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(1)On Frenet's Formula ln a K. ahler Space with Constant Holomorphic Sectjpnql Curvatqt/ .eBy Yosio MUTO-. S,'J ":.1･i .t･,. DePartment of Mathematics (Received May 31, 1965) gl･ Introductieni) The main purpose of the present paper is to construct the Frenet formula. for a complex analytic submanifold of real djmension 2 in a Kahler manifold with c,o,nst.ant holo,morphic sectiQnal curvatyre.. Let M be a Kahler manifQld gf real dim.ension IV=2n and S be a corrpplex analytic submanifold of real dimension 2 in M. For each point P of S there exists a neighborhood U of P such that UAS is given by the equations 6h,.6h(rpi, rp2) 2) where the functions 6h(rp") sati$fy. Oei orpb 4h==ha 06h orpa' 4h and h" being the complex stru,ctures of'M and S respectively. By a suitable choice of rp" we can write. (i.i) . gSlp}h.g;h,, gSi',Pi,h==-gi? and we shall use such parameters rp" throughout the present paper. We define Bbh as usual by. B,hL- gS;･ At each point P of S Bih, B2h are contravariant vectors of M spanning the tangent plane to S, while Bbi, ･･･,Bb" are IV covariant vectors of S. Let C,h, ･･･,CNh be 2n-2 orthonorma! vectors orthogonal to Bih, B,h. 1) Se.e Reference at the end of the paper. 2) We use indices as folloWs, h, i, 1', k, l, m= 1, ･･･,.2n (= N),. a, b, c, d, e, f= 1, 2,. x, y, 2= 3, ･･･,N.. .･'. tL'. '.

(2) 2 Y. MuT6. The reciprocal of the system (Bbh, C.h) is denoted by (B%, CXi), hence. Bai==tgabgihBbh,. CXi==gihCxh. S is a Kahler manifold with the fundamental tensor 'gb. and the complex structure fb" induced from the structure of M; '. 'gb.==BbiBahgih,. , vSt. i. i ･￡. fba==BbiBahEth. In our coordinates (rpi, rp2) we have. (1.2) . 'gba==g6ba ' '. /and ' '' '. Ai=h2=O, A2=-hi==1,. where. 6giOgh 06i06h. g=gih orpi orpi =gih Orp2 Orp2 '. The relation between the connection of M, the induced connection of S :and the second fundamental forms of S is given by. a.3) O. oBrpb,h +B,icB,m{,h.}-/{e,b}B,htHbbh. '. which we can write in the form. "(1･4) 7cBbh=Hl]bh=HbbXCxh･･In (1.3) {,h･i} is the Christoffel from gih and '{,"b} is the Christoffel from 'gb,.. 7 denotes covariant differentiation with respect to the Riemannian connection in M and at the same time with respect to the Riemannian connection in S as it is shown by the following examples. J7kT,h･== Oo{l/'h +{2.}Tim-{,m,}T.h, 11i･. 7dTih=:Baic7icTih if Tih is a tensor field in M. ti,. v,T,a= OoTf2" -LLi{da,}Tie-Belk{hmi}T.,a,. ' 7,s6h= ISrpb,h +B,le{,h.}s,m-c{sb}S,h if n", Sbh are vector fields of Min S for all fixed values of a, b and if na, Sbh are at the same time vector fields of S for all fixed values of i, h.. ' ' ' ' '' ' vdT,a= OoTrpb,a +r{da,}Tbe-/{sb}T," if Tb" is a tensor field of S.. '.

(3) On Frenet's Formula in aKahler Space 3 '. We write here some important equations in the geometry of subspaces. These are the,equations of Gauss, ,. '. <1･5) K]bjihBakBcj'BbiBah-'Ktzcba==Zx(HabxJEL]axTHbbxHdax), ?. a.. the equations of Codazzi,. l･. t. i. (1･6) KkjihBdicBcjBbiCxh=J7dHbbx-J7bHlibx+2y(H}byLdxy-HltbyLcxy),. '. r}clis-, ･L. and the equations of Ricci, ･. '(1･7) ' J7leLbyx'J7bLcyx ' '' '' ==K]bjihBckBbjCxiCyh+HleayHb"x-HbayHb". '. ' +Zz(LczyLbzx'LbzyLczx), ' 'gehfienreedigyh, 'Ktzcba are the curvature tensors of M) S respectively and L.y. is. <1-8) Layx=gihCyi7aCxh･ It must be kept in mind that from the definition of C.h we can lift and lower the indices x, y, ･･･ arbitrarily and. ' ' ' ,, 'vaCxh=lljCrp".h+Baic{khm}CxM '. '. by the definition of V... From (1.4) and (1.8) we get. <1.9) J71tC.h=:-Hizb.B.h'gb"+2yLdy.Cgh. '. S2.Thefirstnormals. .. '. When wh is an arbitrary contrw.a,v=a/;?.zt, .vector of M, dih is defined by. Letusput ' .. "(2.1) '' uh=Bih,. 9' ･-. i.x. . li} ･,lvl. ' fih=B2h.. Makinguseoftheoperators ' ' UVi==SYa, fiiJ71:fi2"7a. weget ' . ' Hi2x== (uZah) C.h, Hix = (UiLuh) C.h,. ,(2.2). Hlii.==(uNiV6uh)C.h,. . Hl]2x=(fiiLUNh)Cxh･ <2.3)ItmustbenOtiCedthat u,ta,.l.fizza,'. .and .. '.

