Generic Submanifolds (*).
Kv,~TARO YA~o - MASAmUO Ko~ (Tokyo, Japan)
Summary. - in this paper we give some examples of generic submanifolds of complex space forms and prove some theorems which give the characterizations of these examples..For this purpose we study the ~'elations between a submanifotd of a K~hlerian manifold and a submauifold of a Sasalcian manifold by using the method of Riemannian fibre br
O. - Introduction.
The theory of submanifolds of K~hlerian manifolds is parhaps one of the most interesting topics in modern differential geometry. A submanifold M of a Ki~hlerian manifold M is called a complex submanifold (an invariant submanifold) if the tangent space T,(M) of M at x is always mapped into itself under the action of the almost complex structm'e tensor J of the ambient manifold ~r, that is, if JT=(M)c T~(M) for all x ~ M. Complex submanifolds of K~hlerian manifolds have been studied by various authors and still suggest many interesting problems. On the other hand, a submanifold M of a K~hlerian manifold M is said to be totally real (anti-invariant) if JT~(M) c T~(M) • for all x ~ M, where T~(M) • denotes the normal space of M at x (see [1], [15] and [21]). Now, a submanifold M of a Ki~hlerian manifold M is called a generic submanifold (an anti-holomorphie snbmanifold) if JT.(M) • c T~(M) for all x e M (see [9] and [15]). For example, any real hypersm'face of a K~hlerian manifold is a generic submanifold. For real hypersurfaces, see [5], [7], [8], [12] and [13]. In [6], Lawson studied submanifolds of complex space forms by using the method of ;[~iemannian fibre bundles and MAEDA [7] and OKIJ~rU~A [8], [9] developed this method of Lawson extensively (see also [21]). The pal'pose of the present paper is to study generic submanifolds immersed in complex space forms by the method of l~iemannian fibre bundles. In w 1, we first of all recall some of fundamental formulas in the theory of sub- manifolds of a Riemannian manifold, give next a summary of known results and prove theorems on submanifolds of a sphere which will be used in the sequel. In w 2, we recall some of fundamental formulas in the theory of submanifolds of K~hlerian manifolds, especially in that of generic sllbmanifolds of Ki~hlerian
(*) Elltr~ta in Redazione il 30 agosto 1978. 60 KE~-~A]~o u - 5IASAm~O K0g: Generic submaniJolds
manifolds. Moreover, we show that any generic submanifold of a Kiihlerian manifold admits an J-structure [16] and study its integrability conditions [2]. In w 3 we give examples of generic submanifolds of a complex projective space CP~ and those of generic submanifolds of a complex number space C% w 4 is devoted to the study of relations between a submanifold M immersed in a Ki~hlerian manifold M and a submanifold h r immersed in a Sasakian manifold by using the method of Riemannian fibre bundles. We prove some lemmas giving the relations between the second fundamental form B of M and the second funda- mental form e of N. We also study relations between the normal connection of M and that of h r and those between the mean curvature vector of M and that of 2;r. Our main theorems will be proved in w 5 which give the characterizations of examples given in w3. We study in w 6 pseudo-umbflicM generic submanifolds of complex space forms. The notion of pseudo-umbilical generic submanifolds is a generalization of the notion of ~-~mbilieal real hypersarfaee. In the last w 7, we give a generalization of a theorem on anti-iuvariant submani- folds proved in [19] and an application of a theorem of Wells [15].
1. - Submanifolds of Riemannian manifolds.
Let 217 be an m-dimensionM l%iemannian manifold with positive definite metric tensor field g and 21/be an n-dimensional submanifold isometrically immersed in 2ft. We denote by the same g the !~iemannian metric tensor field induced on M. The operator of covariant differentiation with respect to the Levi-Civita connection in 2~ (resp. M) wilI be denoted by V (resp. V). Then the Gauss and Weingarten formulas are respectively given by
vZg = vx:g + B(X, Y), VxF = - A~X + 1)~V
for any vector fields X, Y tangent to M and any vector field V normal to M, where D denotes the operator of covariant differentiation with respect to the linear con- nection induced in the normal bundle of M. A and B are both called the second fundamental form of M and are related by
g(B(X, Y), V)--= g(AvX, Y).
