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(2) 2 Y. Mut6 -- 3n+8).. Then, a necessary and sufficientcondition that an An, nl, l.7, with. symmetric afline connection and with n,,on-yanishing projective curvature tensor admit a group of aMne motions of order r::-7 n2---2n+5 is obtained. It. is that the curvature tensor have the form R} .vtu = A" PIL (Pv Qto --Rh) Qv). where AX, Pi, and ttt Qv are covariant constants. This is equivalent to the condition thqt the cQn.nection parameters rh, satisfy. t/. ' r52==K3, other 4, =O. when the coordinate system is suitably chosen. Such connection was already. given by G. Vranceanu [3, 4] as an example, but now it is shown that no other connection is possible.. xl. l. gl. Introduction. '[. ,i. ln a previous paper F2] it wa$ shown that, if an n-dimensional manifold. S. Aq, n >= 7, with symmetric affine connection admits a group of afiine motions. of order r>n2--2n, then the curvature tensor has the form '' ' (1) Rts rwtu == AX Bptvtu +6) (Pvco--Pcav)+SC Rztuny6h Ppv or the form. (2) R}ptvto=(AXP7L+CXQpt)(PvQtu"PulQv) +6x. (p,.-p.,)+6C a.-Sh P.v･ It was also shown that, if nlll8 or if the ineqqality r>n2-2n is replaced with the inequality r>n2-3n+8, then (1) is the only one possible form. In the present paper an An with the curvature tensor of the form (1) is studied. We assume AXBptv. =i=O which means that the An is not projectively eUCiiAdseainn' [2] (i) is subsiituted into xRix..,. .. o where x is the gYmboi of. Mr. the Lie deriv, ative. Then we get. (3) XAX=atAX, ' (4) XBgevtu =-- aBgevtu, (s) xa, =o. which we can treat as a system of linear homogeneous equations in the unknowns 8K, 8: at qn arbitrqry fixed point of 4n. If theorder ofthe group of aMne motion$ adl,n.itted, is more thap u2-2n, then the number of linearly ind.e.pendent equations in (3), <4) and (5) together must be less than 3n. '. 'From (3) we get '. (6) 4atex. -= ct4x (mpd eK),. s. i. 'S'i.
(3) A kind of aMnely connected manif61ds admitting a group of athne motions, II 3. and, as we asstime AX ISF O, there are n---1 linearlY independent equati6ns ifi. (6).Then,asaisobtainedasalinearformofgKandAdie2,Wegbt ' ' Act eS). (7) B.,. 8:+B.. e9+B,,. g: !iiO (mb' d 8K, Of course Bpv. satisfies. ' (8)... .'Bpa(vto)=O,Brikvul]==O･ .,..･, As we assume r> n2--2n, (7) contains at most 2n 'linearly independent equations. Making use of this fact we shall determine the form of B.v.;. '. '. g2. A property of the tensor Bpv.. ;1. At first we remark that, if Bctp.udiuB=O fot every vector uX, 'then B(rw).= O and We get- Brw. =O by vittue of (8). Suppose Bpv. i( O and the vectors Bptpto udi uP, Bctp. vdi vP are linearly depen-. F,. dent for every choice of the vectors uX, vA. We can assume Becp.utu aP ¥O, and if we put Beep.udi uP =C., we get Btup.vtu vB ==a(v) C., hence Blpv).= BpvC. where Bpav = Bvpa. Then, as we get B(pv C.) == O from (8), we easiiy find successively Bpav =B(paCv), B(geCv C.)=O, Bpa==O, B(pav). =O, hence B".v. =±O. contrary to the assumption. '. Thirdly suppose Bpv. )s( O and the three vectors Btup.atuuP, Bctp.zidi,vP, Bdiptoudi vP are linearly dependent for every choice ofthe vectots zih, vX. We can assume Batp.ueo uP and BtuB.vct vS to be linearly independent ahd get Bwp. uat vB == a (u', v) Btus. udi uP+b (u, v) BdiB. vtu vB.. Transvecting with vbl we get a(u,v) ==Ohence . ' (9) ., . Bdip.udivB=b(u,v)Bctp.vctvP, ... or else Batsyuat uBvY =O. But, if the latter is true, we get B(pav).=O hence Bpav. == O which contradicts the assumption. So we have (9). Differentiating (9) partially with respect to upa we get e. (10) B"p. vP =bge (v) BwB.vdi vP, L. and differentiating (10) partially with resipect to v' and fixing the vector vX We find that Brw. has the for'm'. Bptvul = Apv Bto +Cpa Dvol･. Subsbituting this into (10) we get ' Apv v' B. -tLCpt Dv.vV == bs, (v) BctB. vdiivP. and we see that either Agevv' and Cge or B. and Dvto v" are linearly dependent.. Then we get Brw.= qD'v. or Bs,v. == AtptvB.. But from the latter equation we get Bpav.=:O by virtue of Bp(v.) =O' ; Thus we find that Bnv. niust have.
