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(1)[Science. On. Reports. some. of. the. Yokobama. conformally. National口niversity,. Riemann. curved groups. admitting. Sec･. of. Ⅰ, No･. 7, 1958]. Vn,. ･spaces. n≧5,. motions.. By. Yosio. MtJTO. Introd11etion A. group. of. of the. vative vhenever. [3]1) is characterized. motions. fundamental. tensor. in丘nitesimal. an. g)〟. by. to. respect. with. fact. the. that. Lie. the. the丘eld. uK. deri-. vanishes. transformation. point. ′El-El＋vl dt belongs. to. it･. This. vector丘eld,. that. from. we. is equivalent to the condition is, it satisfy Killing's equation vFL;リ＋uレ;〟-0. which. that. be. uK-g-Vα. a. Killing. ,. deduce. can. v);〟,･リ＋R).〃レWVW-0 integrability. the. and. conditions ￡ R).〃ya)-0, V. ￡ (R).FWW;dl;...,･dN)-0. V. In. these. formulae. a. semicolon. denotes. differentiation,. covariant. ￡. the. Lie. V. derivative. to. respect. with. the丘eld. Rl.pリW. and. vK,. is the. Riemann-Christoffel. tensor. R}. From. integrability. the. tenSOr. Curvature. We. ＋〈pay)(alw卜〈paw) (p},),a (a}y) lp)a?,〟. /ww-. ･. conditions get. and. among. the. de丘nition. Weyl. of the. conformal. Others ￡ Cl.F…-0. V. We. also. obtain ￡ Rjル-0 V. RiLリis the. where In also. the. write. Ricci. followlng. tensor,. We. ￡ instead. of. drop ￡ for. R/”-RT/wa･ semicolon no. and. otherfield. write. VFW. appears. instead other. v〃.･y.. of. than. vK.. the. paper+. V. 1). Numbers. in. brackets. refer. to. the. references. at. the. end′of. We.

(2) Y.. 2. integrability. The. and. vG. unknowns them,. the. then The. following. paper. theorem. G,. gnup. was. A. THEOREM.. have. I:hosen. MIFLW. tensor. the. a/〟-g)”. is such. that. M,112-M433｡-･--Mnn_I. M1234ニー2/3 M1256--2/3 M3456ニー2/3. above. ,. ,. M2516-1/3. ,. M4536-1/3. ,. C)I,…. except those which. it have. suitably. derivable. from. the. ones. written. the form. -C[gla, gFLリーgルg/w]. (”-1) (n-2). ＋g〃リ(A}A山＋B^Bw) ＋ BIBy) ]. -. Al,. B). AIAvB/Bw. - A/Aa,BIBv. 1. gαβAaAβ-gαβBaBβlatter and. necessary group. and. -(3n-8),. ]. satisfy. (2) the. ＋ AF,AリBIBw. C[AIAa,BpBy. 2. 2) 3) 4). a. M^[/ww]-0. -glv(A/Aw十BpB,dトgF,a,(A)A,. present. to. respect. to. -一撃才r. a. with. are. c[g}w(ApAリ＋ B/A). admit. 4). C Ml/ww. ,. MIp(…)-0･. that. oy. (1). a. a n≧6, admit the Weyl conformal. ,. M2314-1/3. according. 8,. or. As. the. where. Zero. are. where. [1] Vn･. space is that. components. n-1. ,. Ml〝川-May/A,. if ”-6. Riemann. a. gjW)-(”-1). the. n_1. components. all other. among. ”(n＋1)/2-m･3). paper. previous. ”(n＋1)/2. equations. is. admitted. of. satisfy. ennuPle. orthogonal. irldependent. r>n(n＋1)/2-(3n-ll) the form. c},LV｡-C(glw u'here. that. condition. oF order. C). pv@. tensor. cun)ature. equations. obtained･. necessary. of motions. linearly. m. exist. of motions of the group is a continuation of a. order. present. homogeneous. linear. are. conditions If there v[KTj･2). Mut6. is. condition su丘cient. G, C幸0.. interesting,. more. condition. of motions. that. a. in the. we. Riemann. order･. some. of. Properties. gaβAαBB. ,. large. We are. -. study space assume. beyond. paper･ As See. we. [3]. Indices. have page are. 2)(Kr)-0,. we. can. put. vHT=V[KT]･. 56･ raised. and. lowered. by. means. of gin,. 0. gjLV･. ･. in the Vn n≧5, the. present. paper. (1). satisfying. r>n(n＋1)/2 scope. of. the.

(3) On. some. §1. At丘rst. conformally. Integrability. we. remark亡hat. Transvecting. (1) with. we. deduce. can we. g)a. liilling's. of. conditions. Vn,. spaces. n≧5. (2) from. equation.. (1). as. follows.. get. [1-i((AA)＋(BB))]gpリ. o-(”-1). (”-1) (”-2). [(1-(BB)). 2. A/Aリ＋(AB). ＋(1-(AA)) (AA)-gdβAαAβ,. where. Riemann. curved. (A′pBレ＋AvB〃). BFLBy]. (AB)-gαβAaBB,. Then. (BB)-gaβBαB@･. obtain. we. ′. (AA)＋(BB)-2 (1-(BB)) If the. latter. A/Ay＋(AB). (AF,By＋AyBi,)＋(1-(AA))BIBy-0. is transvected. equation. ,. A”Aリor. with. B〃Bリ,. (AB)2(1＋(AA))＋(AA)2(1-(BB)). -0. (AB)2(1＋(BB)). -O-. we. I. get. 0r. Then. respectively.. obtain. we. ((AA)-(BB)) as. and,. ＋(BB)2(1-(AA)). (BB)]-0. [(AB)2＋(AA)＋(BB)-(AA). ,. have. we. (BB)-(AB)2＋1＋(1-(AA))2>0. (AB)2＋(AA)＋(BB)-(AA). ,. we丘nd. (AA)-(BB)-1 hence. (2). Now. begin. we. ￡ CIpva,-0,. to. from. get. we. a. study. of. group. As. motions.. -gl,. system. (El)obe such. that. an. ￡ (AjAw＋B/Bw)-g/”. arbitrary at. ￡ (A)Ay＋BIB,)]＋. Since. point. have5). (El)o we. Al豊∂)1 The. sign空means. for. the. special. coordinate. that. an. system.. have. we. gαP豊gaβ豊6a@. (4). 5). ￡glFL-0,. ￡ (A^Aw･B^Bu). (”ll) (”-2) Cx 2. ￡ (A)AwBj,By＋ApAyBIBw-AIA,BjiBw-A/AwBIBリ). ×. Let. it satisfies. (1). [g^w ￡ (A/Av･BpBy)＋g〟〃 %-cc如一宇c. (3). and. (AB)-0,. ,. -0. (2) we. can. at. the. take. a. ･. coordinate. ,. ,. B}二聖∂}2.. equation. is valid. only. special. point. (i))o.

