Infinitesimal Variations of Hypersurfaces of a Kaehlerian Manifold

J. Math. Soc. Japan Vol. 29, No. 2, 1977

Infinitesimal variations of hypersurfaces of a Kaehlerian manifold

Dedicated to Prof. Shokichi Iyanaga on his seventieth birthday

By Kentaro YANG

(Received Feb. 10, 1976) (Revised May 27, 1976)

§ 0. Introduction. B. Y. Chen [1, 2, 3], C. S. Houh [2], S. Tachibana [10] and the present author [3, 9, 10] have recently studied infinitesimal variations of submanif olds of Riemannian and Kaehlerian manifolds. (See also [7].) The main purpose of the present paper is to study infinitesimal variations of hypersurfaces of a Kaehlerian manifold, to obtain variations of structure tensors of the almost contact metric structure induced on the hypersurf aces from the Kaehlerian structure of the ambient manifold and to prove theorems on variations of Sasakian hypersurf aces with f -sectional curvature a2. (For the definition of a Sasakian manifold with f -sectional curvature a2, see § 1.) In § 1 we state some preliminaries on almost contact metric structure induced on a hypersurface from the Kaehlerian structure of the ambient manifold. In § 2 we consider infinitesimal variation of a hypersurface of a Kaehlerian manifold and obtain variations of structure tensors of almost contact metric structure induced on the hypersurf ace. § 3 is devoted to the study of what the author calls parallel variation and § 4 to that of variation of the second fundamental form of the hypersurface. In the last § 5, we consider variations of a Sasakian hypersurf ace with f sectional curvature a2 and obtain the condition that an infinitesimal variation carries a Sasakian hypersurf ace with f -sectional curvature a2 into a hypersurface of the same kind.

§ 1. Preliminaries. Let Men+2 (n>1) be a real (2n+2)-dimensional almost Hermitian manifold covered by a system of coordinate neighborhoods {U; xh} and Fih be its almost complex structure tensor and g;1 its almost Hermitian metric tensor, where 288 K. YANo and in the sequel the indices h, i, j, k, run over the range {1', 2', , (2n+2)'}. Then we have (1.1) FitFth==-az and (1.2) FjtFisgts=gji Let M2n+1be a (2nd 1)-dimensional orientable Riemannian manifold covered by a system of coordinate neighborhoods { V; ya} and g,b be its fundamental metric tensor, where and in the sequel the indices a, b, c, run over the range {1, 2, , 2n+1}. We assume that M2n+1is isometrically immersed into M2n+2 by the immersion i ; M2 1-M22 and identify i(M21) with M2n+1. We represent the immersion by (1.3) xh=xh(ya) and put (1.4) Bbh= abxh , (ab= aI a?) . Then Bbh are 2n+1 linearly independent vectors of M2n+2tangent to M2n+1 Since the immersion is isometric we have

(1.5) gcb=gjiBciBb2 We represent the unit normal to M2n+1by Ch. We then have

(1.6) gj1BbiCi=O and (1.7) gjiCjC1 =1 . Now the transform FitBbz of Bbi by Fih can be written as

(1.8) Fit BbI =f baBah+fbCh, where f ba is a tensor field of type (1, 1) and f b a 1-form of M2n+1and the transform FihC1 of C by Fih, being orthogonal to Ch, can be written as

(1.9) FihCI = - f aBah , where f a=f bgba is a vector field of M2n+1,gba being contravariant components of gcb. Applying F to the both sides of (1.8) and using (1.1), (1.8) and (1.9), we find

(1.10) fbefea-~Ub~fbfa, fbefe-O. Applying F to the both sides of (1.9) and using (1.1), (1.8) and (1.9), we Infinitesimal variations of hypersurfaces 289 find (1.11) feafe=0, fefe`=1. From (1.2), (1.5), (1.6), (1.7) and (1.8), we have

(1.12) f cefbdged=gcbrfcfb Equations (1.10), (1.11) and (1.12;) show that the set (f, gebff b) defines the so-called almost contact metric structure on Men+1 (Tashiro [6]). We denote by T the Christoffel symbols formed with g;1 and by F b those formed with geb. Then it is well known that F and T b are related by

(1.13) r ~= (aCBbhI 1 ~iBcb)Bah, where Bc(=BC'Bbti and Bah=BbigbagZh We define the van; der, Waerden-Bor- tolotti covariant derivative of Bb' along Men+1 by