The second fundamental form B is a vector bundle valued symmetric bilinear form on each t~ngent space T~(M) taking values in the normal space T~(M) • and the second fundamental form A is a cross-section of a vector bundle ttom (T(M) • S(M)) where S(M) denotes the bundle whose fibre at each point is the space of sym- metric linear transformations: T~(M) --->T~(M), i.e., for V E T~(M) • Av: T~(M)--~ KE~TAI~O YAI~O - 1V[ASAHIt~O KOI~I: Generic submani]o~ds 61
l%r the second fundamental form B we define its covariant derivative VzB by
(VxB)(~, Z) = Dx(B(]~, Z)) -- B(Vx~, Z)- B(](, VxZ) for any vector fields X, Y and Z tangent to M. If VxB = 0 for all X, then the second fundamental form B of M is said to be parallel. This is equivalent to VxA = 0, where VxA is defined by
(VxA)v 1r : Vx(Av Y) -- A,~v Y -- Av VxY .
A normal vector field V on M is said to be parallel if -DxV ~- 0 for any vector field X tangent to M. The mean curvature vector # is defined to be/~ ----- (Tr B)/n, where Tr B is the ~traee of B. If B----0 identically, then M is said to be totally geodesic and if # = 0~ then M is said to be minimal. If the second fundamental form B is of the form B(X, Y) = g(X~ Y)[2~ then M is said to be totally umbilical. Let/~ and R be the Riemannian curvature tensor fields of M and M respectively. Then, for any vector fields X, ]r and Z tangent to M, we have
(1.1) R(X, :g) Z =-I~(X, Y) Z -- ABr + A~r + (VxB)(:Y, Z) -- (V~B)(X, Z).
From (1.1), we have equation of Gauss
(1.2) g(/~(X, Y)Z, W) = g(R(X, Y)Z, W)- g(B(X, W), B(Y, Z)) + + g(B(Y, W), B(X, Z))
for any tangent vector fields X~ Y, Z and W. Taking the normal component of (1.1), we have equation of Codazzi
(1.3) (_~(X, Y)Z)~= (VxB)(]~, Z)- (V~B)(X, Z).
We now define the curvature tensor /~• of the normal bundle of M by
R'(X, Y) V =- DxD~ V -- DyDx V -- Drx,~l V,
X, :Y being vector fields tangent to M and V a vector field normal to M. Then we have equation of l~icci
(1.4) giR(X, Y) U,y) = g(R~(X, Y) U, V) + g([A~, A~]X, ~)
for any vector fields X~ Y tangent to M and for any vector fields U, V normal to M, where [Av~ A~] = ArAb-- A~Av. 62 KE~A~O Y~o - ~V~ASA~O Koch: Generic submani/olds
If R • vanishes identically, then the normal connection of M is said to be fiat (or trivial). If, for any vector fields U and V normM to M, lAy Av] : 0, then the second fundamental form of M is said to be commutative. In the following we prepare some lemmas on submanifolds of a sphere for later use. Let M be an n-dimensionM minimal submanifold isometrically immersed in a unit sphere S ~+~ of dimension n-~ p. We denote by S the square os the length of the second fundamental form of M, that is, we put
S ~- ~g(B(e~, ej), B(e~, ej)) = ~g(A~ei, A~e~) = ~TrA~, where {e~} denotes an orthonormM frame for T~(M) and {v~} denotes an orthonormM frame for T~(M)• writing A~ in place of A~ to simplify the notation. In the equa- tion above and in the sequel, the indices i, ],/% ... run over the range {1, 2, ..., n} and the indices a~ b, c, ..., the range (1, 2, ..., p}. Then the formula of Simons is given by (el. S~o~s [ii])
(z.5) AS = nS -- ~ (Tr A~Ab) ~ -p ~ Tr [A~, Ab] ~ -~ g(VA, VA), a,b a,b
where g(VA, VA) = ~ g((V~A)~ej, (V~A)~ej). If the second fundamental form of M a~$~ is parallel, then S is constant and (1.5) implies
(~.6) 0 < -- ~ Tr [A~, Ad ~ = nN-- ~ (Tr A~A~) ~ . a,b a,b
If we put Sa~= TrA~Ab and 8~= ~o, then (S~b) is a symmetric (p, p)-matrix and so can be assumed to be diagonal for a suitable choice of v,, ..., %. Thus we have, from (1.6), (z.;) ~b a =nS ~_1 ~(S_S~)~ <1
On the other hand, from (1.4), we see that the normal connection of M is fiat if and only if the second fundamental form of M is commutative. Therefore (1.7) implies
LE~:~ 1.1. - Let M be an n-dimensional minimal submani]old o] a unit sphere $'~+~ with parallel second ]undamental ]orm. I] g>~pn, then the normal connection o] M is ]lat.