(4) 4-･- Y. Mut6 the fo,rm &ta=:C,,Dv., and substituting this into Btw.] == O we get an equq-. tion of･the form. '. ' ' (11) - Brwtu=a(PvQtu--PtoQv)･, tt ' The result obtained can be summarized as. '. '. t tt utuuP, LEMMA 1. if a tensor Bptv. satiistlies (8) and Bpava =l= ' O, lhe veetors Bdip.. t trt. Btup.vtuvB arelinearly independent for general'directions of ihe veciorsenX, 'vX. The threle vectors Bdip. ucathB, BdiB. vtu v'P, Btup. utuvP dre linearly dopendentfor eve71y. choice of lhe vectors uX, vX of ana only of Bi,v. ==a(Pv Q.-Pto Qv)･･. t. g3. The rank of &v.. , Let us a'ssume that the tensor Bpav. can not be written in the form (11).. ,-. ". Then the vectors u}, or2X are taken such that the three･vectors. tt. (12) 'Bdip.u,tuuz3=P.,BdiB.ura,B----Q.,Bop.u2ctuS=Rto i. are l'mearly 'independent. ' '. tt. ii. s. Suppose that we can take two other vectors vX, 'wX such that thett vectors. t t. tt. /t t ttt. (13), , P.,Q.,R.,Bdip.vtuzbP'. are iinearly independent. We write v" ==ui wX=u4X. uiX and u} arelinearly independent but we do not know whether the vectors u}, ui u3X, ui are linearly. independent or not. ･ , From (7) we get . B.py u: uS g' Y. -= O (mod 8K, Act g2, uou. gZ, u:gX),. hence (lo B.p, u: u2ey. --o (mod gK, Adi e;, u,ct eE, ..., u,tus a). tt. ' where a, b :1, 2, 3, 4 andc = max (a, b). If ' we take a coordinate system such. thatatthe'point'underconsiderationwecanwrite ･' '' AX == 6}, u) = u} B}+...+uZS)+u.a"iS).,,2). then we get the following equations from (14). ,. e. B.p,za,dint,Bgy.io(modgK,ef,e>) (x=3,...,n),.. B.pyufzt,BgXsO (mod gK, gK,, 8K,, 85) (x=4, ..., n), ' B.py agu,Bgxr=o (mod sK, g:, gE, ss) (x == 4, ..., 'n),. B.6yagu,Pexso(modgK,gK,,...,gg)(x=6,...,n). ee. As the vectors Bdip. zeitu ui3, Btupcoaf u2B, Btup. za2tu.42P, Bou6. u3tu u4P are asSumed to be. Iinearly independent, the'number of linearly independent equations contained ,,,,lt-.,,,ij,h,.,,l., abOVe equatiOns is 4n-13･ As nlll7 this number exceeds 2n contrary. . 2) Some of ub......, uZ"i paay possib!y be null. '. },･. ･i.