(1.14) VcBbh= a~Bbh+r, Br bBah and that of C h by (1.15) V~Ch=dCCh+r BcjCi. Then equations of Gauss can be written as

(1.16) VCBbh=hcbCh, where hCbis the second fundamental tensor of Men+1 and equations of Wein- garten as (1.17) VcCh=-hcaBah, where hca= hcbgba Equations of Gauss and Codazzii are respectively written as

(1.18) Kdcba= KkjihBdcbh+hdahcb-hcahdb and (1.19) KkjihBdcbCh= V dhcb_V chdb , Kdcbaand Kkjih being curvature tensors of Men+1and Men+2 respectively, where Bdcbh=BdkBCJBb2Bah,Bdcb=BdkBciBb2 and Ch= C1gih. We now assume that the ambient manifold Men+2 is Kaehlerian, that is,

(1.20) V,F~h=0, where V; denotes the operator of covariant differentiation with respect to F. Differentiating (1.8) covariantly along Men+1 and using (1.16), (1.17) and 290 K. YANo

(1.20), we find (1.21) Qcfba= -hcbf a+hcafb and (1.22) Qcf b = -hcef be. Differentiating (1.9) covariantly along Men+1 and using (1.16), (1.17) and (1.20), we find (1.23) Qcf a= hcefea . Equations (1.22)`and (1.23) are equivalent because of the relation f ba-fbehea =-fab~ We now consider the tensor field Scba defined by

(1.24) Scba= Ncba+ (Qcfb-Qbf c)f a , where Ncba is the Ni jenhuis tensor formed with f ba:

(1.25) Ncba_'feeQefba-J i, efea-(Qefbe-Qbfce)fea . When the tensor field Scba vanishes identically, the almost contact metric structure (f b', gcb, fb) is said to be normal. Substituting (1.21) and (1.22) into (1.24), we find

(1.26) Scba'_ (fcehea`-heefea)fb-(fbehea^hbefea)fc Thus if the almost contact metric structure is normal, we have

(f eehea-hcefea)f b (fbehea-hbefea)f c - , from which, using (1.10) and (1.11),

(1.27) f eehea-hcefea = 0 and (1.28) heafe - hfa , where h=hc6 fc fb. When the almost contact metric structure (f a, gcb,f b) satisfies

(1.29) Qcfb-Qbfc=2fcb, the structure is said to be contact. Substituting (1.22) into (1.29), we obtain

(1.30) fcehea+"hcef ea = 2f ca . When the almost contact metric structure is normal and contact, we call the structure a Sasakian structure [5]. For a Sasakian structure, we have Infinitesimal variations of hypersurfaces 291

(1.27) and (1.30) and consequently (1.31) f cehea-f ca , from which, using (1.10), (1.11) and (1.28),

(1.32) hcb=geb+(h-1)fcfb For a Sasakian structure, (1.21), (1.22) and (1.23) reduce respectively to

(1.33) Qcfba--gcbf a+~afb (1.34) Qcfb=fcb and (1.35) Qbf = f ba . But if, instead of (1.29), we assume (1.36) Qcf b-abf c= 2af eb, a being a function, we obtain, substituting (1.22) into (1.36),

(1.37) fcehea+heef ea= 2afca. Thus, from (1.27) and (1.37), we find

(L38) f cehea= of ca from which, using (1.10), (1.11) and (1.28),

(1.39) heb=agcb+(h-a)fcfb Thus (1.21), (1.22) and (1.23) reduce respectively to

(1.40) Qcfba= a(--gcbf a+ ~cfb) ,

(1.41) Qcf b = of cb and (1.42) Qbfa = a fba . From (1.36) and (1.40), we have Qd(Qcfb-Qbfc) = 2(Qda)f cb+ 2a2(-gdcfb~gdbfc) , from which, using the Ricci formula and the Bianchi identity, we obtain

(Qda)f cb+(Qba)f dcT (Qca)fbd =0. Thus if n> 1, we have (1.43) a = constant. 292 K. YANo

For a Sasakian structure, we have, from (1.33) and (1.35),

(1.44) Kcbeaf e= oaf oafc

which shows that the sectional curvature with respect to a plane section con- taining f a is always 1. But for the structure under consideration, we have, from (1.40) and (1.42),

(1.45) Kcbeaf e _- a2(Oafb Ubfc) ,

which shows that the sectional curvature with respect to a plane section con- taining f a is always a2. So we call an almost metric structure satisfying (1.40), (1.41) and (1.42) a Sasakian structure with f -sectional curvature a2 (Okumura [4]).