(4) 生. Y.. 1Ⅳe shall. indices. use. Mute. follows,. as. P,q,r,s,i-1,2; arid, if. we. putスエw-a,. -. a,b,c,d,e,∫-3, in. a幸b. ”-ソニb,. (3), we. ,”,. get. ￡ C-0.. (5) Then. (3) becomes. (6). ￡ (Ap Av＋B/Bv)＋giLレ￡. g}w. A/Aa,BIBリ). -A)AyB/Bwwe. i-a,. put. 0. -. ･. we ”-♪ in (6),. ”-b,ソニC, ￡ Aa豊0,. if. and,. we. i-A,. put. easily. find. We. ￡ B2空=0,. ￡ Al空こ0, Hence. find. ￡ Ba三聖0,. iLエリ-a,. a'-q,. (A/Aw＋BpBa,). ￡ (AIAwB/Bv＋AjAyB)Ba,. ￡ (AIAu＋B)Byト(”-2). -gpa,. If. (A)Aa,＋BIBw)-gル￡. ￡ A2＋￡. Bl空=0.. have. we. ￡ A)-αB),. (7). ￡ Bl--αAl. (6) is completely satis丘ed･ which We drop and write semicolon. by. Al〃-Al;〟 From. (7). we. get. B袖-Bl;〃･. ,. (El)o. at. vα聖二α∂l,. vl^十A}｡. ,. (8) vヨ}＋B血Vα三聖-α∂ll if. and,. we. i-a,. put. we. 2n-4. get. ,. equations. 2)1a≡聖0. (mod. vE). ,. v2a豊0. (mod. vK). ,. (9). where. (mod. of u応. besides. Since. that. 2)”)means have. we. an. a linear form within because of (2), (8) gives. to. only. A,,,=!=0-0, B2〃=竺0--0, A2/J＋B.Iu聖O. (9) only v12＋A2a. As. is valid. equation. is. this. independent. an. equation equations. of. determines. only. which vK,. v[xT]. 2)α-*-0-α.. in (7)･. ther占are. α,. Hence. we. just. get. m≧2n-4,. r≦n(n＋1)/2-2n＋4. From. we. we. ￡Al〃-(￡A1);”-α,. get. pB)＋αB)p,. Aα〃vで1十Al｡V竺〟-Bl〃V12聖α,. (10) If. (7). use. the indices. x,. y,. z. as. follows,. p∂12. hence. (mod vK). I. 2n-4. 1inearly.

(5) On. some. conformally. Riemann. curved. Vn,. spaces. n≧5. x,y,z-4,-,”,. and. putスエ3,. that. is,. in (10),. iL-X. we. get. Az∬u:3＋A33u望x＋A3zV…x聖二O. (∂xzASS-Azx) for. have. we. uFLソニーVyp. and. are. there. 1inearly. ”-4. (mod. independent. be at least 3〝-8. would. to. we. good,. the. same. we. V2K,. we. way. from. get. putス-a,. unless. way. we. From. the we. we. (mod. uca豊0. equations. obtained. above. let. LEMMA. us. l･. vに, ”,K,. and. its fundamental. PROOF･ Ap･. form. except. be. can. Grst. Bp. is Xn-2-forming･6). find. that. (ll), the. vectors. which. in the. case. get. Acl空0.. In. A2P＋BIp皇0,. equations. Acl>AO.. of the. same. AIp豊0,. page. system 81･. of An. CiL Satisfying. B)B〃トAIK/i.. admits. a. u)ritten. as. the. partial. En-旦-field. ”-. ;. satisfying (2) and. (ul,･･･, un) dua dub.. as. we. BIKbBP. spanned. by. the. covariant. get. ,. 〃ニーA)KJA”. differential spanned. CobAふ-CαBα-0.. jield A〟, Bp. En-2一点eld. Besides,. APBl. all. must. ＋B)KF”. (ul, u2) dug duq＋gab. At. we. Vn. space. B/Al.;. [2]. we. for. u2応). we. the. b(a)iL-AIA〟-. Riemann. a. ds2-gpq. 6). Hence system. the. prove. If. (12). from. integrability. get. Bル-. vectors. in the. Then. get. A)〟-a(g)/rAAAp-BIBp). (ll),then. A3z聖0.. ∂ab.. (ll) Now. have. we. ∂ab.. independent 7l-3 1inearly equations we have assumed r>n(n＋1)/2-3n＋8, Ac2豊0, Bcl些0, Bc2豊0･ get. B2〃些0,. ,. coordinate. are as. obtain. equations. any. ,. ￡ B)〃ニーα,jA)-αA}〃. in (10),. ”-1. Ael which But,. u3K). U2K). ,. get. Bab=聖b If. VIK,. r>n(n＋1)/2-3n＋8. for. Aab=聖a In. ul応,. independent. assumption hold good. must. uK,. uE,. (4). We. of. equations. 1inearly. the. conditions, contrary have As this A3z豊0･. (4) holds. because. DKx豊u.Gx. vK,VIK,V2K). (mod. u3z＋A3zuz∬豊0. A3zVzx豊0 which. (mod. by. equations Ap,. Bp. means. an. En-g･aeld. spanned. by.

(6) Mut6. Y.. 6. A” ∂〟′-0. β”∂〝′-0. ,. Al, Bl is by the contravariant vectors spanned by f3,-,′n. If Its ”-2 independent solutions will be denoted ∂〟g, ∂ph, ∂FJ3,-, ∂pfn Ap-α∂pg＋β∂ph, Bp-r∂pg＋6∂ph, the vectors. is complete, x2-forming.6) we. put. linearly. are. E2-丘eld. the. and. independent. the. and. independent. EK variables by putting ul-g, are. u2-h,. functions. ”. functions. we. and. ua-fa,. A. the. new. we. have. are. independLenl introduced. βα-0. ,. Then. system.7). coordinate. coordinates. ”. Ba-0,. Aα-0. for. uK. get. Aa-0,. (13). of. a,h,f3,-,′n. new. (14). we. get. gap-0. for. Aa-galAl-gapAP-0 Ba -galBl This. ,. 0. -. -gapBP. ･. proves. dub un) dug duq十gab(ul,-I, un) dua. ds2-gpq(ul,-, We. also. ･. obtain gpq-ApAq＋BpBq. i. (15) Since. we. 1. APAp-BPBpfrom. get. gPq-APAq＋BPBq. ,. APBp-BPAp-0. ,. (ll), (13)′and Apa-BpKa Bpa-. ･. (14) Aap-0. ,. -ApKa. ,. Bap-0. ,. ,. we丘nd Ap,a-Ap,a-Aa,p-Apa-Aap-BpKa. ･. Bp,a-Bp,a-Ba,p-Bpa-Bayfrom. we. wb.ich. -ApKa. √. obtain. gpq,a-(ApAq＋BpBq). This. Furthermore LEMMA. (ll). ･. a-0. (-. (12).. proves. and. I. we. 2. ike. In. a. get. the. Riemann. space which. a. admits. jield A/A,BJL. gab(ul,-, un) of the fundamental. components. (16). gab,p-2(aAp＋bBp). gab. tensor. ,. hence. (17) 7). gab-ス2(ul,･･･,un) kab(u3,･-, un). In. the. following. we. shall. use. only. such. coordinate. systems･. Satisfying. (2). in (12) satisfy.