(5) '. A kind of affinely connected manifolds admitting a group of affine motions, !I ･5. totheassumptionr>n2-2n. ･.･., ,. We therefore see that the four' vectors (13) are linearly dependent for everychoiceofthevectorsvA,whandget,'.. , <15)･.'I･･,T. ,,.Bptvto;Apt:Ptu+BptvQtu+CptvRto,. .,･''.--'. where P.,Q.,R. are linearly independent. This shows that the tu-rank of. Bavblisnomorethan-3.,.･-.,. .･ .:t, '. Substituting (15) into Bpttv.)=O we immediately find that Bpv,. has the. form. t.. t. Bpavto = Apt (Qv Rtu -- etu Rv)+ Bp(Ry Pto - Rto Pv )+Cpt (Pv Qto - Rt) Qv),,･. and substituting this into Btw.] -- O we.find i. (16> Bpav. = (aia +bi Qpa +ci Rpa) (Qv R,)- Q. Rv). . +(a2 Pge +b2 Qpa +･C2 Rsk)(Rv Pbl-Rto Pv) , +(a3a+b3 Q. +c3 R.) (P, Q.-Pb Q,). '. ' Thus we get the･･'･ ･'. LEMMA 2. thr nllz7'the number of linearly indopendenleauations in (7) is at mosl 2n only when the tensor Brv. laas the form (11) or (16).. '. t ttt '. '. g4. Canonical forms of Bpv... In the preceding paragraphs we find that if we assume T>n2--2n we get (11) or (16). Wd now study the case of (16).. -' We can write (16) in the form '' '' '' 'i ''. ' (17) '. B.v.==aideIlfPIPts3) '' t.t.t '". .t wherePl=A,p3r'Qpt,PE=Rpaand ,' ,''. .'. ''. (18) ai(dk)FO, a[ii'k] :O: i. As we assume Btwv.ieO we have aide ¥O. If we take another set of vectors }. . Pitsuch that . .. (19)'･' ' Plf=aS･Pl,.N.,-' lct/il}eO,･. thenweget '･ , '': '. rv 'V "v' ' ･- B..=ttiylePZPjPts -･. where ' ･･'. 'ii.7･k == almn cti･ ctw aZ･. We observe that the system of equations '. tttt. 3) In g4 the indices run as i, 1', fe, l, m, n=1, 2, 3 and summation convention is used.. '. .t.
(6) 6 Y. Mut6 '. faimn a5 ctri ag = O,. (20) . iatninaiatttct!=1, Kaimn di ct3nt = O admits solutions aS･ such that ctS }( O. Such solutioris satisfylal･l)E O, and with suCh a/'･ we get. a 123=:O, j131==O, jn2=1, tt212==O･ Therefbre we can asisinme that aimn satisfy. (2i, IZ,iig .:- 21, Zi,Z', Z. 2i, Zi,12, :. 6I ta323 = Cl, a331 == ic2, a312 == ---b2. from the beginning. First let us asstime c2 )FO. Then We get it23i=O, at3i =O, an2==1, j23i=O,. "V N. a212 = O, a312 =O if we put. i. h. g･. al==1, ai==O, a?:O, ct>==--(b2)2/c2,aZ==1, a,3ew--b2/c2,. ctg=o, ctg=o, ag==1. This implies that Bpv. can be reduced to the form B@vto = Plk (Pv Qto -Pho Qv) +bi Qp (Qv Rul -- Qe) Rv ) + R. [c!(Qab R. - (?. Rv)+c2(Rv Pa, -- R. Pv )]･'. This form is preserved when Pge, Qpa, Rpa are replaced with paRL,p-2Qp, XRM and we find that the possible forms of Bgv. are. (22) Brwtu = Pik (Pv Qtu pRh, Qv)+Qp･(Qv Rto'Qto Rv) +Rpt (Rv Pa) -d Rtu Pv )+cRp (Qv Rtu -- Qtu Rv ),. (23) Brwtu =: Rh (Pv Qto -Pb Qv)+Rp (Rv Plo --" Rol Pv) ± Rp (Qv Rtu pt Qtu Rv ),. e. (24) Bpvto == Rh (Pv Qto "'Pto Qv)+ Rpt(Rv PU, -Rto Pv )･ If we have c2 :O in (21) we can not proceed as above. But, if we get aimn ct5 ct3M ctr )FO for some solution of (20), we obtain. a123:O, bi3i=O, bn2==1, a223=bl, i231=S2, tt212=O, tt323=cNl, b331=i2)FO, tt312=-S'2,. Hence we can proceed as before and get (22), (23), (24). ' Next we assume that. (25)' aimn a5 cyge (x' ,n =O. N..