§ 2. Infinitesimal variation of a hypersurf ace of a Kaehlerian manifold.

We now consider an infinitesimal variation of the hypersurface M2n+1 in n f2n+2 given by

(2.1)h=xh~hCY)~ h being a vector field of M2n+2 defined along M2n+1 , where s is an infinitesimal. We then have

(2.2) Bbh = Bbh+ (ab h)~ ,

where Bbh=abxh are 2n+1 linearly independent vectors tangent to the varied hypersurface at the varied point (xh). We displace vectors Bbh parallelly from the varied point (.Ch) to the original point (xh) and obtain Bbh^ Bbh-+ ji(x+~)E'Bb~E ,

N (2.3) Bbh_ Bbh+(Qbh)~ , neglecting the terms of order higher than one with respect to ~, where

(2.4) Qbh=~bh-~ I';2Bb~~. In the sequel we always neglect terms of order higher than one with respect to e. Putting

N (2.5) Bbh=Bbh-Bbh , we have

N Infinitesimal variations of hypersurfaces 293

(2.6) ~Bbh== (Qbh)~ . If we put (2.7)h =aBah+ ~Ch , ea and A being respectively a vector field and a scalar function on Men+1,we have (2.8) Qb h=(Qbr_Ahba)Bah+(QbA+hbe e)Ch We denote by Ch the unit normal to the varied hypersurf ace. We displace Ch parallelly from the paint (xh) to (xh) and obtain (2.9) We put (2.10) ~Ch= Ch-Ch . Then OCh, being orthogonal to Ch, is of the form (2.11) OCh _ ~aBahE ~a being a vector field on Men+i Thus from (2.9), (2.10) and (2.11), we have (2.12) Ch = Now applying the operator o to Bb'Cigji=0 and using ;(2.6) and ;ogji=0, we find (Qb ~)CZgjir Bb'~IaBatigji._ 0, from which, using (2.8), (2.13) '2b- -(Qb2+hbar), where 7b=~legeb. Thus (2.11) can be written as (2.14) Ch= -(QaA+hbab)Bah5 , where Qa=gabQb and (2.12) as (2.15) Ch=Ch_ftjCti~-(paA+hbab)BahS, Now applying the operator o to (1.8) and using oFih=O, (2.6) and (2.14), we find Fih(Qb 2)e-(Ufba)Bah+fba(Qa h)~+(ofb)Ch-fb(QaA+hea e)Ba', from which, using (1.8), (1.9) and (2.8), C(Qbee_Ahbe)(feaBah+feCh)-(abA+hbee)f aBahls =(Ofba)Bah+Lfbe(QeEa-Ahea)Bah+fbe(QeA+heaa)Ch16

+(ofb)Ch-fb(QaA+heaee)BahB 294 K. YANo

and consequently comparing tangential and normal parts, we have

(2.16) ~fba=[(Oboe-Ahbe)fea-fbe(Qeea-2hea)

-(QbA+hb ee)f a+fb(Q(LA+heae)]~ and

(2.17) Ufb= [(Qbee-2hbe)fe-fbeCQe~-~hear)]~ . Using (1.21), we can write (2.16) in the form

(2.18) ~fba= [l.fba+ (fbehea__hbefea)+fb(QaA)-(V2)f] baE r where J'f ba denotes the Lie derivative of f ba with respect to the vector field ea in MZn+1[8], that is,

(2.19) -Cf ba_' eevef ba+(V b e)fea-(Qe a)f be. Using (1.22) we can write (2.17) in the form

(2.26) (f b= [.E f b_Ahbefe- fbCQeA]~ where £f 6 denotes the Lie derivative of f b with respect to ea, that is,

(2.21) £fb =eVefb+(Qbe)fe Next applying the operator d to (1.9) and using oFih=O, (2.6) and (2.14), we find __F2h(QaA+hbab)Bai~=-(o f a)$ah _f aVah~ or using (1.8) and (2.8),