(4) 4 -Y. MuT6 '. '. uNiLuNh==fiiL(uleE,h)==(uN.Zuk)ah =r(uiLuNk)f7lah=ul･L(u'V.icE6-h),. herice. (2.4) ai7itih==-uiV,uh,. (2.5) 42x=-Hiix' We define r by. s. "t. gr2= r2gihuiuh==1. We also define A, B,L by. ' A=r2(uiViuh)uh,B=r2(tiiViuh)uh, L==gih(uic7icui)(uj'V,･uh). .. Then. wehave . ' ,.. . ･ r2(uiLu'"h)uN,=r2(uNiLuh)uN, '. =-r2(uNe:LuNh)uh==r2(uiLuie)zah=A,. r2(uiLuh)u",==-r2(uZah)uh -' .=-r2(uNiLuh)uh==-B, r2(uNiL.uNh)uNh,=r2(uNiLuh)uh=B, ' gt,(zaNleJ7ku`)(u'"7'JZfuh)=g:ih(uicJ7kfii)(ujJZtifih)==L,. and moreover '. (uk7),uh) (uNi'lfu,) == (ukVi,uh) (ujLfih) = .Eh,i(ukl7]tuM) (uj71?･u`) ==O. becauseof(2.3),(2.4). ･' ' If we pu,t. vh==uiJ7iuh-Auh+Bah, (2.7). arh=ui'7iah-Afih--Buh,. we get 7h=vZah and. (2.8) v2=T2==L-g(A2+B2), wherev2==glehvivh. wefindal$otu,atruh,rah, Vvh , Tvh are orthonormalvectors.. We assume ' '. (2.9) L-g(A2+B2) tO and fix the vectors C,h,, C,h by the eqyations. ' (2.10). '' C3h=v-ivib, C4h=v-'i7h.. TheSFeroamre (t2h.2e) fiwrest gneotrmals. to S.. ),. e ,gi 1' :.

(5) On Frenet's Forniula' in aKahler Space 5 a13='ts23=V,. (2.11) Hl,,==4i4=V,. it14=4,4="23=413=O. 9. by vjrtue of (2.7) and (2.10). Since C,h (t=5, ･･･,N) are orthogonal to vh, 7h,. s. uh , ah, any other Hb.. vanishes because oif (2.2).. If we have L-g(A2+B2) =O at every point of S, then we have vh=:Th==O, hence. Hbah==O. Thus it is reason'able to assume (2.9).. From (1.5) we get. '. (2.12) iKl,,,=B,icB,dB,ZB,ipKlejih+2v2. ' can calculate iK12i2 direct!y from 'gba and obtain But we iKl2i2={i-[OiOig+O202g-t}-((Oig)2+(O2g)2)].. Thescalarcurvature'Kisthengivenby .' . (2.13) 'K=- gl, (O,O,g+O,O,g)+ gl, [(6,g)2+(O,g)2]. This proves that, if S is compact and g is not a constant, then /K can not have a constant sign in S.. '. '. g3. Equations of Codazzi and equations of Rieci in a space of constant holomorphic sectional curvature Now we assume M to' be a space of constant holomorphic sectional curvature, hence {3･1) Kilrjih=-l}[(gichgj,-gjhgici)+(Eich4i-4hFici)-2I7;bjJEih]･ ). From (1.6) 'we'get '. (3.2) 72Hlix-LHhix+Zy(HiiyL2xy-HhiyLi.,)==O, 1;. 72Hl2x-ViHl]2x+2y(H12yL2xy'it2yLixy)==O' Hereafter we use indices s, t, ･･･ =5, ･･･,N.. Putting x==tin (3.2) we obtairi'. (3.3) L,,3-L,,, =O,-.L,,,+L,t, =O by virtue of Hb.t=:O and (2.11). ' On the other hand we cah prove 4Hiix'ViHlirx==O2Hiix'O'i4ix,. LHI2x'ViHh2x==O2Hi2x-Oi42x straightforwardly using 'gba=gSb.. Hence vtie get. J･. h.