(7) A kind of aMnely connected manifolds admitting a group of aMne motions, II 7. is a consequence of (20). But (20) is solved by taking ct{, ai such that i. (26) almnati(xY}eO, atqltctict3M=O, ag}FO, for, if ad == Bd is a solution of aimn ai aiM ctn = 1, then ctS= Bi+ptai satisfies. (20) when we put X==-aTmnBiaiMB4. We thus find that th.e atmp,p wg are studying has the prQperty that (25) is a con,seguence of (26). Then we rr}ake. useof(21)wherec2==Oandget , (27) ctl=bictl+ciai, ag=b.2ai, ct33--datl--b2ct3i from (26) and. (2s) (- ct5, ct3t2b2 ct3. ag) al +(a5 ctl -- b2 ae ctg - bi a2, ag - ci ag ct,3) al. +(-b2 a" ct;+bi ag ag+ci ag ag) ai. =o. from (25). Hence (28) must be satisfied identically when <27) is ' substituted. into it. We then easily find ･ ' 2(b2)2=cl, bl==O. If b2 =O we get Bptvto == Plh (Pv Qto '-- PLo ev). ' Rpa' with ' which we have already obtained as (11). If b2 )FO we can replace ' b2 Rpa and get. '. (29) Brwto=P7b(PvQtu-Pa)Qv)+Q"(Rvllh)-RtoPv) ' --Rpt(Pveto-PthQv)+2Rp(QvRto--QtoRv)･ Thus all possible 'forms of Bptv. are exhausted and we get the LEMMA 3. if a tensor B"v. satzSflyin' g (8) has ranle 3, then it has one of the .following fopar forms. (i) 4 (Pv Qto '- RD Qv)+Qge (Qv Rth - (9ul Rv) +R@ (Rv Plh) - Rbl Pv )+C RIk (Qv Rto '-' Qo Rv ), b. (ii) a (Pv Qto -- Pho Qv )+ Rge (Rv Ro '- Rtu I?v). (30) ±Rge (Qv Rto-(?to Rv ),. (iii) a(PvQ.--PbQ.)+R.(R.1%-rR.Pv),. (iV) PZL(PvQto--PU)Qv)+Qp(RvPL)-"-Re)Pv) --Rlk (Pv Qtu -ny PL) Qv)+2Rl- (Qv Ral m Qto Rv )･. g5. A neces.sary cQ, nditiop.,.. A.$supae th4t the rank of Bny. is 3. Then Bgev. I,.q$ one of the cano4ical forms (i), <ii), (iii), (iv) pf Lemma 3,. (31) Bny,.= aijle ?ti Pg' Pk...
(8) 8' " Y.Mut6 We can' take p} satisfying. . ' P2p9･=:6S- ･ SUChTtrhaantsvtehcetifnOgUr(7V)eCwt ?.trhS. pP.iiXlo//3le PwtK6 geXtare lineariy independent.. (32)'- ai]'kopS･+aiileop'j+ai]･iop2Ei'tO(mod8K,Atu82)'''''･. if we 'put ･. (33) - ･'--' '. op/'･=pastupp,P･.'-' '. We also obtain. '. (34) aiyi P: 8g EsO <mod 8K, Atu e:, nla). bytransvecting(7)withP:･P,"･.,, ,., .,.'. ,a. As AX, PiX, P2X, P3X are linearly independent we can treat the system of equations (6), (32), (34) separately. Hence, if the number of linearly indepe.n-,. dent equations in each system is got and summed up, we find the number Ql. /1,. .. linearly independent equations in all these systems together. This is easily understood if we take a coordinate system such that at a fixed point (x}).o. wehave AX==S},P,X･=S}･.i･ , Again, if such a coordinate system is taken, we get from (34). aidiPk8.ou-=O(mod8K,ef,...,8:) (x=s,...,n), ･.,,, . and, as the rank of aijle is 3, we have 3(n --4) linearly independent equations,.. Hence, if (32) contains at least six linearly independent equations, then we have