(7) on. pR｡｡F.. If. some. we. putスエa,. Riemann. curved. conformally. in. ”-b. Vn,. spaces. first members. (ll), the. 7. n≧5. the. form. agab,. bgab. take. l. Aab-Aa･b-(aab)Aα-甘gab･pAP. Bab-Ba,a--( of (13), (14). As (16) and (17) spectively, because. Now. let. From. us. the. Ricci. the. re-. tensor.. (mod. RaリVT〃＋Rpa v㌘y豊0 we. to. reduced. is, from. that. ￡RFW-0,. auBa--÷gab,PBP. are members second immediately obtained･. are. consider. I. 〃㍍). obtain R/,,. (18) as. 3'ust. obtain. first equation. the. obtained. we. Rab豊p6ab but,. have. we. since As the. (ll) from. of. a/”. (10)･ Of. we. at丘rst. course. form. of the. relatio血s. ＋rB/A, ＋ p. -PA/Aリ＋q(A/By＋AyB〃). RjW-R,〟,. Rap豊0. ,. we. (18)･. get. of (1) remains. secorl.d member. ,. unchaTlged. Ap,. when. Bp. are. replaced. with A〟cosβ-β〃sinβ we. can. RjW. (19) Lie. the. gFLリ. derivative. p. ･. of (18′)we. get. (A/By＋AyBp)-0. (即)A〃Aリ＋( ￡r)BpBリ＋( ￡ p) gjW＋α(P-r) by. virtue. (7). Hence. of. have. we. (20). α(♪-γ)-0. (21). ￡♪-0 The. condition. for. necessary. and. we. be. can. wbicb 2,. shall The. G,. of motions. existence. derived. a. of order. from. lemmas. is resumed Riemann. of. group. But. from. derived. equations. the皿. of these. obtained. a. condition from. use. l∴ (f. of. ￡β-0.. were. These. equatiQn.. s-u氏cient. make. result. TI‡EOREM. the. ,. ￡γ-0,. ,. (5), (7), (20), (21). eqnation§ of Killing's. ditions. ,. obtain. R-p＋r＋n Taking. ＋p. -♪AjAv ＋rB/Bv. We.also. ♂ suitably･. choosing. 0. (18) with. replace. (18′) by. ApsinO＋BiLCOS. ,. of the丘eld If. motions･. them,. we. must. integrability. the vK we. a. glVe. obtain. a. investigate. equations Lemma 1 and. have Lemma since in §4 instead of the ab6七e we. necessary. to. wa血t. con-. equations･. in the. (1) is satisjled admits hence then (2), (ll) and. space for which. r>n(n＋1)/2-3n＋8,. a. goub (12), (17).

(8) 古. Y.. also■'′aresatisjied･ (18). (5), (7),(20), (21). §2. First. -. is alsorsatisjied. (18′)∼. or. As. for. the, group. of motions,. be satisjied.. must. Curvature. Censor,. the. we. calculate (1),(18′),(19) into. (22). Mut6. BiancIli†s. components. arLd Ricci's. of･the. identities. tensor:. curvature. substituting. R^v] R^〃vw-C^/ww＋去[g^wR”'gI,VR^w-g^vR〃山一g/1 R. 両二打百両we. (g^w a/-. -gルg/-). ,. get. ♪＋γ. (22′). R^/,vw-[c--i). (n12). a/uニーgルg仰) ＋岩](g^w Ay＋B/” By)＋gp(A^. Aw'B^. -宕c[g^w(A〃. Ay＋B). Ba,)-gpo(AI. -gル(A/Ad＋B〃. Bv)]. c[A) Aw B/A＋A/A 十上空=1-L(些二軍-)2 Ay B/”Bw-A/Aw. -AI. AリB). Aw十rB^. Aリ＋rB/A)'g/”(PA^. Aw＋rB/A. Then. we. Bw. BI Bv]. ＋去[g^w(pA/1 --giレ(PA〃. Bw). Bw)-a/”(PA). Av＋rB). Bw). Bリ)].. obtain. (gad gad) [c1高二号)i:-二軒十-”-阜r]. (23). Rabcd. (24). Rapqb-[-宕c一両二書芸=-2T＋議告]gabgpq. -. gbc-gas. Aq＋rBp. ,. Bq) 十元一王宮一gab(PAD Rpqrs-. (25). If the tensor. (gpsgq,-gp, [(9二-212(p=一声) c＋駕壬p1 Rabcp-0. ･(26) curvature. is gpq. is denoted. Vh-2(gab′)whose bave･'from. ,. tensor. by. fundamental. (12)L,(16). Rabpq=0. ,. of. a. Riemann. Rぎq,s and tensor. the is gab. gqs). ,. Rapq,-0.. ,. space curvature. V2(gpq) whose tensor. a. of. fundamental Riemann. space. *. is. denoted. by. R.abed, then,. as. we.

(9) On. some. conformally. Riemann. curved. Vn,. spaces. n≧5. (品)-0, (諺)ニー(aAP＋bBP)gbc ∂諾,. (pad)-(aAp＋bBp) we. ,. get *. Rpq,s. (27). -. Rpq,s. ,. *. (28). (gad gbc-gac. Rabed-Rabca-(a2＋b2). From. (23). we. which. ♪＋γ. is. curvature. --(-i1) (”-2). Vn_2(gab) is. that. shows. ･. obtain. 滋abca-[c-. (28′). gad). function. a. to obtain order identity Bianchi's. In. of. ul,. a. (gaagbc-gac ＋7iIITLa2＋b2]. space. constant. of. god). curvature,. ,. whose. scalar. u2.. necessary. more. conditions. we. precisely,. investigate. must. R)p⊂,a,; ♂] -0. (29) Ricci. and. identities AiLU;a,-AFLW;”--R㌘〃vw. Aα. BiLリ;a-BiW;ソニーR㌘/ww. B｡. (30) If (22') is substituted. (31) where. into. 7T/”. is defined. (33). we. I. (ll) in the. write. B〃リ-b7T/”-A〃. A〃ソニa7r/w＋B〃尻,. form. Kv. by g/”-7C/”＋A〃 Aリ十B〃 By. (32) and. if. (29) and. ,. calculate,. we. get. Ud] (7=J1[w7Cl,,,”-7C/u[w7CTITリ) To]＋2(7r加Al/,[Aリー打,uEwAlスIAリ). B.A,Bリ) Vd]＋2[7rl[w(A(〃IBリ＋B】/u'Aレ) ＋2(7r仙B.〃[Bリー7E/u[w Bv＋B一人EAリ)]Wdコ＋2(AIA[wBJ〃■BリーA/A[wB川Bリ)So]-0. -7C/u[,A(4). where. the. vectors. T” U”. V”. W”. de丘ned. Sリare. by. C(aAv'bBy). ･y-c,リー市PjI)I芸,_7F十浩十(”-1) (PaAu･rbBy). 一石㌔ Uy--二. 〟-3 2. G,”----(. ---. ,. C b Bv. -芝葺-hPty2十空言-!--. np=芸)i-:･iL21). (”ll) 2. CbBリ. rb ”-2. β=.

