no multlPle root. A necessa7:y and suL197cient condition that g be numerical is that the eigen-. values of g be constant and its AIZ7'enhuis tensor N(g) vanish.. If we take a Jordan canonical form BGigi･hAha=ppSBct, the functions. ppr= BYOjpp .vanish for B and r such that pp4pr (M. KuRiTA [3], p. 142). Hence we get. B7ajRs=O ifp.7tR,.. ,. On the other hand we have Traceq=gl･i==pi+ ･･･ +p. :miRi+ ･･･ +mpRp.. Hence we get. B70j(Traceg)==m,BYajR, ifpr==Rs. Then, if Traceq is constant, R, is constant, too. Thus, we obtain the. THEoREM 4.2. Assume that the minim.al polynomial of g has no multiple root. 27 Traceg is constant and the ATZ7'enhuis tensor of g vanishes, then g is numerical.. From Theorems 1.1, 1.2, 4.1, and 4.2 we obtain the THEoREM 4.3. Assume that the minimal Pollrynomial of g has no multiple ptoo4 ana that Traceg is constant. Then the threeconditions (i) Al(g)==O, (ii) g be numerical, (iii) M admit a symmetric g-connection are equivalent one another.. h.

(11) On some (1,1) Tensor Field and Connections 11 References [1]. FRdLicHER, A. and NiJENHuis, A.: Theory of vector valued differential forms,. Part l. Indag. Math., 18 (1956), 338-359. ' [2]. IsHiHARA, S.: On a tensor ￠ satisfying ￠P=±1. T6hoku Math. J., 13 (1961), 443-. [3]. KuRiTA, M.: Tensor fields and their parallelism. Nagoya Math. J. 18 (1961),. [4]. NiJENHuis, A.: Xl,-rforming sets of eigenvectors. Indag. Math., 13 (1951), 200-. 454.. 133-151. 212.. [5]. TAcHiBANA, S.: A remark on linear connections with some properties. Natural Science Report of Ochanomizu Univ. 10 (1959), 1-6..

(12)