Next we prove
LE~c~A 1.2. - Let M be an n-dimensional submani/old of S ~+~ with fiat normal connection. I] the second ]undamental ]orm o] M is parallel, then the sectional curva- ture o] M is non-negative. KEyr~tr YA~o - MASAHI~O KOl~: Generic submani/olds 63
P~oor. - Since the normal connection of M is flat, the A~'s are simtfltaneously diagonalizable at x eM. Let 2 ~ (l A~R(e~, e~) ei = R(e~, e~)A~e~ = 2~ R(ei, ej) ei. Thus the eigenspace T"i corresponding to 2"i has the property _R(e~,e~) T ~i~_ - T ~i for all a = 1, ..., p If 2 ~ va 2,. for some a, then we have i g(R(ei, cj)e,, ej) = O, which means that the sectional curvature K,- of M spanned by {e~, ej} vanishes. On the other hand, from (1.2), the sectional curvature K, of M is given by Ki~=l@ 2~2, ~ + ....+ 2~;~ If ~ ve 2j for some a, then K~---- 0 and if 2 ~--, 2~~ for all a~ then K~j ---- 1 ~ (~)~ @ ... -[- (2~)"2 > 0. Thus the sectional curvature K~j spanned by {e~, e~) is non-negative. In the following, we prove that the sectional survature g(R(X, Y)Y, X) for any orthonormal vectors X and Y is non-negative. First of all, from (1.2), we obtain g(R(X, Y)Y, X) = 1 @ ~ g(A:X, X)g(A~ Y, Y)- ~.g(AoX, Y)~ . a a We now put X=~,e, and Y=~/~e~. Then we see that 9 2 a a 2a a a,i,~ a,i,~ - 1 + y.(K~-- 1)~8 ~- i,~ 1,i Since we have ~a~fl~2 2 = 1 and ~a, fl,----. 0, the equation above becomes ~,~ e(R(X, ~) ~, X) = y. K.~y-- y.K.~,~3~, = ~. g.(~3~-- ~)~. 64 KE:NTAI%O YA:NO - M:ASAtt~O KO:N: Generic submani/olds We have already seen that K~>O, and hence g(R(X, ~) ~, X) >i O . Therefore the sectional curvature of M is non-negative and the lemma is proved. We now give an example of n-dimensiona] minimal submanifold of a sphere S ~ (see ITS]). For integers p~, ..., p~ such that p~, ...~ p~.>~l, p~-... + pN: n, regard R "+~ as /~ ~+tX... X/V ~+~, where iV : m-- n + 1. Then S~'(rJ x ... x S~(r~) -----{(xl, ..., x~)e/~+~: xt e R~,(rt), t ----- 1, ..., iV}, where S~(r~) c/9 ~+~ (t : 1, ..., iv), is an n-dimensional minimal submanifold of essen- tial codimension m-- n imbedded in S ~ if r~ ~- ... d- r~-: 1, rt = ~/pt/n (t = 1~ ..., iV). Now YAI~O-ISt{I]gA]~A [18] proved the following TB:I~OI~E~ A (YA~O-ISlIIKAI{A [18]). - Let M be a complete minimal submani/old o/ dimension n immersed in S "~ and with non-negative sectional curvature, and sup- pose that the normal connection o/ M is fiat. I/S is constant, then M is a great sphere o/S ~ or a pythagorean product o/the ]orm (t = I, ..., iv) and with essential codimension iv--1, where pl, ..., p~v> ], p~ ~- ... + p~v---- n. From Lemmas 1.1, 1.2 and Theorem A we obtain Tm~o~E~ 1.1. - Let M be a complete minimal submani/old o/dimension n immersed in an (n ~ p)-dimensional unit sphere S ~+~ with parallel second /undamental /orm. 1] S>~pn, then M is a pythagorean product o/ the ]orm x... x r, = V' ln (t = 1, ..., iv), where n >Pl, .