(10) Y.. 10. vy-. Mute. ca. Av. ＋岩宜十P:ii -p:iic,リーii_-f払＋岩i (”-1)(”-2). ba. ^ A=. CaAy-二㌔”-2 ,1_^. 〉L”▲y. 2. wy-宕を尻, sv. -. [1-. c,レ＋旦-ihr_･ITtP-,一旦1)＋i-n=113;(哩]. (”I. If (33) is transvected Hence. -7TαレBα-0.. (34). Sy. because. A[vBa,So]-0 of A”. Bレand. (普)AvI (a)Bv - c,リ＋旦-とj-hr111'}･-上-. -班)i-聖二3). (潔)means. we have with which in flrSt term the member. factor. some. Then by. AIB”, we get with is linear combination a. the丘fth. transvecting. the. remaining 7Cl[v. If this. is transvected. with. (7rlvAw-7EIw If this β〟.. In. Bw.. If the. similar 丘rst. Vu-(普)Aa,＋(*)Bw,. Aw. Uo]＋7E)[v Bw. Ad. and. if. we. with way. gル,. that. we丘nd. member. of. andfinally. Uy-(普)Ay十u. find. we. (33) is we丘nd. obtain. ･. concern.. of (33) vanishes, A/A. with. Aリ) Ul＋7CIw Uv-7rlv. is transvected a. terms. no. of 7CaリAα. we. and. we. obtain. Wo]-0. WdAd-Wl,. UdAd-Ul,. write. We. get. Uw＋(7TルBa,-7=)a By) Wl-0. that. Uw. is. Ww. is also. a. a. linear. transvected. with. that. write. we. can. Vy-2). B=. Ay十(貴)B,. of Aw, of A山,. combination. linear. combination gルBFLBd,. we. obtain. ,. Wy-uAレ＋vBリ. Then,. if p幸r,. we. can. write Ky-lAリ＋m. (35) and. hence. (36). Bリ. get. u--E=をl,. ”-砦をm,. (1一宇)c,リー諾続芳三喜㌻＋芝㌻.畠＋(i)Ay ･宇cb By--坦坦cbBv一志By一詔-llBv-0, 2. (37). (1-P;iL)c,〟--下表要式Inr竺訂十岩T･藷＋Eilca Av･(普)B湖･. -､(里二!1(竺二見caAリー忍Ay一芸壬m 2.

(11) on. The. last two. some. are. equations. spaces. Vn,. if ♪-γ although. we. Riemann. curved. conformally. even. valid. 11. n≧5. do. not. (35). get. then. On. (34), (36), (37). accoun亡of. we. 7El[w7=I/A. Then,. as. we. (38). consider. n≧5,. we. ,. To]-0. ･. hence. Td-0,. get. (33). C,リー(n空言敵十岩÷+-(”-1)C(aAv＋bBy) Thus. we. have. if ♪幸γ,we. over,. have. of A”. combinations. -去(paAリ･rbBy)-0･ (34), (36), (37), (38) from. obtained. (35). The丘rst. (36), (37), (38) is that. the Bv.. variables Now let Taking. u3,･･･,. us. vectors. gradient This is. because. consider. (32) into. account. B〃…-B/w”-7C〃レ(b,. we. get. a,-b. w＋a. -7r/”(∂, y＋a. we. obtain. hand. other. A,”. the. r,”. the. More-. (34),. equations. all linear. p,″are. determinant. the. the. of. made does. co-. vanish･. not equations γ, 〟 do C, not ♪, good. above (13) holds. identities. Ricci. -7Epw(a,リーb. the. from. contain. un.. A/”;”-A/w”-7Epレ(a,. on. C,”. identity･. Biancbi's. deduced. thing. in vectors e丘cients of these gradient in the coordinate Hence system where the. from. get. (30).. from. (31). Ka,＋a2Aw＋ab. Bw). Ky＋a2Aレ＋ab. Bリ)＋B〃(Kyw-Kwリ). Kw＋ab. Aw＋b2. Kv＋ab. Aリ＋b2 Bv)-A,i(Kvw-Kwy). (22′)is substituted. into the. ,. Bw). second. members. ･. of. (30)･ Then. Kp･'a2Av･ab. ＋て品)Aレ] Bv-(字c-芸当〈漣二旦㌍二旦c＋旦妄字〉A/A] B^[K/1y. ”-b 花^/1[a,. ＋. -. P＋p. /rb範＋a2A〃＋ab Bjr〈字c-方}v[a,. ”-1. r. .. ＋ L. (”-1)(”-2). -i--〉A〟]. c･駕三才)AリB〃]-0 -B^[Ky/r(一転旦㌍ ,. 花}/”. [b,y＋a. Kv･ab. ♪. Aリ＋b2By-〈字c-三笠＋ ih=i-)一軒打〉By]. Bリ] -A^[K/uy -〈也3)2(吐乳c･器字)Ap ♪. c-盟-軒i)両〉Bp] -花}y[b, p＋a範＋abん＋b9Bp-(--'2=1.