(6) 6 Y. MuTo. (3.4) 02v-vLi34=O, aiv-vL243=O by putting x==3 in (3.2). (3.3) and (3.4) are all that we can obtain from (1.6). (1.7) is equivalent to. e. <3･5) OiL2yx-02Liyx=HlayHli"x-HliayHi"x ' k.. +Zz(LigyL2zx-L2zyLizx)+ le2g IilthC"Cxh,. t. which contains four sets of equations. (3･6) OiL2ts'02Lits=￡z(LiztL2zs-L2ztLixs)+' le 2g 4hCtiCsh, (3･7) OiL23s'02Li3s==Li43L24s-L243Li4s+1￡t(Lit3L2ts-L2t3Lits), (3･8) ･ OiL24s-02Li4s=Li34L23s'L234Li3s+Zt(Lit4L2ts-L2t4Lits),. ' (3.g) o,L,,,-O,L,,,=-2 v2+x,(L,,,L,,,-L,,,Lit,)+ le2g ･. ' ' Substituting (3.3) into (3.9) we get '. '. (3.10) a,L,,,-a,L,,,== :-v2+ le2g-2,[(L,,,)2+(L,,,)2]. The vectors C3h, C4h satisfy the equation. C4h==a,h. We now choose Cth (t= 5, ･･･,N) so as to get. - tvC2mh=C2m-lh m==3, "'. ,n.. Then we have (9'ii) . L.:il,:li=-'L-"LY.1i,'.X,1i.Iify,xareodd and especially. (3.12) L13s=L146== L236=-L24,, . L136= -L14s= --L23s= -L246･. ' g4.' The Frenet formula Now let' us proceed to the Frenet formula.. As we have defined r by gr2==1, we get. A=-r,/r, B==-r,/r where r. :Or/Orpa. From (2.7) we immediately obtain. rukL(ruh) = r2vC,h+r,ruNh, '. ruicL(rfih)==r2vC4h-r2ruh,. <4.1). raic7ic(ruh) = r2vC4h-rirah, ruNk7)kr(ruNh) =-r2vC,h+riruh.. `.

(7) On Frenet's Formula in a Kahler Space 7' Since ruh, rah are orthonormal tangent vectors of S, r2v is a scalar which may be interpreted as playing the role of the first curvature.. The relation between the first curvature r2v and the curvature scalar /K e. of S is obtained by substituting (3.1) into (2.12), hence. (4.2) ' 'Kl,i,==2v2-kg2, .'K==-4rv2+2le. i. -- In such a set of formulas as (4.1) the second and the fourth ones are ob-tained directly from the first and the third ones by operating f. Hereafter we shall not write such superfluous formulas.. Multiplying (1.9) by r and putting x=3 we get. ruk7kC3h==r(-Hlb3B.h'gb"+ZyLiy,Cyh), rfiicJ7ibC3h= r(-4b3B.h 'gb"+ ZyL2y,Cyh) ,. which becomes on account of (1.2) and (2.11). ruk7)bC,h=-r2vruh+ZyrLiy3Cgyh, (4.3). rfik7]bC,h=r2vruNh+2yrL2y3Cyh. Hitherto the vectors Cth (t=5, ･･･,N) were chosen to be orthogonal to uh. and orthonormal. Now C,h is chosen in such manner that the six vectors zah, tih, C,h, CN ,h , rukLC,h, C,h are linearly dependent, hence. fih, C,h, CN ,h. (4.4) . L,,,==O t=6, ･･･,N. Since we have (3.3), (3.11) and (3.12), w,e get. (4.5) L,,,==L,,,=L,,,=L,,,=O t=7, ･･･,N, (4.6) L136=-L14s=-L2,s=-L246==O, hence uh, fih, C,h, CN ,h , fik7icC,h, C,h are also linearly dependent.. By such choice of vectors C,h (t==5, ･･･,N) we can write (4.3) in the form. rukl7kC3h=-r2vruh-rLi34C4h-rLi3sCsh, (4 .7) n. }. ruNle7kC3h=r2vruNh-rL234C4h-rLi3sC6h･. tv are completely We assume Li3siO and take Li,s<O. Then Csh and C,h=C,h determined by (4.7). These are the second normals of S. Again, we get from (1.9). (4.s) I rUicVkCsh=rZyLiysCyh=rLi3sC3h+rztL,t,c,h, t raicVkCsh=rZyL,ysCyh=rL24sC4h+r2tL2tsCth for we have Li4s==L23s==O･ Putting s==7, ･･･,Nin (3.7), (3.8) and taking (3.3), (3.11), (4.4) and Li3s,`O. lnto account, we get. (4.9) L,,,=--L,,,, L,,,=L,,, t==7,･･･,IV. We choose Cth (t =7, ･･･,N) so as to get. L,,,=O t=8, ･･･,N..