(V A+hbe b)(feaBah+feCh)E = -(of a)Bah-f e[(Qer_Ahea)Bah+(QeA+hecEC)Ch]~

from which, comparing the tangential parts,

(2.22) of a= [(DCA+hbe~b)fea-fe(Qer_2hea)]~ . Using (1.23), we can write (2.22) in the form

(2.23) of a= [aL'fa-f-CDe~)fea~~fehea]~ where l'f a denotes the Lie derivative of f a with respect to r, that is,

(2.24) £f a = ecvcf a_ (Qer)f e . Applying the operator 6 to (1.5) and using 6g,1=0, (2.6) and (2.8), we find

(2.25) Ugcb- [°Cgcb-Z~hcb]~ where £gcb denotes the Lie derivative of gcb with respect to ea, that is, Infinitesimal variations of hypersurfaces 295

(2.26) -Cgcb `- Qcb+Qbc s (b -cgcb)

Thus summing up we have THEOREM 2.1. Under an infinitesimal variation (2.1) of a hypersurf ace of a Kaehlerian manifold, the variations of structure tensors of the almost contact metric structure induced on the hypersurf ace are given by { ba-CCfba+2(fbehea-hbefea)+fb(QaA)-(Qb2)f ajj ofb - C-~fb-~hbefe-fbeQe~~~ (2.27) of a =[-Cfa+(Qe2)fea+2f ehea]r ~gcb = [-Cg~b-2Ahcb]r. COROLLARY 1. Under an infinitesimal tangential variation xh=xh+rBan~ of a hypersurf ace of a Kaehlerian manifold, the variations of the structure tensors of the almost contact metric structure induced on the hypersurf ace are given by their Lie derivatives with respect to r. COROLLARY2. Under an infinitesimal normal variation xh=xh+2Chs of a hypersurface of a Kaehlerian manifold, the variations of the structure tensors of the almost contact metric structure induced on the hypersurf ace are given by

f a_ a,S Ofb = -C2hbefe+fbeVe2lr, , (2.28) {a = C(Qe2)f ea+Af eheaj Ogcb 2AhCbs. From (1.27), (1.28), (1.39) and Theorem 2.1, we have COROLLARY3. Suppose that the almost contact metric structure induced on a hypersurface of a Kaehlerian manifold is a Sasakian structure with f -sectional curvature a2. Then under an infinitesimal variation (2.1) of the hypersurface the variations of the structure tensors are given by

Ofba= C-Cfba+fb(Qa2)-(Qb~)f aJ~ Ofb = C-Cfb-2h fb-f beVC,Js , (2.29) O f a = C-Cf a+V e2)fea+Ahf a,s ~gcb -- C-~gcb-"4~ {agcb+(h-a)f cf b~~E .

§ 3. Parallel variations.

We consider an infinitesimal variation (2.1) of a hypersurf ace of a Kaeh- lerian manifold. When the tangent space at a point (xh) of the original

E. 296 K. YANo

hypersurf ace and that at the corresponding point (x") of the varied hypersurface are parallel, we say that the variation is parallel. From (2.5), (2.6) and (2.8), we have

N (3.1) Bbh= [oo+(Qbr_Tba)~JBah+(QbA+hbeEe)ChS , and cosequently we have THEOREM 3.1. In order for an infinitesimal variation (2.1) of a hypersurface to be parallel it is necessary and sufficient that

(3.2) Qb2+hbeee= 0. COROLLARY1. In order for an infinitesimal normal variation xh=xh+2ChE of a hypersurf ace to be parallel, it is necessary and sufficient that 2= constant. From Theorem 2.1 and this corollary, we have COROLLARY2. Under an infinitesimal parallel normal variation xh=xh+AChs (2= constant), the variations of the structure tensors are given by

Ufba- 2(fbehea-hbefea of b = -~hbef e~ , (3.3) ~ f a = ~heaf e~ ogcb= -2Ahebr, 2 being a constant. COROLLARY3. Suppose that the almost contact metric structure induced on a hypersurf ace of a Kaehlerian manifold is a Sasakian structure with f -sectional curvature a2. Then under an infinitesimal parallel normal variation of the hypersurf ace, the variations of the structure tensors are given by

fba=O, ~fb = (3.4) of a = Ahf ar

Ogeb= -22{agebH-(h-a)fcfb}S 2 being a constant.