(12) 12. Y.. Mut6. (-Ln二等竪ゴL Bβ] ･A^[Ky/c十莞三半〉AL 0. -. ,. hence. (39). K/uリーKy〃-[也二等㌍二13i c･砦] (A〃 Bリ-AリB/). (40). a,リーb. Ky十[a2--NIB:lc＋砦-fln=i1)1r(-あ=12T]. (41). b･. Ky･ab. Ay･. §3.. Riemann. y･a. Av･ab. 2. Lemma. we. By-0. ,. By-0 [b2-一字-c･-rn-＋=pl--一両二i-)?w] ･. motions In. ,. spaces. admitting. under. consideration.. groups. of. obtained. gabニス2(ul,-･,un) kab(u3,-, un). The. results. in §2 aBirm. obtained. (42). that. has. gab. the. form. ga｡ニス2(ul,u2) kab(u3,-, un). in the fouowlng. shown Atfirst we consider p幸r･ Then, 1, Proof) (see Lemma and moreover. as. Bp. depend. only. ul,. upon. also only upon i has the same. ul,. let. If at. least. us. to. one. of. b is not. a,. zero,. 0. 0-B,}sin Then. we. we. of. replace. (16),. can. Ap,. b depend. consider. that. A/” B〃 with. obtain ∂-0. (41) becomes. and. (〟-3). a. is. which. a. By-0. Ky-[字c--fh=-iう両-♪-岩丁]. special. If a-b-0,. we. case. get. df (35). Hence from. we. (16) gab.p-0,. get. is Thus. satis丘ed,. a. special we. then. see we. case. that,. (42).. ､. of (42);スエ1. if. in. a. ba-∇e (12) and. ds2-gpq(ul,. ,. hence. gab-kab(u3,･-, un).. (43). scalars. we. that a,. A〃sinO＋B〃cosO. ,. α幸0,. This. (35), we血d. of. (40), (41) the. solution. Bp,a--ApKa. ♪-㍗.. consider. ♂ suitably.. choose. Ap,a=BpKa,. because. property.. Next. A〃cos and. (17) is the. have. we. Ka-0. According. u2･. As. u2.. as. Riemarln. (42). so. space. Vn,. n≧5,. (1), (ll), (18). that. u2) duP duq＋ス2(ul,u2) kab(u3,-･,un) dua dub.. are.

(13) On. As. of constant. its scalar. and. Riemann. curved. Vn,. spaces. 13. n≧5. to (28′)a space Vn_2(gab) is according fundamental tensor Vn_2(kab) whose space. If its curvature. curvature.. by. curvature. K. K,. is. K. a. is. tensor. constant. is hat, is also. denoted. K.aむcd. by. have. we. and. constant. of. space. Riemann. the. space. conformally. Riemann. the. curvature, a. some. kbe-6ca kbd)-K.abed-R.abed (∂da. (-h二否-)ih二笥. ♪＋γ. -[c-. 〟. (∂諾gbc-βea gad) ＋a2＋b2]. -ト ”-i. (”-1). ,. hence. (44). K-(”-2) Thus. have. we. LEMMA has. metric. of. 3.. u2. This. that. reveals. leave. which. the. have. We. G,. of. Vn admits group ul, u2 invariant. formulae. many are. results. If. motions. of Vn_2(hat,) b by. a,. p,. of motions. from. Vn, n≧5,. where. space. of order. r,. are. tensor. starting in the. gathered. Riemam. a. C, A,. scalars a. variables. 2.. TIiEOREM. (1), (ll), (18). Vn, n≧5,. space. u,ith the. the. obtained The. /213n＋8. group. is related. and. I. (43) u)here kab is the fundamental K The scalw curvature of. curvature.. ul,. upon. Riemdnn. a. the form. constant. P. the. obtained. If in. ±L______ [c- (-ii二1) (”-2) ＋岩＋a2＋b2]. (”-3)12. a. Vn_2r(kab). space. does. depend. not. (44)･ (”-1) (”-2)/2. of order. the. the. satisjied,. assumption. r>n(n＋1). (1) is satisjied admits. a. then. r>n(n＋1)/2-3n＋8,. --nin2i:i)--2n･1) r≧一連二!)2(聖二旦( -. and. the. of. constant. has. metric. V2's. and. gonal. to. constant-. the form Sn_2. curvature a. system. the. spaces. ul,. of. (”-1) (”-2)/2 for u)hick From tion. we. this get. theorem. 3.. 8) metric. system･. scalar. curvature. The. u2.. Vn_2. each. the. and. except. Sn, Rn, fundamental. Zf C)/”o, has was. Which. -(”-1)CMl〃レW not. the. Vn. one. spaces of. Moreover. a. admits. the. a. of. system. …n-2. of are. system. Vn-2's. space. ar?. ortho-. SPaces of. Vn. is. admits. an. theorem. a. inuariant. group. of motions. Variety. in [1] and. obtained. cited. order. of. in lntroduc･. the. TIiEOREM. occur. other. tensor. fundamental. (”-3)[c-て前書㌶二訂＋ n告＋a2･b2]. k-スー2K-(”-2). u)hick. Of Rn_2.8) Hence Vn_2's such that the. u)ith the. is afunction. is the. 0r. the. of. curvature. (45). -2. of. (43) lPhere kab. Cn. the. mean. case. a. tensor,. given. of. space a. C)/ww-C(g^a,gFW-gルg/”) special form in the theorem just mentioned and u,/u'ch can. not. ”-6,8,. the. in order. of constant. conformally. that. an. curvature,. Euclidean. Riemann. a. Vn. a丘ine. respe`ctively.. space. space. Vn,. with. n≧6,. a. sym-.

(14) Y.. 14. is not. which. Cn. a. is necessary. a. admit. that. r. G,. group. that. Vn. the. have. are. Vn. Necessary. is determined The. glVen･. of order. i[. r>n(n＋1)/2-3n＋11. ＋1 ≧r≧旦(y2t-I)-ーーーー2n. the property. §4. This. of motions. satisfy 2 -ninjL-2n＋4. and. Mut6. scalar. and. sudicient. Vn-2. curvatureぷOf. (1).. corLditions. A, and. function. V2(gpq), the. when. 2･. in Theorem. mentl'oned. 2. in Theorem. the. K. constant. is obtained. as. K=Kス￣2. Let. us. already. were. here. write. in. given. of. the. (23), (24), (25), (26)･ They. are. the. again. components. Raゎep-Rpqra. Rabpq-0. -. Raゎca- Q(gad gbc -gae. tensor. curvature. which. ,. gad). ,. (46) gqr-gpr Rpqrs-号(Eps. Rdpqb-Rpabq. gqs). ,. Ppq. -gab. where. ♪＋γ. (47). Q-C-Tn-二iう両十妄Ifli-. (48). 号ニー坦-誓空理c十b-;-;i-p-. ,. ,. Two a. (Q-字c)gpq･ぅ左(pApBq). ppq-. (49). then. constant,. take. are. cases. the. Aq＋rBp. possible we. can. such. a. to. respect. that. way. function. the. A.. Ifスisnota. (a)スエ1.. consider. in. ul. variable. with. have. we. ･. As. then. constant,. if i is. ス幸0,. we. can. (b)スエexp(ul).. (a)A-1. In this. case. immediately血d. we. from. (16) that. Then. a-b-0.. (40), (41). become γ. i. ･50). and. we. On. get♪-r.. -0. ,. -p諜c十担-w)r6ij2jn-1. -. ♪. c＋. -p:2=3- -r-tQIn11 -てh-lT市二打the. other. hand. we. 0. have. ds2-gpq(ul, u2) dug duq＋gab(u3,-･, un) dua dub hence. the. Vn. is. a. product. space. V2×Vn-2.. Thenwe. get. ,. Rapqb-0,. Ppq-0. and.

