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(2) ,. 2 ･' ' '' Y.Mut6 (7) Cpa ;,--a;.=a7 (Apa C, -Av CD. '. , +27(B- Cv "b-Bv CD +a (Apa Bv -Av Bpa ),. (8) ,. ary2==---1, '. for T3. (g) (BA:,;,v == alacb,?p Cv, ' ' (10) ik Cpa;v-Cv;pa==a(AtwBv-AvBpt). The groaPs for Tl. t tt t. t/. '. e.. I. (11) '' ,XApt=O,Xa=:O, ' '. r = n2, transitive, of ct is constani, r =n2---1, intransitive, of ec is not constant,. ' 'tt '. '. fbr T2. (12) XAp==O, XBM=BApt,. r==n2-n+1, transitive,. for T3 , (13) XApt:-abBBpa, XBp==BAp, ,. r :n2---n+1, transitive. :tt. In these equations the curvature affinor Rbpav. is given by. '. '. t/. ' 'l':to T:v (14) R}rwto == TpXv, to-ritto,v +l'ffv l')tu. and Xis the symbol of the Lie derivative, hence (15) Xriv=:8",lk,v+8tuI'ltv,cb---I-'lv8",di+Z-'l;v8at,pa+I'ittu8at,v. for the connection parameters rhv and (16) XThill =8blI7it:::;at'-Tptui:: 8h--''''+T}1::8pact+'" for an afiinor Th:1, the semicolon denoting the covariant derivative. As we cons'lder that the connectfon is without torsion, we have. (17) '' 8i:8N,,,,'. t+. l ,v.
(3) l 1. l i. Onn-dimensionalprojectivelyflatspacesadmittingagroupofaMnemotions 3. l tl. and, a group of affine motiQns being-a group. generated by infinitesimal. l i. transformations with the vectors eX satisfying. I. (18), XI'in=O, '. ' we have [5]. '. '. (19)' ' ･' 8it;v='-R>gevdigi'diL '. ''. tt. tt. The spaces of type Tl stated in this theorem were studied by I.P. Egorov and also by the present author [2, 3, 4]. Hence we study the spaces of type. ' T2andtypeT3inthepresentpaper. - 1 '. tt tt. e1. The spaces of type T2. For the spaces of type T2 we have (6), (7) 'an.d (8). But we find from (2) that the vector B. is not uniquely determ'ined, a transformation of the. form t.v (2o) .(j3'": -:- BA,:+' xA. be'fng adrnitted without chang'mg the second member of (2). Then, as we get. isBpt;v=:Bge;v+NAge;V+X,vApa -v fV tNJ .N.Av = A.(C,+aorM,-7N2Av+27NB,+h,v)+a7A.B.---orBpBv, .we see that we must put. .v (21) C. =C.+X,.+(a7N+7x2) A.+27NB. in order to have (6) for the new Vectors A'V ., g., Ci,.. Then we get. Av Av (22) C. == a7B, ' if we take a solution x of a system of part'ial d'ifferential eq'uations '. (23) N,. == -C.-7N2A.+(aor---2 7X) B., '. '. ' which is integrable. ''. Hereafter the vectors A"V,,, BA". thus obtaEined will be simply denoted by A.,. B.. Then we get.
(4) 4 Y. Muto". (245 (2:;; .:-- Z,"Z.A:.A,".".or.(.A.pt,B-v J.A tt.pa )'. We find that these eq'uations are valid when Apa,Bp.7 are replaced with -Apa, --Bpt. --or. As the second member of (2) is not changed then, we find that the same connection is admitted for the two roots of 7 in (8). The connection satisfying (2) and (24) being uniquely determined by giving a and 7, a real space of type T2 admitt'lng a group of affine mot'ions of order. r =n2-n+1 is nothing but the space found by Egorov [1] ' (2s) {eTrv..=.,;6itor, tr.B-}.,", "'", =... .. v. .. o. '. It seems then that there is nothing to do with this space. But we shall deriv,e, kh,g,c&".";c.tao¥,Ee5D.bi, t.re,a,taeg,,,.`,2`8id ,.,. 't. (26) ' B. == g; .. tt '. Then e-2Yg Ap is also a gradient, and we can put. (27) A. ::.-r e2YgL.. Now, the space being projectively fiat, we can take a coordinate system such that '. (2s) Ii).=--sh v, --s} v.. '. Then we get from (24). (29) Lpa,v== PptLv+PvL sk where. .. '. '. ". (30) pge=-iPn. ---7gy pt+a7e2YgL ... we put ' (31) h= e-7g,. Again, if'. '. '. '. we get. (32) h, pa,v= Ppa h, v+Pv la, pa`.