(8) ,8 Y. MuT6 'Th6n vvh ･obtaih. L(4.10) L,,,==L,,,=L,,, ==L,,,=Oi t=='9, ･:･,N'. ,. Llss=-L167=-L2s7=±-L2-6･sj=O,. <4.11). '. Lts7==L16s==L2ss=-L2･67.. ' we can write (4.8) in the form Taking th'ese fesults' in'to' account g. ' 7uktikbgh=rL,,,t,h-L:rLi,,66h'LLrLis7C7h,. <4.12) -'-. nikVicCsh==-rLi3sC4h-rL2s6C6h-rLis7Csh･. If we assume Lis7<O, C7h and Csh==a,h are completely determined by (4.12).. These are the third normals of S.. ' If we put t==s,6 afid s==g', ･･･,N in (3'.6) and tdke (4.10), (4.11) and Lis740 '. mto account, we get. <4.13) L,,,=-L,,,, L,,,=L,,, t==9,･･･,N. We choose Cth (t=9, ･･･,N) so as' fo' get. L,,,==O t'=10, ･･･,IV hrid 6:bthi･fi･. nticVicC7h=rLis7Csh-rLi7sCsh-rLi7gCgh, (4.14). rfileVicC7h==-rLis7C6h-rL27sCsh-rLi7gCioh･. Such process can be continued tintil We get. (4.15) L,,,..,,,==O t==2v+3, ny･･,IV or until we get rukJ7leCN-ih=rLi,N.-3,N-iCiv-bh-rLi,N--i,NCArh,. <4.16) ,･･･ , ･. rakVicCN.-ih= -rLi,N-3,N.-iCN.-2'h-rLifu-i,NCNh. '. If we get (4.ls) at every pdirit o'f' s, hii covafi5rit dekivatives of ruh, ra'h,. .-. C3h, C4h, ･･,r, ,･ {i,;2,+2h are line'ar cbtnbii atio'ns of the tangeiht vectors and the first. .. 2v normal vectofs.. ' Frenet formula Thus we have the '. ":, -t. tlkVlet2p+lh='Lpt2p--lh+Mlpt2p+2h+Lp÷lt2p'÷3'n,'. <4.17). t2kl7kt2p+'ih==Lpt2ph+Mlipt2p+2h+Lp+tt2p÷4'h,. tlkJ7]ict2p+2h=-Lpt2ph-Mltpt2phh+Lp÷ll2p"4hj. t2icVlet2p+2h=-Lpt2p-lh-Mllpt2p+lh-Lp+lt2p+3h. p=:O, ･･･,n-1. where tih=rzah, t2h=rfih, t.h=IC.h,. Lp=-rLi,2p-i,2p÷i p=2, ･･･,n-1, Map ='rLa,2p+i,2p+2 p= 1, ･･･,n-1,. ./. 1.

(9) On Frenevs Formula in a Kahler Space. 9. Lo=O, Li==r2V', L.=iO,. Mio=r2, cao=-ri, Md.==O. v. Il!lrestrewlaetihOanvebeftrWoemen(ih4)e COefficients Layx or Lp, Mdp is obtained as foiiows. '. 'Li34=02logv, L234=-Oilogv.. e. Substituting (3.12), (4.5), (4.6) into (3.10) we get. Li3s=-[r2v2+ 4ler2 -{i-(aiL234-02Li34)]VIF .'. '. {. Putting x=3, y=5 in (3.5) we get. .. i,li. ' ' 02Li3s=]Xz(Li3zL2zs-L23eLizs)--Li34L24s'L236Li6s,. lhence . 1og Li3s･ ' Lis6=Li34+02 Similarly we get. L2,6 = L234- Oi log Li3s ･. Putting s=5, t=:6 in (3.6) we get O,L,,,---O,L,,,==2,(L,,,L,,,-L,,,L,,,)- fe2g =- k2g --2(L,,,)2+2(L,,.)2,. ' Lis7==--[(Li3s)2+' 4ler2 --il'(6iL2s6-ahLis6)]-ll', Continuing such process we get Li,2p･-i,2p=Li,2p-3,2p-2+02IOgLi,2p--3,2p--i,. <4.18) L2,2p-i,2fo==L2,2p-3,2p--2-OilOgLi,2p-3,2p-i,. '. ,. Li,2p-i,2p÷i= - [(Li,2p--3,2'p--･i)2. + 4ler2 ---li-(aiL2,2'p'-i,2p-02Li,2p･-i,2p)]S'. '. (p =3, ･･･,nLl),. <4･ig) I2,i::".::J:::2:::#:,3:::.:±g:lg.:2i,:2,:-1'z::n.zll. ' ' We can calculate L.,. step by step by these equations. We get another formula by putting s==2n, t=2n--1, nainely, OiL2,2n-,2'n-O'2Li,2n･-i,2n = 2r2le+2(Li,2'n･-3,2n-i)2' But, after alli this' is an identity.. As for the coefficients Md, and L, we get. w. repF ' ti.