at least 4n-7 linearly independent equations in all, which contradicts. the assumption r>n2-2n for n}ll7. Thus we find that we have less than in.e,a:gy,t':,d,eP,ehn.dg,ng.e,q,.".a8' i.O,"g.11.n.a3g,)',.,,i., {i, j, le} = {i, i,2}, {2, 2, iS)" six i. {1,1,3}, {1,2,3}, {2,1,3}, {3,113}, {3,2,3}, {2,2,3}, in (32). As ,. a123==O, a131=O, al12=1, a212 =O e. the matrix of the coeMcients is obtained as follows.. ･opl di op?･.,op>,'opz opg, nl op3 opg. 2. o a312 ,o .1 o o. {1, 1, 2} {1, 1, 3}. o o. ' a213. {2, 1, 3}. a213. a223. {2, 2, 1}. oo oo. O a223, --1 ,O a321+a231 O. a313 o o o. o o･ {1,2,3} O a323 a223 o o a213 a313. ,O 1O O -1 O O oo. a213. {3,-1,3} a313 a323 , O O O･ O' Oa213+d3r22a313 {3, 2, 3} O O O a313 a323 'O a321･'a223 '2a323 ' {2, 2, 3} O ' a213 '"2a223 o o a223 a323 oo. k.
(9) A kind of affiriely connected manifolds admitting a group of affine motions, II 9. '. '. If Brw. has the form (i), we get. -. a223=1, a231 ==O, a312==O, a323==C, a331=1･ Hence we find that the rank R of the matrix written above satisfies R}}) 6 by examining for example the determinant obtained from rows 1, 2, 3, 4, 5, 6 and columns 4, 5, 6, 7, 8, 9. This contradicts the assumption r>va2--2n. If B`,v. has the form (ii), we get. ' an2=1,a323=±1,a33i==1,otherai7･le=O,. hence R;}l 6. ' If Bpv. has the form (iii), we get. an2= 1, a33i=1, other aiJ･k =O, .. hence Ril:6 again. If Bpv. has the form (iv) we get. : aii2=1, a23i==1, a3i2=---1, a323=2, i. and in this case we have R= 5. Then we have 4n---8 or more linearly in.dependent equations in all and as a consequence get r;5{n2-3n+8. It is possible that i>n2-2n may be satisfied for n == 7.. We thus obtain the .. ' tttt t THEoREM 1. if an n-dimensional manofbla An, nllll7, zvith symmetric a;0ivae connection whose carvature lensor RX.ev. has the form (1) admits a. gromp of aO7ne motiovas of onde7 r>n2-2n, then the R}ptv. has the form ' (35) . R}sLvtu =AXPIk(PvQto-'PtuQv)+Sit(Pvco-'Rtov)+6,"Pptco-SX.Ppv or the fbrm (36) RX. pvbl = AA[Plk (Pv Qtu -Rt) Qv )+Qpt (Rv Pto -Rto Pv) +2Rp (Qv Rto -Qto Rv )- Rge (Pv Qtu -PU) Qv )]. +6it (Pv.-Ptov) +6) a.-Sh Pptv. The latter z's Possio'le only for r;iS n2-3n+8, hence for n =7.. For sufEiciently large n we have only (35). Moreover it is not dificult b. to understand that the tensor 4v .is expressible in terms of the vectors a, Q... ' g6. A necessary and sufficient condition for r = n2-2n+5. Let us look for a necessary and sufficient condition for an An to..admit a group of afline motions of order r.== n2-r2n+5.. From Theorem 4 of [2] and Theorem 1 of the present paper we get (35) as a necessary condition. Substituting this into XR} ptvto == O we find as in [2]. (37) XPnv =O, (38) XAA=aAX,. (39) X{4 (Pv Qul '-- PL) Qv)}='aPZL (Pv Qco '--Pto Qv )･.
