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(1)Qn Some Minimal Varieties in Sphere '. By '. '. '. Shigeo AKIBA and Toshihiko MAEDA tt. (Received May 31, 1974). '. ' gO. Introduction. '. ' ' Let Mn be a riemannian manifold and let f:Mn-->Sn'i be an isometric minimal immersio' n. H.B. Lawson has proved in [4], theorem 1 and 2, that if the scalar curvature of M" is identically equal to (n-2)/(n-1) or if the Ricci curvature of Mn is covariant constant, then, up to rotations-of S""i, r(M", f) is an open submanifold of one of the minimal products,. Sle(Vfe/n)×S"-le(V(n-k)/n) for le=O,･･･,[n/2]. ' ' idea of Lawson's proof of the theorem is that if the'second The central fundamental form is covariant constant, then (M", f) is an open submanifold tof one of the minimal products given above ([4] Proposition 1). Now let x==(xi,･･･,xr)==(xl,･･･,xei'i;･･･;xl,･･･,xer"i) be a vector of. r'. .Rn+r= Rlei+i × ･･･ × Rfer+i, where 2 lei== n. Submanifolds of SntP of codimen-. :sion P== r-1: ' i==1. ' 'i=1,･･･,r}. ' ' {xERn+r;leS+i(xy)2-lei/n,. j=1 -. :aredengtedby'. '- . ' ' Mle,,-,le.==Ski(Vle,/n)×･･･xSkr(Vle./n), ' twher"e each ki>O. These are minimal submanifolds of S"'P, P=r-1, intro-. ducedbyChern,doCarmoandKobayashi[1]. ' ,. The main purpose of this paper is to prove theorem 1 in g2 which. ･characterizes minimal submanifolds Mk,,,..,k. of Sn'P of codimension P up to. .rigidmotionsinS"'P. -･ . ' ' ' ' '. '. '. tt '. '. gl. Definitions and notations ' ' ' ' '. n+1 ' 'where E"'i'is the (n+1)Let S"(r)--{(xi,･･･,x.+i)EEn"i;2x2k=r2}, k==1 -. dimensional euclidean space. Let ip:Mn.Sn'P(1) be an isometric immersion. We shall introduce the notations developed in the paper of J. Simons [5]. The tangent and the normal bundles on Mn are denoted by T(Mn) and N(M"), Tespectively, and the inner products in these bundles are indiscriminately.

(2) 6 S. AKiBA and T. MAEDA ' written as <,>. The riemannian connections on M", S""P and En'P'i aret. NI and7 respectively. ' denoted by V,V. Let xET.(M") and let veAll.(M"). Extend v to an arbitrary vector' field V in Sn'P, such that V is normal to di(Mn) in a neighbourhood of m. Then the second fundamental from A of the immersion is the cross-section. in the bundle Hom(N(M"), S(Mn)), which, at m E M", is given by. ' iN' - , AV(x)=-(V.V)', . ' '. '. i. where S(Mn) is the bundle of M" whose fibre is the space of symmetric. Iinear transformations of T.(M"), and where ( )T denotes the orthogonaL. projectiontothetangentspace. - ' , Let x, yE T.(M"). We define B(x, y) E Nh(M") by <B(x, y), v> = <A"(x), y)> .. , Let (ei,･･･,e.) be an orthonormal framein T.(M"). The immersion di is.. called minimal if the mean curvature ' ' n 2 B(ei, ei) i=1. isNow,identically on Mn. . ' 'zero we present some fundamental equations and formulas which are;. usedinS2. ･ '7.y == 7.y+B(x, y) (1) '. '. '. '. fi.v= --AV(x)+7-.v (2) <R.,x, w> = <x, w><y, 2>-<x, 2><y, zu>-<B(x, z), B(y, w)>. +<B(x, w), B(y, 2)>, (3) where R is the curvature tensor of M". These are the Gauss' formula (1), the Weingarten's formula (2), and the: Gauss curvature equation (3). Now, we define some operators on the cross-sections of N(M"), which is. used in g3. For the detail, see [5]. Let V b.e, a cross-section of N(Mn) such. that V(m)=v. We define a Coo cross-section A of Hom(Al(M"), N(Mn)) which,.. at mEM,AJis given by ' ･ n' <A(v),w>=i=,<AV(e,),AW(ei)>, for wEAipt(Mn). ' where (ei, ･･･,en) is an orthonormal frame in T.(M").. We set ,' 'N "' AX. .t. '. R(V) = iE.l], (Reivei)N ,. where RN is the curvature tensor in S"'P, and where ( )N is the orthogonaE projection to N(Mn). We define an operator 72 on the space of cross-sections･. of N(Mn) such that '' '. e. `.