~ p~>~l, Pl -~ ... ~ P~ = n, p -: iV-- i. P~ooF. - From Theorem A the essential codimension of M is N- i. On the other hand, by (1.7), we have S~ ---- S~ for all a, b (---- 1, ..., p). Since S ----- pn, M is not totally geodesic and hence B~ ~-S/p ~ O. Thus we should have p----iV- 1. The other statements are trivial consequences from Lemmas 1.1, 1.2 and Theorem A. Fttrthermore YA~O-Ism~A~A [18] proved the following theorem. Tm~o~]~ B (Y~t~O-ISmt{A~A [18]). -- Le{ M be a complete/ull submani]old o/dimen- sion n immersed in an m-dimensional unit sphere S ~ and with non-negative sectional curvature. Suppose that the mean curvature vector o/ M is parallel and that the normal KE:NTARO YA~0 - ~V[ASA_g~0 Ko~: deneric submanifolds 65 connection of M is fiat. I] ~ is constant, then M is a small sphere, a great sphere or a pythagoren product of the ]orm ~'(rl) x ... x S~(r~), r~ + ... + r~ = 1, ~ = m- n + 1, or o] the form ~(rl) x...X~'(r~v,)c~-~(r)cS '~, r~@...+r~=r2 From Lemma 1.2 and Theorem B we have Tn~o~]~ 1.2. - Let M be a complete n:dimensional submanifold of S ~ with flat normal connection. If the second fundamental form of M is parallel, then M is a small sphere, a great sphere or a pythagorean product of a certain number of spheres. More- over, if M is of essential codimension m- n, then M is a pythagorean product of the form ~'(r~) x... x S~(r~), r~ + ... § r~ = 1, y = m- n + 1, or a pythagorean product of the ]orm S~,(r~)x...xS~'(r~.,)cS~-~(r)r ~ , r~ -~... -~ r~,~----- r~< 1 , N'= m- n . If the normal connection of a minimal submanifold M of g~ is flat, then (1.5) implies AS = ng-- ~ (Tr A~Ab) ~ ~- g(VA, VA), a,b >~nS-- S 2 ~- g(VA~ VA). Thus~ if M is compact, then we have (1.8) ~S(S - n)* 1> 0. M Using this, KE~o~sv [3] proved T]~EORE~ C (KE~[0TSU [3]). - Let M be an n-dimensional compact minimal sub- manifold of S ~ with flat normal connection. I] S-~ n, then there exists an (n-~ 1)- dimensional sphere S ~+~ containing M as a Clifford minimat:hypersurface I~E~AnK. -- For submanifolds with parallel second fundamental form immersed in real space forms, see WALDEN [14]. 5 - Annali di Matematica 66 KE~TAt~O Y.a~o - MASAIaL~O Ko~: Generic submani]olds 2. - Submanifolds of Kfihlerian manifolds. In this section we assume that the ambient manifold M is a complex m-dimen- sional (real 2m-dimensional) K~hlerian manifold with almost complex structure J. Let M be an n-dimensional Riemannian manifold isometrically immersed in _M. For any vector field X tangent to M, we put (2.1) JX ~ PX -~ ~X , where PX is the tangential part of JX and FX the normal part of JX. Then P is an endomorphism on the tangent bundle T(M) and F is a normal bundle valued 1-form on the tangent bundle. If JT~(M)~ T~(M) at each point x of M, then M is called an invariant sub- mani]o~d (complex subraani]old) of ]g. If M is an invariant submanifold of M, then /~ in (2.1) vanishes identically. If JT~(M)r T~(M) J- at each point x of M, then M is called an anti-invariant submani/old (totally real subnani]old) of M. If M is an anti-invariant submanifold of M, then P in (2.1) vanishes identically. If the ambient manifold _M is of constant holomorphic sectional curvature c, then 2~ is c~lled a complex space form, and will be denotes by /~(c). The l~ie- mannian