§ 4. Variation of the second fundamental form.

To find the variation of the second fundamental form we have used in [9] the formula C~QeBbh-Qef~Bbh=KkjihkBcbE-(OF~b)Bah,

where OF b is the variation of the Christoffel symbols Feb of the hypersurface. But here we shall use another method to find the variation of the second Infinitesimal variations of hypersurfaces 297 fundamental form. For the varied hypersurf ace, we have

(4.1) VcBbh= hebCh. Thus denoting by ['+o[b' b the Christoffel symbols and by heb+oheb the second fundamental form of the varied hypersurface, we have

(4.2) acBbh+rja(x l ~)Bc'B1 -(! (!g+U cb)Bah=(hcb+ohcb)Ch Substituting (2.2) and (2.12) into (4.2), we find by a straightforward com- putation (4.3) (VcV b h+Kkjih kBcb)s= [U b-hcb(V a2+ heaee)~~Bah+(ohcb)Ch• On the other hand, using (2.7) and (2.8), we have

(4.4) VcV b h+Kkjih kBcb= CVcV br-V C(Ahba)-hca(VbA+hbeee)l Bah +LV cVbA+V c(hbeS~e)+hce(V b e__2hbe)1Ch +KkjihBdcbd+2KkjihCkBcb• Thus from (4.3) and (4.4) we obtain

(4.5) orb-hcb(V cA+he(ee)S _ CVcV br-V(2h) -hca(V b2+hbe e)+KkjihBdcbhed T kjihC kBcbhJ~ where B~bh=BC~BbiBah and

(4.6) ohcb= LVcV bA+V c(1 be e)+hce(Vbee-2hbe) +KkjihBdcbChd+2KkjihCkBcbCh1E

Thus using equations of Gauss (1.18) and those of Codazzi (1.19), we have from (4.5) (4.7) UTb= [CJab-Vc(2hba)-ab(2hea)+Va(2hcb)~~ , where £l cb denotes the Lie derivative of E, that is,

(4.8) £L eb=VCVbS"+Kdcbad, and from (4.6)

(4.9) ohcb= C°ehcb+VcVbA+A(KkjihC kBcbCh-hcehbe)~~ , where £heb denotes the Lie derivative of heb, that is,

(4.10) -ehcb-dVdhcb+hceV be+hebV ce . 298 K. YANo

Thus we have THEOREM4.1. Under an infinitesimal variation (2.1) of a hypersurface, the variations of the Christoffel symbols and the second fundamental form are respec- tive given by (4.7) and (4.9). COROLLARY1. Under an infinitesimal tangential variation xh=xh+rBahs of a hypersurface, the variations of the Christ offel symbols and the second- fundamental tensor are given by their Lie derivatives with respect to r. COROLLARY2. Under an infinitesimal normal variation xh=xh±2Chs of a hypersuaface, the variations of the Christ offel symbols and the second fundamental tensor are respectively given by

(4.11) ~T b= -Cac(2hba)+Qb(Ahca)-Qa(Ahcb)1s and (4.12) ohCb=[QcVb2+2(KkjihCkB~bCh-hcehbe)1E.

§ 5. Variation of Sasakian hypersurf ace with f-sectional curvature a2. We now assume that the almost contact metric structure induced on the hypersurface is a Sasakian structure with f -sectional curvature a2. Then we have (1.39) and consequently

(5.1) hCehbe=a2gcb+(h2-a2)fcfb Thus (4.9) reduces to

(5.2) ohcb=[a(-Cgcb)+(°Ch)fcfb+(h-a){(Cfc)fb+fc(Cfb)} +QcVb2+/~{Kk jihC kBcbCh-a2gcb-(h2-a2)f cf b}~s . On the other hand, we have ~h = o(hcbfcf b)- (Uhcb)fcf b+2hcb(of c)fb .