(15) On. (51). some. conformally. ”-3. C＋. -. Ti-7TT--oT. 2 -qlT-31〉'. is obtained. which We. can. from. (52). (”-1)(”-2)yl. in C, A,. K. p. 0. A ＋二旦丁”-1. to. according. (”-3). 応2=(n-2). C十坐±迦 n-1. (48) and. (45),. '. 2p. (53). c-て蒜二打(蒜二町十一L ]･ (n13)[ n-1. K-(”-2). Conversely,. C, A, we. example. be. can. p. in. expressed. K,. K2. C=. If 〝2-K-0.. us. a. consider ￡R1〃va,-0. from. get. be. must. which. Let. '. ;nT. (”-1)(”-2)2(”-3)一. ,. re)pecled. of. group. As. motions.. Rpq,12):a-0 RabeeV:p-0 If K幸0,wehave hence ”pa-0,. hence. 2):p-0,. yap-0,. agaln･ 2)ap-0, ”:p-0 is also a direct product.. motions Thus. For. account.. is, if. that. C-0,. (51) into. 2ぷ. .. ＋ 7T--TTT_. (”-1)(”-2). Rabcd-Rpq,s-0 then. by taking. K2. obtain. (54). have. we. THEOREM Vn. 15. l㌔, n≧5. spaces. (50) also.. K2,. express. Riemann. curved. the obtained If i is a constant. 4. 2. in Theorem. a. Product. that. betu)Gen. the in. shown. this. scalars. is not. space. in. (43),. is. a. we. space,. product. ,. ,. D:a-0･ a. way. u)e. V2×Sn_2. that. we丘nd. think. can 0r. If K2幸0, wehave. that. V2×Rn_2.. the. A-1.. Since. u)e. ”:a-0.･ group. of. Then. the. have. as-. G, of motions is a the combletegroub admitted product G(n_1) (n_2)/2×Gs, u,here G(n_1) (n_2)/2 is the complete group of motions in Sn-2 0r Rn_2 and Gs is a coml)lete group in V2. The relations of motions sumed direct. Vn. is. vpa-0, Insuch. Vn. the. a. C, A(-r),. Cn,. p. in (1), (18′)and. the. curvatures. scalar. (51),(52), (53).. (b)A-eul. As. we. have. gpq-gpq(ul, u2). ,. (55) gab-e2ulkab(u3,･-, un) we. ,. get. (qp,),a-0,. (bac),p-0. (品)-0,. (品)-0,. (bay)-∂夢∂pl ,. ,. (品)--gPlgaゎ｡. x2,. K. are.

(16) Y.. 16. These. formulae From. be. will. (46). we. ,. Rpqra”-0. ,. i. ･57). constantly.. get. Rpqrs;a-0. (56). used. ) (J21gpr-Ppr)∂｡1] [(号gqr-per ∂pl-. Rpq,a; ｡-gab. Rabcp;e-(gil Ptp-Q∂pl). i. ･58). Rabcp;￡-0. Taking. Mute. (gaegbc-ga,cgbe). ∼. ･. (57) into. (46) and. ,. we. account. Rpq,a;eve＋Req,a. from. obtain. ”:p＋Rpe,a. ￡Rpqra-0. ”:q＋Rpq,lV:a-0. ,. hence. (59). (va. Taking. -o ∂ql＋vqa)(号gpr-p”) ∂pl＋vpa)(号gqr-pqr卜(va ･. (58) into. (46) and. account. Rabcp;eVe＋RLbcp. from. obtain. we. ￡Raむcp-0. V.ta＋Ratcpv…b＋RabeeV:p-0. ,. hence. (60). (P.I. where. P.tp-gis. ∂p.)(uagbcIVbgac)＋gbe(P.i p via-Qvpaトgac(P:. p-Q. Suppose. pvtb-Qupb). -0. Psp･. va. If ♪♯r, we. ∂pl＋vpa幸0･ get. 号gpr-pprj=0 I. Then. taking. into. account. that. gpr. Pp,. and. are. symmetric,. we. get. from. (59). (Lp幸0) 121-gpr-Ppr-lLpLr. (61) and Fence. (Ma＋0).. ∂pl. vpa＋va. -LpMa. (60) becomes gbc(P三pLtMa-QLpMa)-gac(P三pLIMb-QLpMb)-0. from. we. which. get. P三pLt-QLp. (62) As. we. Since. have Ppq. ,. p幸r,. assumed. is given. by. 与-Q -. we. (49) and. Can. ･. consider. that. gpq-4pAq＋BpBq,. c十In-ri -n12Ii,. C. have. Lp-Ap. or. Lp-Bp･. get. -. Or. 号-Q一宇c＋右左,. we we. c-. -(-nl?-%--I-21)-2γ. (nrl) (”-2).

(17) On. by. (∋ are. must. be. Riemann. curved. conformally. we have as of (61), (62) according by (47), (48). Consequently we given. virture. and. some. Vn,. spaces. Lp-Ap get. 1丁. n≧5. Lp-Bp.. or. ♪-C-0. or. But. γ-C-0. 応2/2 which. rejected.. If 1,-r,. we. from. Ppr-Pgp,. get. (49), hence. ppr-%gp, from. (59). Then. we. from. obtain. (60). 号IQ-0, hence Ppq-. (!gpq. ･. (49) becomes. -9ii- c＋石生-o (47), (48) give. while. 2p. c-て右iうて右打-A-i)坦c十票･ 2. Consequently This. we. ♪-0, C-0,. obtain. to. contrary. the. assumption. C幸0.. proves. (63). vpa＋va6pl-0. Then. we. obtain 2)pa-. -Va∂pl. 2)ap-Va6pl. ,. ,. thatis,. )vb-2)pa-. vp,a-va∂pl-Vp,a-(pba. -Va∂pl. ua,”-va∂pl-Va,p-(abp)Db-Cap-ua∂pl. ,. ,. implies. which. (64) for. vP-vP(ul, u2) have. we. Now,. the. 2)a-ua(u3,-, un). ,. ,. (55). vectorfield. 2)”. must. satisfy. ul,p,v＋(A),ぴva-(I,ay)vス,｡＋(i,)vα,p＋(”)a)uα,,-0 ･. If. we. putスエP,. (65) If. we. (66) But. ”-c,. we. get. (bye),tvl＋(bPc),eve-(bee)vP,t＋( epc)ve,｡＋(bye)De,a-0. putス-a,. FL-b,. ”-r,. we. get. (ba,),tVt＋(ba,),eve-(be, )va,e＋(. (66) becomes. (67) while. ”-b,. vl,,-0,. hence vl. (65) becomes. )ve,b＋( bat )vt,r-0.. ea,. -constant. ,.