curvature tensor /~ of _7~(c) is given by (2.2) _~(x, ~) z = = ~c[g(Y, z)x- g(X, z) :Y + g(Jy, z)jx- g(JX, z)JY -~ 2g(X, J[Y)JZ] for any vector fields X, Y and Z of _M~(c). From (1.1) and (2.2) we have (2.3) .R(X, Y) Z = = ~c[g(Y, Z)X-- g(X, Z) Y -~ g(aY, Z)JX -- g(JX, Z)JY ~- 2g(X, JY)JZ] + § A.,,~,z)Y § (V~B)(X, Z) -- (V~B)(Y, Z) for any vector fields X, Y and Z tangent to M. Comparing the tangential and normal parts of the both sides of (2.3), we have respectively (2.4) R(X, Y) Z = = ~c[g(Y, Z)X-- g(X, Z) Y + g(.P:Y, Z)PX- g(.PX, Z)_PY + 2g(X, .PY)PZ] + ~- A~(y,z)X -- A~(x,z) :Y , and (2.5) (v:B)(]~, z)- (VyB)(X, Z) = = ~c[g(t)Y, Z)FX-- g(PX, Z)~Y + 2g(X, P:Y)FZ]. I~ESTTAtr YA~O - MASAI=IIRO KOST: Generic submani]olds 67 From (1A) and (2.2) we also have (2.6) ~c[g(FY, U)g(FX, V) -- g(FX, U)g(F•, V) + 2g(X, PY)g(JU, V)] = = g(R'(X, Y) U, V) + g([Av, Av]X, Y). We next study properties of the second fundamental form of M. Let X and Y be vector fields tangent to M. From the Gauss and Weingarten formulas we have (2.7) JB(X, Y) -~ (VxP) :Y-- A~.xX + B(X, PY) -~ (VxE) Y, where we have put (VxP) :Y = Vx(PY) -- PVx~ and (VxF) :Y = Dx(~'Y) -- FVxY. Since B is symmetric, (2.7) implies (2.8) (V~P) Y-- A~rX + B(X, 2Y) + (Vy)Y = = (VrP)X-- AFxXz -[- B(Y, PX) -t- (VrF)X. Comparing the tangential and normal parts of the both sides of (2.8), we have respectively (2.9) (VxxP) Y -- (VyP) X -~ AFyX -- AFx Y, and (2.10) (VxF) [Y -- (VrF)X = B(2PX, Y)- B(X, PY). In the scquel~ we consider generic submanifolds of K~hlerian manifold which have been studied by O~A [9] (see also WELLS [15]). A submanifold M of a Ki~hlerian manifold M is called a generic submani]old (anti-holomorphie submanifold) of 2~ if JT~(M)'c T~(M) for all point x of M. Especially, if JT~(M) -L = T~(M)~ then a generic submanffold M is an anti-invariant submanifold such that real dim M = complex dim 2~. The tangent space T~(M) of a generic submanifold M is decomposed in the following way: Y~(Jl) = H~(M) | " at each point x of M~ where H~(M) denotes the orthogonal complement of JT~(M)" in T~(M). Thus we see that JH~(M) = H~(M) ~- T~(M) (3 JT~(M) , that is, H~(M) is a holomorphic subspace of T~(M). 68 KE~A~0 YA~o - ~[ASAmlr Ko~: Generic submani]olds If dim T~(M)Z: 11 that is, if M is a re~l hypersurface, then M is obviously a generic submanifold of M. Applying J to the both sides of (2.1)I we obtain -- X = JPX d- JFX = P~X + F.PX -}- JFX. Since JT~(M)• we see that JFXeT~(M). Thus we have (2.11) F.PX = O , (2.12) P~X : -- X--J-~X . :From (2.7) we also have (2.13) JB(X, :Y) = (VxP) Y-- AFrX, (2.18) B(X, P:Y) = -- (VzF) :Y. Let U be a vector field normal to M. Then we have VxJ U : J VxU : -- JAeX + J.Dx U : -- PAvX-- FA~X ~- JDz U, ~xgu = v~v + B(X1 Jr). Comparing the tangential and normal parts of the both sides of these equations, we obtain (2.15) V~U = -- PAvX + JDxU , (2.16) B(X, JU) = -- FAgX . We now prove the following LEI~L~A 2.1. -- Let M be a generic submani]old of a Kghlerian mani]old M. Then we have (2.17) A~rZ = AFz Y for any vector fields Y and Z in JT(M) • P~ooF. - Since PJT.