Thus substituting the third equation of (2.29) and (5.2) into the equation above, we find

(5.3) oh = [l'h-F (DeVd2+2Kk jihC kBich)f ofd+2h21s . Now in order for the variation to carry the Sasakian hypersurf ace with f -sectional curvature a2 into a Sasakian hypersurface with f-sectional curvature a2-F0a2, it is necessary and sufficient that Ohebgiven by (5.2) is equal to

(5.4) Ohcb- ~Cagcb+(h-a)fcfbJ that is, to Infinitesimal variations of hypersurfaces 299

(5.5) ohcb-C~a)g~b~aC~gcb)-ECdh--~a)fcfb~Ch-a){C~fc)fb~fcC~fb)} Substituting the second and the fourth equations of (2.29) into (5.5), we see that (5.4) is equivalent to

(5.6) ohcb-Ca(.Cgcb-22ageb)-22(h2-a2)fCfb +(h-a) {(.Efc)fb+fC(J1'fb)-(fcefb+fbefc)(Ve2)}J€ +C~a)(gcb`fcfb)+(~h)fcfb Comparing (5.2) with (5.6), we find

[VCVb2+2{KkjjhGkBcbch+a2gcb+(h2-a2)fcfb} +(h-a) {f cefb~ f befc} (V,A) -(oa)g cb-(oh-J hs-oa)fcfb=6 from which, using (5.3)

(5.7) CVcV b2+2{KkjihCkBcbch+a2(gcb-fcfb)} -(V CVdA+2KkjihCkBedCh)fefdfcfb+(h-a)(feefb+fbefe)(V e2)1s -(oa)(g ~b-f cf b)= Q. Conversely if 2 satisfies an equation of the form

(5.8) VCVb2+A{KkjtihCkBcbch+a2(gcb-fcfb)} -(V eVd2+2KkjihCkBedCh)fefdfcfb +Ch`a)(fcefb+fbefe)(Ve~)-a'(gcb-fefb) ~ a' being a function, then (5.2) becomes

~hcb- CaC~gcb-2~IXgcb)-22(h2-a2)f cf b + (h-a) {(Ifc)fb+f e(°~fb)-(f cefb+f befc)(V e2)} +(VeVd2+AKkjihCkB)dCh)fofd+~Ch+2h2}fcfb which shows, together with (5.3), that (5.6) is satisfied, if we put oa=a's. Thus we have THEOREM5.1. In order for an infinitesimal variation (2.1) to carry a Sasakian hypersurf ace with f -sectional curvature a2 into a Sasakian hypersurf ace with f sectional curvature a2---oa2, it is necessary and sufficient that the function 2 satisfies (5.7). COROLLARY1. In order for an infinitesimal normal parallel variation 2h= 300 K. YANo xh+2Ch~, A=eonst. *0, to carry a Sasakian hypersurface with f-sectional curvature a2 into a Sasakian hypersurf ace with f -sectional curvature a2+oa2, it is necessary and sufficient that the original hypersurf ace satisfies (5.9) AL{KkjihCkBcUCh+a2(gcb-fcfb)} -KkjihCkBedC.hfefdf cfb1~-(oa)(gcb-fcfb)=0 . COROLLARY2. An infinitesimal normal parallel variation xh=xh+2Chs of a Sasakian hypersurface with f-sectional curvature a2 of a flat Kaehlerian manifold carries it into a Sasakian hypersurf ace with f -sectional curvature a2+5a2, if and only if Aa2s=oa. COROLLARY3. An infinitesimal normal parallel variation never carries a Sasakian hypersurface (with f-sectional curvature 1) of a flat Kaehlerian manifold into a Sasakian hypersurf ace. Suppose that the ambient Kaehlerian manifold Men+2 is of constant holomorphic sectional curvature k, that is, k S g Kkjih= 4 [okg _ ki+FkhFj~-FjhFki-2FkjFih] , where Fj2=Fjtgt1. Then (5.9) gives

a, k +a2 E-~a (gab-fcfb)=0. 4 Thus we have COROLLARY4. If an infinitesimal normal parallel variation xh=xh+ACh~ carries a Sasakian hypersurface with f -sectional curvature a2 of a Kaehlerian manifold of constant holomorphic sectional curvature k into a Sasakian hypersurface with f -sectional curvature a2+oa2, then we have

oa = 2 k + a2 s . 4

COROLLARY5. If an infinitesimal normal parallel variation carries a Sasa- kian hypersurf ace (with f -sectional curvature 1) of a Kaehlerian manifold of constant holomorphic sectional curvature k into a Sasakian hypersurf ace, then we have k=-4. The author wishes to express here his sincere gratitude to the referee whose suggestions improved the paper very much.

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Kentaro YANo Department of Mathematics Tokyo Institute of Technology Oh-okayama, Meguro-ku Tokyo, Japan