(5) Onn-dimensionaiprojectivelyflatspacesadmittingagroupofaffinemotions 5 The vectors Ap. B. being linearly independent, f and h are independent functions. As the eq'u.at`ions (29) and (32) show that the system of partial differential equations. '. tt ' ･ ' ･ X;rw=PMXv+PvXpa. '. admits at least two solutions X=f and X= h, we find that we must have. ' ' .(33) Ppa,v--PptPv:O,' --'' hence. ' .. {tttt '. (34) ' p=-log(KIxtu+Ki).. Therefore we get ' '. '. '. '. (3s) (f, -:- :(,K.Z:lx.Z +.K.2,3 /, 2K.ziI;`I#.+.K.1)s.. '. '. If we effect a coordinate transformation. ' (36) xM = (K}SL xtu +Kx)/(K9 xat +Ko), '. with. ' xii=(K:xw+Ki)/(K}9xtu+KO), x2, == (K3xw +K2) /(K2`ce+Ko) naturally, we find. '. (37) . xV=--eYg, x2i==feYg, ' hence from (26) and (27). Ai,= x2i, A2t == --nfti, A3t == ･･- =- A.t =O,. '. By=1/7,2fi', B2t == B3, == ''' == Bnt == O･ , iin is changed, while the form of (28) is not changed by the transformat'ton (36).. This shows that, if we take a suitable coordinate system, we have,. besides (28), ' (3s) (BA; .-'--. fP',.(?2 =B=,-----Xb',.A.3.r'.-.L"Bi :lno7 0'. andmoreover ''. '.
(6) 'Y. Mutod'. 6･. g==(1/7)log(-xi), f=-x2/rd, by virtue of (37).. Putt'ing pa =a(= 3, ･･･, n), v:,=1 in (29), we get pa=O, hence Va ==O from (30). Putting ibe ==v =2, we get NEt2= -a)txi, and., putt'mg IL= 1, v= 2,. we get ftIni= a7x2. Hence we get. (39) "i=a7x2, V2=-a7xi, th3'=･･･ :Vn=O･ As we have a = -1 for the real space, we have the connection of Egorov by tak'mg the root r>t = ---1. Thus.we get the THEoREM 1. A necessa7:y and su;07cienl covadiZion llaal a Pwfectively flal. manijCbld An, n>=3, with symmetric qptne connection and with asymmetric. t,. Ricci lensor admit a grouP of aLO7ne motions of order r = n2---n+1 is that the connection parameters satiSt2y, `. Ii.X, - --sh v,' -6,X th.,. Vi :a7x2, th2= --ayxi, th3=･･･=Vn=O with. a72:--1, a=±1 in some coordinate system. As regarcls the real space we can consider that. a=-1, 7=-1. In the following this space is denoted by Rl. It seems as if we can cons'lder only the reg'ion xi < O because of g=: (1/7). log(--xi). Besldes, (38) shows that the po'mts xi =Oare singularpoints. But these are not singular points of the space and a point with xi <O is taken. into a point with xi=O or xi>O by a transformat'lon of the group under. .. cons'ideration. As'we get. Rii= (n--1) (x2)2,. Ri2= n+1-(n-1) xi x2,. , R2i=:--n-1---(,n--1)xix2, ,. '- R22 == (n--1) (xi)2. i' ' '' '' '. '. from (25), hence Apt =:O for. '. (40) .xi=O, x2=O, the points with (40) are sing'ular po'lnts in the sense that the formula (2).. `.
(7) Onn-dimensionalprojectivelyflatspaees.admittingagroupofathnemotions 7 does not hold good. We shall find in a5 that (40) is invariant by the trans-. formatlon of the group, and it is not unreasonable tG remove such points frOm Rl.. ; e2. The space ef type T3 with a==b-- 1. ' For T3 we have (9) and (10). In the case of a= b == 1 the second member. of(3)'isaninvarlantofthetransformations' '.. t-v A. = cos e A, +sineB,,, AV B.= -sineAttcoseB.,. and we get .N., c. == c. -e, pt. 'Ni Bpa, .v NCM. If e is a solution of by replacing A., Bp, q w'ith Apt, t. (41) e,. == c. --th cos e---B. sin e, we get c"". =A"".. (41) 'is easily found to be completely integrable. Con-. sequently we can regard Apa. Bpt as satisfying. (42) .'. (2:li -:- .;B;i[f4v･ ,. instead of (9), (10).. tt 'we put BM =g, .. Then e-gA. 'As we find ,from (42) that B"s a grad'ient,. j isalsoagradientandwehave ''' '. ,43) . '-. ･(s:.:-gft.pa･. ' system such that If we take a coordinate. 't. tt. (44) rh,=--sh v, --s} "., vixe get frQm (42) (`5) (Z pa;,V, :-=--:- (:N),lt"g',g-'-pty'n,]igl :lif g2g"]"i 7". ."'4 "'. If we put .,. f.