(M)J-= 0 and JPT~(M)c H~(M), we have (VxP) Y : VxPY-- PVzY : -- PVxY eH(M) KE~TA~O Y)~O - MASAwr~O KO~: Generic submanifolds 69 for any vector field X tangent to M and 17 e JT(M) • From this and (2.13) we have g(AvrX, Z) = -- g(JB(X, 17), Z) = g(B(X, 17), JZ) = g(AFz 17, X) for any Z efT(M) • Thus we have (2.17). Applying P to the both sides of (2.12), we obtain PaX ------PX for any vector field X tangent to M. Thus we have (2.18) pa _.}_ p ___ 0. On the other hand~ the rank of _P is equal to dim M-- codim M = n-- (2m-- n) --= ~---2(n--m) everywhere on M. Consequently, /) defines an f-structure of rank 2(n-- m) (see [2], [16]). We now put ' (2.19) .L = -- .p2 T = .P2 Jr- I We can easily see that Z and T are complementary projection operators. Thus there exist complementary distributions ~f and ~-" corresponding to the projection operators L and T respectively. Since the rank of P is 2(n- m), ~ is 2(n- re)- dimensional and Y-~ is (2m- n)-dimensional. The distributions s and ~ are defined also by ~9~. = {X e T~(M): FX = O} and Y~ = {X e T~(M) : PX = O} respectively. We now study the integrability conditions of ~ and 3-. First of all, we prove P~oPosITIO~ 2.1. - Zet M be an n-dimensional generic submanifold o] a complex m-dimensional K~ihlerian manifold .M. Then the distribution ~ is completely integrable and its maximal integral submanifold M' is a real (2m- n)-dimensional anti-inva- riant submanifold of M. PlcooF. - Let X and Y be vector fields in ~'. Then (2.9) and Lemma 2.1 imply P[x, y] = Pv~y- PV~,x = - (V~P) 17 + (vyP)x -~ AFx Y- AvrX = O . Therefore the distribution Y is completely integrable, and we see, from the con- struction, that the maximal integral submanifold M' is a real (2m- n)-dimensional anti-invariant submanifold of M. If the distribution ~qz is integrable and moreover if the almost complex structure induced from P on each integral manifold of ~o is integrable, then we say that the /-structure _P is partially integrable (see [2], [16]). 70 KENTA~O YA~O - MAsAm~o K0~: Generic submani/olds PI%OPOSITIO~ 2.2. - Let M be an n-dimensional generic submanifold of a complex m-dimensional Kiihlerian manifold M. Then the ]-structure P is partially integrable i] and only if (2.2o) B(PX, ~) = B(x, P Y) for any vector fields X and Y in s PRooF. - Let X and Y be vector fields in ~. Then (2.10) implies that FIX, Y]= FGY FV~X=- (Gf)Y + (G.F)X = B(X, PY) -- B(PX, Y). Therefore, ~ is integrable if and only if (2.20) holds. Then the integral manifold 3/o of s is an invariant submanifold of ]I and hence 3/0 is also a Khhlerian mani- fold. Thus the almost complex structure induced from P on 3/o is integrable. Thus P is partially integrable if and only if (2.20) holds. P~OPOSITIO~ 2.3. - Let M be an n-dimensional generic submani/old o/ a complex m-dimensional Kiihlerian manifold M. 1/2m-- n >~ 2 and M is totally umbilical, then M is totally geodesic. PgooF. - Let # be the mean curvature vector of M. Since JT~(M)'c T~(M), we see that J/~ is tangent to M. Thus we have g(A X, X) = -- g(AFzJ#, X), X