(3) On Some Minimal Varieties in Sphere 7 ' tt ' ' ' ,n-- . 72(V)=,2=,(t7e,(t7V))(ei) ,'. '. '. .. v. ' (T(Mn), N(Mn)) induced s Yrhoemrefie- denotes the connection on N(Mn) oF on Hom '. '. '. '. tt .tt tttt t. '. '. g2.Localrigiditytheorem -' . .. ' ' .Let M" be an n-dimehsional riemannian manifold and let di:Mn.Sn'P be an isometric minimal immersion. We'assume the following conditions. on the immersion di:Mn-sn+p: ' (i) thesecondfundamentalftomAiscovariantconstant. , (ii) there exists a global parallel normal frame field (Vi,･･･,Vp) on ￠(M"), i. e. the holonomy group of normal frame bundle of (M", di) contains. onlytheuniteiement. ' '' :'-. ,.,h(i lih).,ag.S,OhM,e, i,Oie.nt,ge.E.M,.n',lgh,efie.,gl,i,StS,fa,hf,ra,M,,e,ki'iu'.'de.n.) ,O.ft.T/miCl,Ii2). A"foranynormalvectorvofdi(M")atm,i=1,･･･,n, ･,' . REMARK. Under the condition (i) and (ii),"at some point" in (iii) implies. ss i. )J ･. at 2IVHeEroYRPEOMMIt.' under the condition (i), (ii'5 and (iii), we have that di(M") is'. anopensubmanifoldofoneoftheminimalProdzactsMki,･･･,k.･ . Note that if P == 1, this theorem reduces to proposition 1 of H. B. Lawson. [4]. ･ .' '.,'･ r･. PRooF. Fix a point m E M and let (ei, ･･･,e.) be aframe of T.(M) which. satisfies the condition (iii), and we fix a normal parallel frame field (Vi, ･･･, Vp). onceforall. Put , ' ' '. ' A"iej=2S･ej i=:1,･･･,P,7'=1,･･･,n,atmEM, ' '. where vi == V,(m). And put ' '' ' tt･2i=(･Rl,･･･,R,P･)'i=1,･･･,n. ''. .t. ' and denote by Ele the Let {1"i, ･･･,1".} be a maximal set of distinct tuples, 2ljeghoSwPasCe C9rreSPOndipg.to 2"fe･ , Then the simiiar gonsideration to that'in. .t. '. tttt tt. <Reiedei,ej>==O for eiGEiandejEEjwheni#1'... , LEMMA 1. 1]IC A"-LO for any vit O at m, then there are exactly P+1 distinct tuPles ' 2" ,,. ･･･,R" ..,,. and ]2"il'2= ]Sb (21･')2=ni/fei, where lei== dim.Ei and ni. .. j'=.1･., ., .. PROOF. Let (ei, ･･･,e.) be an orthonormal frame fixed at m. The Gauss. curvature equation (3) reduces to ･.. '. '. O=-<R,,.i.e,,e,>=1+<1""2",>,ifiiE1', ..