(8) 8'' ･, Y.Mut6' ppa = --"ge -g p, h == e-2g +f2, (45) becomes .f; pm == PM v+PvL ,L, h, pt,v == pp h,v+pv h, ge･ As .L pa and h,. are 1inearly independent, we get. (46) p:--log (K2 xct+Ki), (`7) (', -Z. (,K.2gX.at.'.K.l',C(,i,i,ii X.Z '.Kti,'S. '. '. If we effect a coordinate transformation of the form. '. `. xX' =(K2xct+K'")/(K:lxtu+KO),. ' we get xi' == 1/7; xn' = h/lf: Hence, if the new coordinates thus obtained are simply denoted by xX, we get '. ' tt (48) xi=1zax2==hff;f==1/xi,g=logxi-･glog(xix2--1) and. 1 fAi= -xi(2eix2--1)v2, A2 == ''' " An =O, <･49). ''. ' IBi:2xiftxX'/,-2,l=1)...'..Bb.M-'.."o.2'(X':l:i)'. (44)remai"svalidandweobtainfrom(45) ' Xl (so) gNPni= 2(.i .2-1), X2 92 =: 2(xi x2li-1),. K v3==･･･,:v.=o･ In the real space the coordinates must satisfy. (51) xi x2 -1> O. tt Thus we get the. '. THEoREM 2. A necessazy andsztzfiicient condiZion that a Pro7'ectively 77at. .. -.
(9) '. Onn-dimensionalprojectivelyfiatspacesadmittingagroupofaffinemotions 9. manlpld An, nll3, with symmetric a;07ne conneetion avad with symmelric Ricci tensor wlaich is non-positive admit a comPlete gTozip of aOine motions of order r= n2--n+1 is thal the connection Pdramelers satz:svi)2 . '. '. I'itv == ---6h V, v-S} ", p,. .t . V=-l-log(xix2-1) in some coordinate system. in lhe real space the coordinates must satikij3, rd x2-1> O.. Such space will be denoted by R2.. e3. The space oftybe T3 with a=b== -1. In the case of a = b = ---1 we find from (9), (10) that the system of partial differential eq'uations. ' : ' ' or +B`, cosa), (s2) fcr, pa = Cfr +-}(eP'-e-P)(--Apa sin (e,. == --i" -} (eP+ehP)(Age cos a+Bi,'sin a). is completely integrable. Then, if we put '. ･ A""pt ---- --Apa s'm a+Bpa cos ct, .-.i Bpt = Apt cos a+Bpt sin ct, cA", ,. -c. +a, ge,. tv eNJ eNJ As-vCp we find that we can replace A., B,, Cge by A., Bp, Cpt ia (3), (9), (10). .'v we get = (1/2)(eP-e-P)A., ANpt;v = -- S- (ep--e-p)BN. A"",., B"".,, .. -} (eB---e-p) A"". A"",.. B,v==--}(ep+e"P)'B"'.. . We can think that we had such vectors A'"., B'"pt as the vectors Ap. B.. from the beginning. Then we get. ' , (Age;v=---}(eP-g-P)BptAv. (s3).. .. izpt;';.:-tl･Liiel',,e;i'-e;.4;:...
(10) 10･ ･Y. Mut6 ''. As Bpa is a gradient, we can put B.= g;pand get. (54) eP =: -tan (g12). Then, as Apa/ sing becomes a gradient, we can put .(ss). , ., (L.,xZ.-[bOSe.CgApt･･. ' TakiAg a coordinate system such that. '. , T),- -6,X,9, -S} Y7),. we get ･ ;, '. e. '. (66) I`,,, :, ", 'L (:,llll".:,(-'OJ,gk ee,`. k",.`,'.",' ,-ig.ot gt' g' "`" ,. '. '. '. Now, let us try to find the functions X(L g) ofLgsuch that. (57) Xp,v=: T,k X] v+ rrv -X; p, '. ' wherezahavetheform t'. '. '. (58) T.=-iPn. 't ----N.IC; .-- pa ,gt .. '. Then we find that X==Fand x=G where., '' ,. , F=tanL G=secfcotg , are two independent solutions of (57) if we put , .. N= ---tanL , pa=:cotg:. Hencerrptmustsatisfy ･ .. ,. ･ Tpt=:7r,ge,T=--log(KSxtu+KO),. and we get ' ･. '. F=:(IC:xdi+Ki)/(K:lxtu+KO),. G = (K3 x`" +K2) / (K19 xtu +KO).. ' , Effectingacoordinatetransformation ･. '. '. ..