(10) 10 Y. MuT6 Ml ,p+i = Mlp- ra2 log Lp÷i+ r2 ,. (4.2o) da,pa+i=asp+rOilogLp+i-ri, L.÷,=[(L.)2+-ii-+-il-(o,ca.-o,M,,)+ r2lllClpIliriuap ]-}. '. p=O, 1, -･･,n-2 which we can also derive from (4.17) directly.. g･. '. g5. Change of local coordinates in the subspace Let us consider a change of variables. ca=4a(tyi, rp2). where 4a also satisfy .. g2Zi .E,h. g{,h.. Then we have the Cauchy-Riemann equations. 04i-042 OC2-OC2. Orpi Orp2' Orpi Orp2' Let us write. wh=06h z2>h=,06h. ' a42. O ci '. '. and denote the normals which play therole of C.h by D.h. Then we can. put. wh=auh+Pah,. (5.1) D3h=a2C3'V+P2C4h, D2n-lh==crnC2n-lh+PnC2nh,. where. Orpi Orp2 Orp2 Orpi a= OCi ; 042 ' P== 04i =- OC2 and. cr.2+P.2=1 P==2, ･･･,n. If we denote any linear combination of vectors uh, fih by (*), we have. w`Lwh==(aui+Pfii)7,(auh+Pah) ' ==a(crzai+Pu"i)Luh+P(crui+Pfii)7pth+(*). .=(cr2-p2)uipzuh+2crPuZfih+(*), from which we obtain. a2-P2 2aP a2== a2+ p2, P2==a2+p2･ If we denote any linear combination of vectors uh, fih, C.h (x=3, ･･･, 2p> by (*),.we have. -.

(11) On Frenet's Formula in a Kahler Space. ll. wiLD,,-ih=(aui+Pai)Z(cr.C,,.-ih+P,C2.h). ==ap(aui+PuNi)LC2p-ih. +P.(aui+Pfit)LC,.h+(*) e. ==(cr.a-P.P)uiLC,,-,h +(a,P+P.a)uiLC,,h-(*), b. from which we obtain ･ crp+1==. a.cr-P.P. cr.P+P.a. Va2+p2 ' Pp+i== vcr2+p2 '. Putting .. (5.2) '' ai==v.,a+p,==cos' 0, Pi=v.2P+p2==SinO,. we get ･ (5.3) a.==cospO,. P.=sinPO.. If we write. ' ri,= 1 'vigihwzwh ' t{h==r'wh, t5h==r'cbh, tSh=Dxh, we get Lfo==(t{ic7kt6p-ii)t2p+ihgih. from equations corresponding to (4.17). Applying (5.1) we find. (5･4) Lfo=Lp･ We also obtain. Mlp = (t(ic7ict6p+ii) t5p÷2hgih. =r'[(auk+Pfiic)Vk(crp+it2p+ii+Ppu-}-it2p+2i)]. (-Pp+it2p+ih+crp+it2p÷2h)gih. ,. ). - r.' (aMl.+pas.)+r'(--p.+i Oocr4P,"i +ap÷2 OoPEfi. and similarly '. Ms..,, ;' (-pM,,+evua.)+r'(--p.., 06a4P,"i +a.+i OaPcP,"i ).. Since we have ' rl =. r Va2+p2. and (5.3), we get ･. Ml.==Mi.cos0+capsin0+ v.,r+p,,(P+1) oOcOi ,. (5.5). Ms..==-M,.sino+Mh,coso+ v.,r+p, (p+1) oOcO, ..

(12) 12 ' Y. M' '. '. uT6. ' Thus L. are invar'iantS' of the' s;uibsipace S,' while. Mdp are not lnvarlants,. the law of transformation being given' by (5.5).. '. ,' l Reference. K. YANo, Differential geometry on complex and almost Press, 1965.. '. complex spaces,. Pergamon G. 'a. f.

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