(4) ' r8 S., ' AKiBA and T. MAEDA ' ' A･A == 2 P' 2ig･ 2e･. 'where ,,<22j> First, we tuples. When .is valid for .i tL then by by 2" ,,. ･･･, Z". '. '. '. k=1. prove by induction on P that there exist at most P+1 distinct P=1, the assertion is trivial. We assume that the assertion P<q. Let 2" b ･･･,1" ,+i be q-tuples which satisfy <2"i, 1" ,･>=-1 for induction assumption, the dimension of-linear space V spanned. , is q-1 or q. If dim. V=:q-1, we have <T(1" ,.,),. Z"i> == <Z" ,.,, 2" i>. ,. == -1. 'fori=1, ･･･, q, where n is orthogonal projection:Rq-> V. But, this contradicts the induction assumption. If dim. V =q, 2" ,+i must satisfy the q independent. `. linear equations <2"i, 2" ,･> = -1, i= 1, ･･･,q, so 2" ,+i is determined uniquely; i. e.. we cannot choose any other distinct tuple. This completes the induction. Now, we prove that there exist exactly P+1 distinct tuples 2"i, ･･･,ipu+i. Assume that there are only r<P+1 distinct tuples 2" ,, ･･･,Z" .. Since M is minimal, leiZ" i+･･･+le.A.･==O.. Put v==k,v,+･･･+k.v.tO, then AV==O,at m, ' which contradicts the assumption of lemma 1. ' Now, we prove the last part of the lemma.. P+1 ･･ ' i=1 ･. Since Z kil"i== O, and since <Z"i, 2" ,･> == -1 for itj, we have. ' ' R",> == -(tti le,- le,)+k,<2",, O = <ki2i+ '" +kp+i2"p+i, Z",> . J'=1. Thus we have. <z",, 2",> = ni/ki.. '. This completes the proof of lemma 1.. LEMMA 2. (IC AV=O.for somevtO at m, then ￠(Mn) is immersed in a geodesic (n+P-1)-sPhere. PROOF. Let V be a parallel nornial vector field on M such that V(m)==v. :and let r be an arbitrary regular curve through m. Then,. 1N tv -N. 7r V == 7t V+B(dr, V)= -AV7+7r V+B(7, V), where BN is the second fundamental form of c:Sn"pDEn'r. since BN (7, V)= "<7, V> and since 7 and V are mutually perpendicular, BN (dr, V)=-O. Sin,ce. V is parallel in,IV(M), 7fV-O. Since, moreover, A is covariant constant,. AV=-O. ThusLV=-O. So, ￠(M) is contained in a euclideanhyperplane perpendicular to V. This implies lemma 2.. From now on, we assume that A"tO for any non zero vE ATin(Mn). LEMMA 3. Each AVi has the same eigenvalues with the same multiPlicities at all Points of Mn, for i= 1, ･･･,p.. PROOF. Let m' be any point of M and let r(t) be a regular curve such that r(O)=m and r(1)=m'. Let e",･ be a parallel vector field along r such thate". ,･(m)=ejandput ' ,, ' ' n AVie"J･=2)pt/ike"k i==1,･･･,r,1'==1,･･･,n.. Then we have. h=1. ;.

(5) ' OnSomeMinimalVarietiesinSphere ' 9 '. '. , 7r(AVie",･)==(7rptS･le)e"k+ptS･kVRk i=1,･･･,r,7'=1,･･･,n. (4) Since V, is parallel, since A is covariant constant and since e",･ is parallel. aiong r, (4) means , , , -'7fpt/ile=O, i=1,･-,r,Lk=1,･･･,n. '' '. '. ' Thus, rd･le is constant along r and since ptZ,'le(m) =23･Si the assertion of lemma ;. 3 is obvious.. By the lemma above, we can extend Ele over M to define'a distribution on Mn and we denote i't by the same notation Eh. That is,. Ele(m)={xET.(Mn);AVix==R//x,i=1,･･･,p.} k==1,･･･,r. '. LEMMA 4. Ek is an involutive distribution on Mn, fe= 1, ･･･,r. PROOF. Since Ek is invariant by the parallel translations and since the parallel translations are Cco, distribution Eh is COe. Let V(Ek) be the･set of Cco vector fields X.such that X(m)EEle(m) for all m M. If X, YE V(Eip),. '. AVi(7.Y)=7.(AVi}7') =7.(2%' Y)=li 7.}-, ,. since Vi is parallel and A is covariant constant. ･･ ' '･ Thus, VxYE V(Ek), and so [X, Y]= ViY-7yXE V(Ele), which proves. lemma4. - '- ･. LEMMA 5. The maximal integral submanipld of the distribution Ek is a. totally geodesic submanijold of Mn and has constant sectional curvature n/ki.. -. ' PROoF. Since Ele is invariant by parallel translations, the first half of the lemma is obvious. Since Yi,･･･,Vp are parallel, the inner,products <Vi, Vj> are constant. So, a linear transformation of constant coefficients transforms (Yi, ･･･, Vp) to a normal frame field which is everywhere orthonormal. Since <B(x, y), v,> =<AVtx, y> ==2Z<x, y>, B(x, y)= tr.,2%<x, y>v,, where. vi== Vi(m). Then the Gauss curvature equation (3) implies:. <R.,y,x>=(1+ni/kt)(<x,x><y,y>-<x,y><x,y>). ' Thus the sectional curvature of (x,y)-plane is (1+n,/kD ==n/lei. This completes the proof of lemma 5: By the decomposition theorem of de Rham, there exists an immersion. '. .,ip:Mn---->Mki,-ler, '. whichisalocalisometry. - '. Put di=io￠, where i:Mki, ,k.-Sn"P is the standard minimal imbedding <see gO). Then (M, ￠) is an minimally immersed submanifold of S"'P satisfying the conditions (i), (ii) and (iii). Let Ur be another isometric minimal immersion which satisfies the conditions (i), (ii) and (iii).. LEMMA 6. SuPPose T(m)=:<Z>(m) and (dZP').=(d<Z>)., then ZPr= ￠> on Mn. PRooF. Let (zi, ･･･,zn) be a local coordinate system around m, and let. r(t) be a regular curve through m. Put ￠o== dior and ￠j= 6o￠ 2cl･. '. or, ]'=1, ''', n･.