(11) On n-dimensional projectiveJy fiat spacesadmitting a group of aMne motions 11. '. we get. ' ' xx' == (Kh xx +KK)/(K9xat +Ko), ttt tt. xi'==tanL t2'=secfcotg; ' Hence we can think that we have. (59) Xt==tanL x2==secfcotg and consequently. , (-Li=1+(lxi)2,L2=='''=Ln=O, (6o) .. <l,gt,=E(1+(.i),),,,(PCIitX+2.(x,),i:(x,),), , lg2= Fii,2.+,S,X+)2(t'X)2,,,&3== ･･･ --.gn=o. ,. '. tt. We get also. 'tf] N-----xi, pt--± (1+(Afi)Z)1!2 , hence. (61) ($l 'L i'--(Xi)X."2.'.(opt)2' th2 =i+(rd)`C'2+(th2. from(58)or(56).ASforthevectorsAp,Bge,wehnd -. ' '1 Al (1+(xi)2)Y2 = (1+(xi)2+(x2)2)ii2 ' A2 == ･･･ = A,, == O,. (62) Bi=. xlv2 .. (1+(Jtii)2)1i2(1+(xi)2t(x2)2) ,. (1+(xl)2)1i2. B2 = -1+(rd)2+(x2)z, B3 == ''' = Bn = O.. Thus we get the THEoREM 3. A vaecessa2zy and spa17icient covadafion that a Pro]'ectively 77at. mandeld An, n 2il 3, with symMeVric ofne connection and wafh symmetric Ricci. tensor which is non-negative admit a comolete gTomp of atO7ne motions of order r == n2--n+1 is that the connection Parameters satztsi12y.
(12) 12 Y. Mut6. ,,,, {'$.-.,' ;,l;6iggage,#:,',*?,:,==,,Y,'"'. in some coordinate system. S'utch space will be denoted by R3.. 'If complex n:umbers are adm'itted, this theorem can easily be derived from Theorem 2 by a transformat'lon of the form. ui=,xi+ix2, u2 :-xi+ix2.. o. ' a4. ThespaceoftypeT3witha=-1,b=1. This space is obtained by replacing the first equation of (49) by. .. ' Ai='-xi(1-lxix2)ii2'' ' (50) remains unchanged, but in the real space the･ coord'inates must satisfy. xi x2--1 < O.. Thus we get the THEoREM 4. A necessazy ana suLO7cient condition that a Proiectively 77at. man2ifbld An, nll3, with symmetric afivae connection and with symmetric Ricci lensor which is indofnite admit a complete grouP of afi7ne motions of order r= n2-n+1 is that the connection Parameters satiSij2y I',X, == -6iiPn,-S", ,Pn., NPn. = ain,.,. V=: -} log (1---rd x2) '. in some coordinate system. Such space will be denoted by R4.. a5. The group of affine motions. For the space Rl we have. ' zav==--Sh"v-S}"., "i=x2, th2=-xi, "3==･･･=:"n=O '. ..
(13) s On n-dimensional projectively flat spaces admitting a group of aMne motions 13. 8X=xi 6i , x2 S},. .. xi 6iX --x2 6i ' S}･ (i=3, ･･･, n), 8,X･xw ,･ , (a=1,･･･,n) .. as Egorov has shown [1]. The finite equat'ions of the group are. ' (64) . (i';21.t-=Z/1..xXi+aa.',:i}2i ' ,,=,,...,.,,. '. '. where. ' al a2== 1. a? a22. 11 .. For the space R2 the group of affine mot'ions is obtained from. (65) XAsk=-BBp,XBge==B4, ' ' where B,v=XCv. As we have Cpa=Age in a2, we get B,v== --BBv. Hence '. ' ' 'B==ii(xix2-1)y2 '. , tt/t t tt. by virtue of (49). ' Because of (42) (65) becomes '. ･ (Apt8ct );y-Aat8di BsL +Btu8di Apa +B Bpa == O,. ･ , (Bdi8di);p--BAge=O.. Ifweput ･. .t. , X= A.edi,- Y== B.8tu, we get a system of equations. ' 2Y; "XBtt YA. +B B. =: O,. '1,. . . Y;.-BA.=o, which admits the solution th,. v.