e JT~(M) -L . Since M is totally umbilical, the second fundamental form A of M satisfies g(A , X, X) = g(X, X)g(/~,/t), g(AFxJ#, X) = g(J#, X)g(#, FX) , from which g(X, X)g(#, #) = -- g(J#, X)g(#, FX) . Since, by the assumption, 2m-- n>2, we can choose X in such a way that g(Jtt, X) =- 0 and hence # = 0. Thus M is minimal and hence M is totally geodesic. The following result is to be noted. Ptr A (Ctt-EN-OGIuE [1]). - Let M be an n-dimensional submani/old o/ a complex space from M'~(c) with c :/= O. Then M is an invariant submani/old or an anti-invariant submani/old o/ M~(c) if and only if M is an invariant submani/old K]~a~o YA~o - lVIASAI-IIRO KOI~: Generic submani/olds 71 with respect to the curvature trans]ormation o] M'~(c)~ that is, ~(X, ]~)T~(M)c T~(M) for all vector ]ields X and ~ tangent to M and ]or all point x o/ M. If a submanifold M of a complex space form ~]~(c) (c r 0) is totally umbilical, then M is an invariant submanifold with respect to the curvature transformation of 217~(c). From this and Proposition A, there is no totally umbilical generic sub- manifold with 2m--n>2 of a complex space form M~(c) (c V: 0). Moreover~ if 2m- n ---- 1, from (2.5), we easily see that there is no totally umbilical real hyper- surface of a complex space from M'*(c) (c :/: 0). Thus we have P~OPOSITIo~ 2.4. - Let M be an n-dimensional (n~3) generic submani]old o/ a complex space ]orm M~(c) (c =/= O) such that m ~ n. Then M is not totally umbilical. 3. - Examples. In this section we give some examples of generic submanifolds and anti-invariant submanifolds of complex space forms. We denote by C ~ and CP "~ the complex number space and the complex pro- jective space of constant holomorphic sectional curvature 4 respectively. The com- plex number space C ~ is identified with the real number space R 2"~. ExA~P]~E 1. - Let M be a 1Biemannian product manifold of theform C~ • where Mq is a real q-dimensional anti-invariant submanifo]d of C q r C ~+q. Then C ~ • M ~ is a generic submanifold of C ~+~. As an anti-invuriant submanifold M=~ we can take a real q-dimensional flat torus (see [19]). Thus r x ~(r~) x... x S~(r~) is a generic submanifold of C ~+q. EXAMPLE 2. -- Let S~(r) denote an m-dimensional sphere with radius r. We now consider an odd-dimensional unit sphere ~2~+~ in C ~+~. Then S 2~+1 admits a Sasa- kian structure (9, ~ ~, O). Let v be the position vector representing a point of S 2~+~ in C ~+~. Then the structure vector field ~ of S 2~+1 is given by ~ = Jv,w here J denotes the almost complex structure of C ~+~. We consider the orthogonal pro- jection amd put 9----~r We denote by ~ the 1-form dual to ~ and by G the standard met1 ic tensor field on S 2"+1. Then for any vector field X tangent to S 2"~+~, we have px ---- Jx q- v(X) v. 72 KEI~TAI%O YA~O - )r KO1N: Generic submani/olds We now consider the following immersion: S m* X...XS m~ -->S n+k-~, n = ~ mi. i=1 We assume thatm~...