(10) 10 Y. MutoAs we get from (39). (40) Xa =BPI,, we can put. (41) X(?. =7Ppt-(a +2B) Qp. Since we assume that the space is not projectively euclidean, Pik, Qpa are ,,linearly independent and we can take a coordinate system such that AK, Pk,, Qpt satisfy. (AX:S},, .. (42) , iR,= S!.A+S2,.,, ' 'Q, ==Sk Q,+63. at (xX)o.. ". We get from (38). ,,,, (S,1 t' O-- .(M,O.d g,K)lr.,, .. ) 'L. (44) Pi 8k+8k E!i BPi 6i.+B62. (mod 8K), and from (41). (45) Q! 8k+83,, -= yPi Sk+7SZ---(ct+2ie) Qi 6k -(at+2x9) 6E (mod 8K). Putting pa =2 in (44) we get. B Ei!.; 8g+P, e5 (mod 8K). and making use of this equation･we get. (i) 8{fO(modeK), ' (ii) Pi (81-8Z--P,e5) ii! O (mod 8K), (46). (iii)eZ+A8h!i-;O(mod8K)" ' '(a:3,...,n),. , (iv) 2(?i (83+P, 85) --- (gij+Qi gS) R, (mod 8K),. (v) eg+Qi85+2(8Z+P,8>)-81giO (mod eK), '(vi) 82+(?i8!FnyO (mod 8K) (x == 4, ..,, n). The number of linearly independent equations contained in equations (i),. (iii), (v), (vi) is 3n-5. Hence the remaining equations (ii) and (iv) must bQei.gZIiSfied. bY virtue of the equati,ons just menpioned.. We thus optain pi= o,. Therefore we get. (47) PF,=62,,, 'Q.= 83,, ' and. e. L.
(11) A kind of' affinely conn3:t･avd manifolds almitting a group of affine motions, II 11. 8i EE!O (mod 8K),. g2. -=O (mod gK) (a == 3,...., n),. (48). eiEiO(mod 8K) (x=4, ..., n),.. gl-28Z-2egiisO(mod 8K) ,. at (xX)o. (48) must be a complete set of equations satisfied by e.n at (xX)o in the sense that no other'equation is possible;. NowcbnsidertheequationsNPI,v=Owhichwereduceto. ･ Pdiv8:+Ppa8ty i-i=O (mod eK). ' ' We get Pouv82tu+P2tu89=-= O (rnod 8K) and, if'v )F2, 8s is contained only in the. term Pkv8s. But we find that 8," appears in (48) only in the form el-2g3-833.. Hence we get Ptuv==O forv)F2. Similarly we get atu ==O,for pabF2. Then {.. we obtain P2tue,di+Pbe2eab2 -- 2P22e3-=-.O (mod gK), from which we get P22 == C}, F. (4g) a,=o. Differentiating (40) covariantly we get. Pdi;v8.tn+P,-;atg.tu -=--'BPI,;v+i8vPIL (mod 8F).. Putting Le = 3 and making use of B -i･822 (mod gK)'we get Ibe,ve,tu+Pb,'. eY--Pb,,83 --O (mod eK),. which become by virtue of (48) (Pi,,+ S3, P,,i) e5+(.P.i.+ S3. Pb,.) 8g+6e Rs,igl+6,2 PIi,3 g:. +S.2 P3;xgi+6'Y, Pb;i8} +6; Pb;x ee+(Pb;v+S,3 P3;3)(81-283) +S} 1)9,i gl+6; Pb,2gZ-Pb,,8Z -. O (mod gK).. The indices x, y run 4, ..., n and summation convention i's used for them.. Comparingthese with (48) we find P2;x=O, Pb;3==O, Pg;i= O, IZx;v= O, R!,v=O successively. Moreover, if we put v :2 we get Pb;.2 (81-2gZ)!O (mod 8K), hence Pb,2== O. We thus find that 4;v==O for pa#2. As we have Pl,= SZ we can put. (so) =4pv. From a,v (38) we get J . --Atu,,g}+Ax,.g3+Ax,.el m===.=- at,, AA (mod sK).. ThenputtingN==i(==2,...,n)weebtain '' ' --Adi;v8zi +Ai, tu.8,di+Ai,v81=-O (mod gK). Putting ny == 2 and.making use of the last equation of (48) we get. -A2,28S-A3,2g6----AX,2 gl+Ai,.8,ot+2Ai,2eZ+Ai,2eg -=O (rpod 8K).. Comparing this with (48) we find Ai,2==O and then Ai;ol = O. TherefQre we can put.