(14) 14 '. 'Y.Mut6 X=i (rd t2-1)'ii2(-- k2 -k' + kS' 'xi +'k22 x2),. y=k2Xl+･ letp for we have (49). Th6n 8i, 82 are obtained at once as '. ' (66) ''. ' Si=-k2i(Ki)2+k,(1.Xi2X2)+kt.,, te2.. k,(1--- li2X',2)-- g2 (x2)2-ktx2, . q. and we get. (67) "w8tu == "i ei+V2g2=: -- rS-.(lei xi+k2 x2)･ , .. tt. Now let us consider the equations (18). We get. '. ieoCnasU ,SReOfge{]it". tb. .t. '. ' = 6it (",,Pn. {l･ di), ' 8X, y.,v ' v+6,X' (llf'. 8ca),.. '. :--Shik,'-SvXSP",pa' Then, substituting (67) into these equa.. ' (68) ･ 8a=--lll-(leixi+le2x2)xa+Klf'xpa+Ka (a=3,･･･jn). ' (66) and (68) g'Lve the vectors of infiniteSimal transformation.. The( 8g)fitegoenq9uraattieoSnsa. ogfrOaUguObfgrtoruapnSEOernlll7SitieOdnSb;n a two dimensionai space.. 8i=2-rdx2; 82=-(x2)2 is. '. '' is. xti= xi+t2 v2+2t tJrfi+1 ' x'2 = "ix"'f'-3'-Jr". Hence the group generated by (66) contains a subgroup of project'Lve trans-. formations. - '. We can easily show 'that xix2-1=O is an invariant of the group.. The equations (66), (68) are valid for R4, too. ･' '.
(15) "M. '. On n-dimensional projective!y flat spaces admitting a group of aMne motions 15. '. Similarly, we get for R3 gi == -illl'- d+'(rd)2)+}' xi x2+k' xi2,. (6g) '. ' e2=-II;Lxix2+-IIItLa+(x2)2)--ktxi,. ' ' 8a=S-(klxl+k2x2)xa+Kftxp+Kiz. . .t. ' '. '. , . , e6. ConclusiQn.. t tt t. From Theorem A and Theorems 1, 2, 3, 4 we get the. ･.. THEoREM 5. Consider amanijZ)ld An, va;ll5, oraProjeclively.17atmanijbld An, da)3, with`symmet7ic aZ77ne connection. A necessai:y and suLO7cient condilion that the manlpld admit a complete gpeozip of aLfi7ne moiiovas of oider. r=n2-n+1 is thai it be one of the spaces Rl, R2, R3, R4. , As the-purpose of the present. paper is to study only local properties, the. global properties ･of the spaces are left open. But we find that ln any of them "pa are zero except "i, V2 which depend ohly upon xi, x2. Therefore, if a suitable structure is assumed in the large, we cap construct naturally a fibre bundle with (n-2)-dimensional euclidean spaces as fibres and a two dimensional manifold M2 aQ the base space. A p,oint of Mle is 'denQted by (xi, x2) and a point in a fibre over (xi, x2) is denoted by (xi, x2, x3, ...,xez). As 8i, 82 depend only upon Jti, x2, a transformation of the group takes a fibre. over (xi, x2) into a fibre over (xii, xt2). Hence the gro-up is imprimitive with. a system of imprimitivity composed of the fibres. The group of the bundle. is the group of affine transformations. M2 of Rlcan be regarded as a euclidean space from which a po'int is removed. M2 of R3 can be regarded as a projective space IDle. ca of R2 can be regarded as the inside of a conl'c. inaPL], while that of R4 as the outside. ., ･, L '. '. E;] E.g,gr,O.V.;. [31 M. (Ult. gOpt. '. i,:,,Pi.l'?,O,k,i.?d,Yg. '. References. , ,. ?,k,a,d,･), ty,a,u-k,,(,N.･S･)･ sssR, s7 (igs2), gg3-6g6. ,. g4g; '1-iC.ienCe Reports of the Yokohama National University, sec. I, No. 3. [4] Mut6,Y.: Ibid.,No.4(1955),1-1.8... . ･ "GrOupS of transformations in generalized spaces." Akademeia p;ess. [51 Ya(nlOg,4gK):: t. '. ' '. tt tt'/t.t t .t: '. t. ... / ... ..t... tt.' t,.,./.t. ' '. '. '1 ,- .. , - ., -.,-- .t--,:.
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