(18) Y.. 18. gpl,tgbcVt-2gPlgbcVl. Since. Mut6. -gPlgbc,eVe＋gtlgむcVP,i-gPlgecve,a-gPlgbeue,a-0. ･. have. we. (68). :. vbe＋vcb-gbeV. :a. c＋gecv. -gbeVe,c＋gecue,b＋(†fc,b) ＋(fb, c)) Df＋2gゎcvl Vl. -gbeue,c＋gecve,む＋gbc,e ue＋2gbc -0, we. get. (69). -gPl,lut＋gtll)P,t-0 We. take. can. the. in. u2. variable. ･. a. such. that. way. have. we. g12-g12-0. Then. we. from. get. (69). (70). gll,i vt-0. gll,tVt-0. ,. and. (71). 2)2,1-0. hence. v2-v2(u2). Now. u2. ,. in. consider a. such. way. that. we. have. that. we. get. for. 2)2-1. equation. Then. v2幸0.. we. take. can. this丘eld. (72) The. forwhich. vx. afield. ･. becomes. v22-0. (73). g22,とVt-0 find. we. and. from. (70) and. the. (73) that. form. fundamental. be. can. in. written. form. the. (7チ) k. where. ds2-I(ul-ku2. )(dul)2＋g(ul-ku2). is. andf,g. a. constant. (duョ)2＋e2ulkab(u3,I-, un) dua dub. functions. are. of. As. ul-ku2.. aspecial. ･. case. we. have. may. ds2-kl(dul)2＋k∃(duョ)2＋e2ulkab(u3,-, un) dua dub. (75) kl, k2. where. If the for. are. COnStantS･. fundamental. everyfield. ,. form We. vK.. get. can. not. (73) again. be. in the. written and from. gll,1-0,. form. (70), (73). (74), then. we. v2-0. get. g22.1-0. hence. ds2-I(u2) (duュ)2＋g(u2)(duョ)2＋e2ulkab(u3,-,un) dua dub. (76) unless the tions. we. have of the. order in (63) and. ul-0 group vl-0,. for. every丘eld. if vl-v2-0, we have 2n-1. But,. vだ.. is (”-1) (”-2)/2, for v2-0,. v12-0.. Hence. we. get. the. we. get v12-0, and independent equa-.

(19) On. TEEOREM. some. 5･. Ifスis. Riemann. curved. conformally. Vn,. spaces. in (43), then. 19. n≧5. think. thatスエexp(ul). 2 is a Cn can be not of the Vn in Theorem which in or as the form loプ曙aS it admits a (74), (75), (76) u)ritten group of motions of (”-2)/2. order r>(”-1) Then. a. constant. Necessary. and. the fundamental. If. u)e. can. form. §5. Case. not. su瓜cient. (2).. conditions. (b) is still considered.. we. (68) and. consj.der. put *. )ue vaむ-2)a,む-(aeb we. get *. *. vab＋vbaThis. that. means. in. have. only. found. form. -2Dlgab.. be. Vn. in. written. form. the group. independent. 1inearly. We. us. get. from. that. consider. the. Vn. has. have-ul-0. On. the. have. 2n-2. which. other. Sn_2,. proper. not. of. then. constant,. We. (76).. or. in. admitted. order its now. case.. each. I. in. (63), (77), we. get. (n12)/2＋2. fundamental. the. vl-kD2-0 we. motion. form. (74) with. f'＋0.. (70). (78) hence. a. motions a. (74)I, (75),. equations. rc≦n(n＋1)/2-2n＋3-(”-1) Let. of. of motions. v12-i 2n-3. are. If Vn_2(gab) is. -2)1.. (gl,,2Vl-g22,1V2). there. homothetic. a. vl-0. a. the order re of the complete consider As･we have (67), we get. (77). induces. consideration. a gi･oup admits if, furthermore, in (43)よis. and. can. in. constant. when if. that,. r≧(”-1)(”-2)/2＋1 fundamental. Vn. motion the homothetic. is admitted. this We. in. a. Vn_2(gab) with. then. As. ,. 1inearly. ,. independent. is equivalent. (78) only. to. Moreover,. equations. k=0.. when. ･EenQe.:We. rc≦(”-1)(”-2)/2＋1. if. K-0,. rc≦(”-1)(”-2)/2十1. if. K＋0,. k-0,. re≦(”-1) (”-2)/2. if. K*0,. k幸rO.. hand. it. is. easy. to. and. a. group. of. -motions. r-(”-1). (n12)/2＋1. if. K-0. ,. r-(”-1). (”-2)/2＋1. if. K＋0. ,. r-(”-1)(”-2)/2. if. ･. K幸0,. G,. such. k-0. k幸0,. ,. if K*0, get. that. we.

(20) Y.. 20. Mute. Thus中e丘nd if. K-0. rc-(”-1) (”-2)/2＋1. if. K＋0,. rc-(”-1) (”-2)/2. if. K*0. (”-2)/2十1. rc-(”-1). i. ･79). for. (74) with. k･-0, k＋0. ,. ′′幸0.. let. Next. ,. us. the. consider. We. of f'三0, g′幸0.. case. from. get. 2)22-0. ･. v2,9＋ig22(g22,1Vl＋g22,2V2)-0 into. Differentiating we. this. partially. to. respect. with. ul. (71). taking. and. account･. get. (1ogg22),1,1Vl＋(1ogg22),2,1V2-0. ,. hejnce. (1ogg)′′(vI-kv2)-0 on. account. have. we (74). If (log首)′′幸0,. of. If (1ogg)′′-0, then. we. get. (78) and. (2mu),. g(u)-k2eXP. (79) again･. consequently. hence. ds2-kl(dul)2十k2e27n(ul-kuB)(duョ)2＋e抑kab'd,･-, un) duo dub be. can. which. to. reduced. form. the. ds2-(dul)2＋e2仇ul(duョ)2＋e2aulkaゎ(u3,･･･, un) dua dub. (80) Then. ,. (m＋0,. a＋0).. have. we. If K-0,. a. rc≦(”-1) (”-2)/2＋2. if. K-0,. rc≦(”-1) (”-2)/2＋1. if. K＋0.. a＋m. ,9). motion 'ul-ul＋t, /u2-e-mtu2 I. ′ua-ha(u3,･･.,un; i) last equation denotes is constant -α, admitted. a. the. where thetic. A. Vn. homothetic we. and. if. K-0. rc-(n-1). (n12)/2＋1. if. K*0. form. fundamental. Rn-2. an. is (75) is. a. Cn. a＋m. ,. ,. ･. if K-0,. for. a. ds2 -e￣2ul(kl(dul)2＋k2(duョ)2) is. an. R2.. If. we. have. (75) and. (82) 9). K幸0,. rc-(”-1) If K-0,a-m,. we. get. anSn･. we. homo-. With. have. (”-2)/2＋2. whose. in. motion. re-(”-1). i. ･81). ,. get. (n-2)/2＋1. vl-0 ･. and. (77),hence. V2. With.