(12) 12 ' ' '. Y.Mut6 (51) A";v= AX av.. As we can take the point (xX)o arbitrarily, the equations (49), (50), (51). hold good throughout the space. We have moreover. (52) Pl, Apa =Q. Apa=O.. Now we can make use of the Ricci identity AK;v;to---AA;to;v = R} pvtu Apa･. The second member vanishes because of (52), and, if (51) is substituted into. v. the first member, we get av;. --a.;v == O. Similarly we get from a;v;.--Ppa;.;v =-Riggev.Ptu =O. p,;.-p.;, :o. Hence av, pv are gradient vectors. As the curvature Mgev. determines the. 't. vectors AK, PZ, only to within scalar factors, we can think that they satisfy. (53) A";v == O, Pkh;v == O,. i tt. ). that is, they are covariant constants.. Differentiating (41) covariantly we get. (54) Qct:v8:+Qpa;ate.at-7,vPZk+Q";v(283--81) +(a,,+2B, ,) Q. E!io (mod gK).. Then we have (?ct;y84ct+Q4;at 8,tu+Q4;v(2eZ-81) !E'!:O (mod 8K).. If v ¥4, e4" i's contained only in the first term Qct;vg2. Hence we find Qge;v. =O for pa )F2,3 and v¥4. Similarly we get Q.,v==O for pa ¥2,3 and v¥5. Therefore we can put. (55) Qpt;v ==arv +Qpt qv･ We also ..obtain from (54). . Qdi,v8,e+Q3,. ee+Q3,,(2e:-81) -O (mod gK) for we have '. ' ct,v:O, B,v=O. o. because of (38), (40) and (53)･ Then we get . )/. q.egifo (mod sK) by virtue of (55) and (48). Comparing this with (48) we find qv =O, hence. (56) Qp;v= P@ rv･. Substituting (56>. into Qp;v;.---ep;.;v =-Rptv.Qtu whose second member. vanishes because of (52), we get rv;.-r.;v== O. As QM isdetermined by (35). AV = Qpt -- r4, wecan think that only to within transformations of the form Qge it satisfies Qpa;v = O･. i..
(13) A kind' ef affiffely connected manifolds admitting a group of ･athne motions, !!. 13. We thus obtain a necessary condition. (57) R} pavto= AX PIi (Pv Qto -- Pw Qv ), where AX,Pl,,Q"arecovariantconstantssatisfying. (58) Adik=O, AdiQ.==O.. It iseasy to verify that this is a suthcient condition for we can take'a. coordinate system such that ･ AX=Sl, Pk,=6E, Q.=62. An An with connection parameters. ･ r!2 == x3, other rh, ==O , was already given as an example by G. Vranceanu [3, 4]･ t. Thus we get the THEoREM 2. A necessary ana spta7icient condition that an n-dimensional manlpla An, n;lil 7, with symmetric ajOine connection which is not Proiectively. eaclidean aamit a gTozip of aLfi7ne motions of order r== n2--2va+5 is that it.. satiSf!y (57) and (58). t. References. [11 Mut6, Y.: On the curvature aMnor of an affinely connected manifold An, nL 7, admitting,a group of affine motions G. of order r> n2-2n, Tensor, N. S., 5 (1955), 39-53. [2] Mut6, Y.: On some properties of a kind of aMnely connected manifolds admitting a group of affine rnotions, I, Tensor, N. S., 5 (1955), !27-142.. [3] Vranceanu, G.: Sur les espaces a connexion a groupe maximum destransformations en eux m6mes, C. R. 229 (1949), 54･3:545.. [4] Vranceanu, G.: Groupes de mouvement des espaces a connexion, Acad. Repub. Pop. Romane, Studii pi Cercetari Matematice, 2 (1951), 387-444.. 6 '. L. ..
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