(6) 10 S. AKiBA and T. MAEDA ' we identify di with eodi, where c:S""P(1)->Rn'r is the For the convenience,. standardimbedding. Bydefinition: ･' tt t. tt. ipoi=;,II]=,ipt'M, ,where"dot"standsfor" ddt "'. ' ' (5). The Gauss' formula (2) implies:. '. '. '. '. ' ttip'i={(Lk･.jor)￠le+(he･jor)ip.+le+<ipi,ipd>ipo}7', ￠n+le=Vkgr, (6). where T-leij are the coeMcients of the connection 7 on Mand hB･j are the com-. ,. ponents of the second fundamental form of the immersion (M, ￠). The Weingarten's formula (2) implies:. .. ￠n+k==-(h,k"'or)￠jii, k=1,･-,P'. (7). If we regard (5), (6) and (7) 'ordinary differential equation with unknown functions ipo, ipi, ･･･,ip.+p, then the solution of these equations is unique up to the initial condition. On the other hand, ￠o, ￠i, ･･･,￠.+. defined similarly by W and by its parallel normals U,, ･･･, U. satisfy equations (5), (6) and (7).. Since the initial conditions for (￠, Vb ･t･, V.) and (ZIT, Ui, ･･･,U.) can be･ identified by rigid motion in S"'P, ZU' =-￠ around m, up to rigid motion in Sn+p.. For any m' E M, let ri(t) be a regular curve such that r'(O) == m and rX(1>. .. =m'. Then the standard "open and closed" argument with respect to the parameter t of r' implies that LYE￠ on r' and that ZP' (m/)=￠(mi). This･ completes the proof of lemma 6. The Proof of Theorem 1 is, thus, completed. g3. The index and the nullity of Mki,･･･,ler. Let (Mn, di) be a compact minimal submanifold of S""P, and let V and M7 be cross-sections of N(M"). We set. '. '. ' I(V,VV)=S.<-72(V)+R'"(V)-4(Y),W>*1,,' (8). where *1 denotes the volume element of M". This is the second' variation of (M", ￠). (See [5]). The f611owing fact is well known:. PROpOslTION 2. I is a symmetric bilinear form on the sPace of crosssections in N(M"). I may he diagonali2ed with resPect to the standard inner prodzact, and has distinct, real eigenvalues pt, such that leti < ,et2 < ･･･ < pei < ･･･ . oo. Moreover, the dimension of each eigensPace is finite.. DEFINITIoN. The index of (M", O) is the sum of the dimensions of the eigenspace corresponding to the negative eigenvalues. The nullity of (M", di> is the dimension of the O-eigenspace. Norma! vector fields on' di(Mn) which. t-.