(21) On. Now. we. some. Riemann. curved. last. the. conside･r. (76). We. case. Vn,. spaces. to. this. reduce. can. 21. n≧5. the. form. ds2-f(u2) (dul)2＋(duョ)2＋e2ul kab(u3,-, un) dua dub. (76′) and. conformally. need. we. to. the. case. Then. f'幸0.. consider. only. of. rc-(”ll). (”-2)/2＋1. if. K-0,. rc-(”-1). (”-)2/2. if. K＋0.. we. get. besides. v2-0. (77), hence. i. ･83) Gathering THEOREM Then. the 6.. results. Ifスis. is. case. as. constant. Vn_2(kab) is. r>(”-1)(”-2)/2. curvature. not. The. the. state. in (43), then. u,e. 2. form. form. K.. a. can. of the Vn in Theorem (74), (75), or (76) if it admits. the fundamental. ”,ritten in the. we. obtained. a. Cn. a. motions. of. curvature. of. (ul)･. thatス-exp. is not. which. group. space of constaht. of the ￣comPleie group. rc. order. a. think. can. be. of order the. with. motions. can. admitted. scalar in each. follou)s,. ]垂+ j車J. ＼…‥‥.‥‥…=‥=‥…‥…‥(79) /. ト--+. (1ogg)′′幸0. ′′-0. /. g′幸0. t (log首)′′-0. -ぎjfg･･････-･･･---･･････-･･･(81). I三重j l垂l‥‥.…‥.‥‥…‥‥‥……‥‥‥….…‥……‥……‥.‥‥(82). i車卜[車fI麺L･.-･.-+--･･-････-･････.･･･-=･--･ From and of. 2, Theorem. Theorem. su氏cient order. is the. TEEOREM. 7.. A. u)ritten. one. of VnlO). of wder. the. complete. which rc. is not. u)hen. of the following. 6. w-e. can. (1) admit satisfyilig even We￣ can replace n(n＋1)/2-2n＋2,. ”(n＋1)/2-2n＋3, order. of motions in. Theorem. Vn. r>n(n＋1)/2-3n＋8.. γ｡. group. that. condition. -”(n＋1)/2-2n＋4, where. 4, and a. and. group a. Cn. of motions,. a. a. obtain G,. group. the. necessary. of motions by. inequality. rc. ”(n＋1)/2-2n＋1,10) and. the. get. a satisjies (1) admits complete be its fundamental form can u)hen. and. only. forms. for rc-”(n＋1)/212n＋4 (i.). ds2-(dul)2＋(cos. mul)2 (duョ)2＋kab(u3,-,un). dua. dub. (m幸P). ,. i. (io). ds2-(duュ)ヨ＋(duョ)2＋kab(u3,･･･, un) dua dub. (K幸0). (L). ds2-(dul)2＋e2肌滋1(duョ)2＋kab(u3,･･･, un) dua dub. (m≠0). ,. ,. for re-”(n＋1)/2-2n＋3 (ii) 10) -3n＋8,. ds2-(dul)2＋e2mul(du2)2＋e2aul((du3)2＋ It must. be--七eptin. -2n＋2>-3n＋8,. mind or. that. ”. -2n十1>13n＋8. is to. -. ＋(dun)2). (α幸0, ∽幸0,. be such that -2n＋4>-3n＋8, is satisfied respectively.. α幸∽), -2n＋3>.

(22) 22. Y.. Mute. for rc-”(n＋1)/2-2n＋2 (iii). ds2-(dul)2＋g(ul). (duョ)2＋kaむ(u3,-, un) diladub. (-÷告＋÷(3gl)2幸constant) ,. (iv). ds2-I(ul-ku2). (Ⅴ). ds2-I(ul) (duュ)2十g(ul)(dug)2＋e2ulkab(u3,･･･,dua dub. (vi). ds2-(dul)2＋g(ul-ku2). (duュ)2＋g(ul-ku2) (du2)2＋e2ul((duョ)2＋＋(dun)2). (f'幸0). (f'幸0,. un). ,. K幸0),. (duョ)2＋e2aul((du3)2＋･･･＋(dun)2) (a幸0, (1ogg)′′幸0) ,. 1. (vii). ds2-. (du上)2＋g(ul)(duョ)2＋e2aulkab(u3,.･･, un) duadub. (viii) ds2-. (a＋0, (logg)H＋0, K＋0),. dua duわ (duュ)2＋e27n?:(duョ)2十e2aulkaゎ(u3,-, un). (ix). ds2-(duュ)2＋(du2')2＋e2aulkab(u3,-, un) dua dub. (Ⅹ). ds2 -I(u2) (duュ)2＋(duョ)21#ul((duョ)2＋＋(dun)2). (a*0,. K＋0),. mj=0,. (a＋0, K*0), (′′幸0). ･-. ,. for rc-”(n＋1)/2-2n＋1 (Ⅹi). ds2-gpq(ul, u2) duP duq＋(A(ul, u2))2kab(u3,-, un) dua dub which. In this condition Applying. functions. theorem. lest C. can. constants. and. become. should. be written. not. in any. must. form. satisfy. ”)ritten above.. naturally. some. other. zero.. we the cited in Introduction obtain 8･ If C)〃川has the special form not C)/wu-C(g)ug/”-a)vg/1α) a,as in the theorem just mentioned, in wder that a u)hick given -(”-1)CM)/uレW is a Riemann Cn G, Vn, not n≧6, which admit space agroup of motions of o71der. the. theorem. TⅡEOREM. it is necessary. r>n(n＋1)/2-3n＋11, can. be u'rittenb in. of the forms. one. and. Y･. Mute,. On. of. motions. G,. conformally of. curved. the. fundamental. form. 7.. rLCeS. Ricmann. otrderLr>n(n＋1)/2-(3n-ll),. that. in Theorem. stated. :Refere [1]. sujncient. spaces. Vn,. Journal. of. a n≧6, admitting Math. Soc. Jap., 9. group. (1957),. 38-61.. [2] [3]. ∫.A. Scbouten, a The Yan?, Publishing. Ricci-Calculus, Theory. Coり1957.. of. Lie. 1954. Springer, edition, Derivatives its Applications, and. second. North･Ho11and.

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