(7) OnSomeMinimalVarietiesinSphere ' - 11 is'. ' the orthogonal projections of infinitesimal isometries of Sn'P along M are. contained in O-eigehspace, and are called Killing-Jacobi fields on M. The Killing nullity of (M", di) is the dimension of linear space of the KillingJacobi fields on M.. LEMMA 7. For any n-dimensional minimal variety M in Sn"P(1), we have. '. rv V) = -n V. R(. whereVisanycross-sectionofN(M). '- ･･-conslllllinOtOFc'urvLaetturXe' iY, and Z are vector fieids of s"'p(i). since sn+p(i) is of. RN..Z=<Z, Y>X-<Z, X>Y. . ･. ' ' (See [3] Vo･1. I, p. 203.) We may prove the lemma atapointmEM. Let (ei, ･･･,e.) be an orthonormal frame of T.(M), and let v= V(m). Then, .･ .. ' ' ' ,'R'"(v)=]$l)(R'V,,.e,)N=lib{<ei,v>ei-<ei,ei>v}"=-nv.. i=1 i=1 ' ' ' LEMMA 8. Let ￠:Mk,,-.,k..S""P be such as in gO. Then there are P normal vector fields V. ･･･, Vp such that each Vi is Parallel and (Vi(m), ･･･, Vp(m)) forms a normal frame at each Poilnt of Mh,,..,k.. In other words, the. normalbundleisabsoluteParalleli2able. . ' ''. ' PRooF.Wedefine'' ' ･ ' ' v,ll/2{grad.(le. tt. tTtl,i(xg)2)-<grad.(fe.2`.'.,i(x,")2),x>x}. ''. Then, these Vi, ･･･, Vp satisfy the' requires of the lemma.. LEMMA 9. Let M== Mh,,...,le., and let V be a cross-section of N(M). Then,. A( v) == nv. PRooF. Easy calculations. '. By lemma 7 and lemma 9, (8) reduces to. l( V, 17V) = S.<-P2V- 2n V,.17V>*1.. THEOREM 2. Let M==Mk,,...,k. be the minimal submanijbld of S"'P given in gO. Then theindex ofMis P(n+P+2), the nullity of M is P,E.j(let+1)(lej+1). andtheKilling-nullityofMisi.,.(ki+1)(lej+1). '. PRooF. By lemma 8, any cross-section of N(M) has a form V= t?.,gtVz,. whereg,arefunctionsonM.Since 2.achV,isparallelinAl(M), ... --V2V-2nV=-,l.2)=,(V2gi+2ngi)Vi･ .'' of the metrices on S lei, i= 1, ･･･, r, Since the metric on M is the product metric the Laplacian 72 on M decomposes to V?+･･･+V?, where 7;'･ denotes the Laplacian on Slei. On functions, -73･ has 1-dimensional O-eigenspace consisting of the constants, and has (le,+1)-dimensional eigenspace corresponding.

(8) 12 ' S.AKIBAandT.MAEDA ' '. ' to the eigenvalue n, which ' consists of linear functionals on Rlez'i. The other eigenvalues of --VZ･ are all strictly greater than 2n. Thus on cross-sections. of Ai<M), negative eigenspaces of -72-2n consists of V=,￡giVi, where each gi is a linear functional with respect to at most one set of variables ,(x},･･･,xe･j") and constant with respect to the others. O-eigenspace of -V2-2n consists of V= :ll)AVi, where each A is bilinear functionals with i =1. respect to exactly two sets of variables (x}･, ･･･,xole･2'+i), (x},,･t･,xkh'i) and con-. ,. .. stant with respact to the others. Thus the index and the nullity of M are. asannounced. ･ '. The subgroup of the isometry group of S""P 'which leaves the submani-. fold Mle,,-.,k. invariant is isomorphic to O(ki+1) × ･･･ × O(k.+1). (See [3] VoL I, p. 240.) By Hsiang-Lawson ([2], p. 240.), the Killing nullity of M is. '. ' tt ' (n+r)(n+r-1)/2-,,ll.il,(fet+1)kz/2F=il.Ii).,(lei+1)(fej+1) This completes the proof of theorem 2. '. '. '. '. Bibliography. tt. [1] S.S. CHERN, M. Do CARMo and S. KoBAyAsHi, Minimal submanifolds of a ' spherewithsecondfundamentalfromofconstantlength,Functionalanalysis. -. and related,fields, Proc. Conf. in honor of Marshall Stone, Springer, Berlin, 1970.. ,[2] W.Y. HsiANG and H.B. LAwsoN, Minimalsubmanifoldsof low cohomogeneity,. J.Diff.Georn.5(1971),1-38. ' . ,. ,[3] S. KoBAyAsm and K. NoMizu, Foundations of differential geometry, Vol. I. -(1963),Vol.II(1969),Interscience,NewYork. , [4] H.B. LAwsoN, Jr., Local rigidity theorems for,minimal hypersurfaces, Ann.. of Math. 89 (1969), 187-197. MiniMal Varieties in Riemannian manifolds, Ann. of Math. ss (lg6s),. [5] J6･2.SIio¥9NS,. ''. '. `.

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