~m~ are odd. Thenn+k--1 is also odd. Let v~be apoint of S~'(Vm~) on /~m,+l = C(,~,+1)/2. S~,(~/m~) is a real hypersurface of C (~+1)12 with unit normal ~n/m~v~. Thus v = (v~, ..., vT~) is a unit vector in R~+k= C (n+~)t2. This defines u minimal immersion of M~ ...... ~r = ]~ 8~(~/m~) into 8 "+~-~. We restrict the almost complex structure of C (~+~)/z to C (~'+~)/z. Then each dv~ is tangent to S~'(%/m~). Thus Jv is tangent to M~ ...... ~. We then consider the normal space of Mm...... ~ in S ~+~-x which is the orthogonal complement of the 1-dimensional space Let w~, ..., w~_~ be an orthonormal frame for T~(M~ ...... ~)~. Then w, is given by linear combination of v~, ... ~ vk. Thus Jw~ is tangent to M~ ...... and henc% by (3.1), ?wi = Jw~ ~ ~(wi) v = Jw~ . Therefore ~w~ is tangent to M~ ...... ,~ for all i-~ 1, ... k--1. Thus we have (3.2) ~T~(:~,, ...... m~)• c T~(Zg~ ...... ). We now consider the following commutative diagram where N m...... ----,~(M~, ...... ~) (for a l%iemannian fibre bundle, see w 4 and YA~o- Ko~ [21]). We denote by G the metric tensor field of S ~+k-1. The induced metric tensor field of Mm ...... will be denoted by the same G. We denote by g the metric tensor field of Cip(~+k-2)I~ as well as the induced metric tensor field on s ...... ~,~. We denote by * the horizontal lift with respect to ~. Then we have G(X*, ~*) = g(X, Y)* , G(cfX*, Y*) ~ g(JX~ Y)* , (JX)* ~ q~X* K]~ARO u - MASA~O Ko~: Generic submani]olds 73 for any vector fields X and :Y on CP ('+~-e)/~'. From this and (3.2), we see that N~ ...... ~ is a generic submanifold of Ct )(~+~-~)/~. We see moreover that ~ ...... ~ is a minimal submanifold of CP (~+~-~)/~. For the retail, see w 4. EXAMPLE 3. - Consider an immersion S~'(r~) x... • ~(r~) -+ ~+~-~ n-~ ~m~, i=l where r,~ -F ...-/-r~ = 1. Then we can see, by a consideration similar to that used in Example 2, that ~(S~(r~)x... • is a generic submanifold of CP ('+~-2)/2. Moreover we see that ~(S~'(r~)•215 has parallel mean curvature vector and is with flat normal connection because of the fact that S~'(r~)x ... XS~(r~)has parallel mean curvature vector and the flat normal connection (see I~emmas 4.7, 4.10 in w 4). EXA~T,n 4. - If m ~ = m t ..... roT0= 1 in Example 3, then ~(S~(r~) x ... x g~(rk)) is an anti-invariant submanifold of CP ~-~, and n = k. Moreover S~(r~)• • and Sl(r~)• •215 are both anti-invariant submanifolds in C ~ (see YA~o- Ko~ [19], [20]). These examples have parallel mean curvature vector and the flat normal connection. 4. - Riemannian fibre bundles. Let /Y be a (2m ~-1)-dimensional regular Sasakian manifold with structure tensors (% ~, ~, G). Then there is a fibering ~: 7~ -->_~/~ =/~, where /~ denotes the set of orbits of ~ and is a real 2m-dimensional K~hlerian manifold. We denote by (J, g) the Ki~hlerian structure of _M. We denote by * the horizontal lift with respect to the connection ~]. Then we have (4.1) (JX)* = q~X* , G(X*, ~*) = g(X, Y)* for any vector fields X and 17 on M. We denote by V (resp. V) the operator of covariant differentiation with respect to G (resp. g). Then we have (4.2) (v-~17)* = - ~2 vx.17* = v~.Y*- v(vx.]~*)~. Since we have ~(Vx*17*)=- G(qX*, Y*), (4.2) reduces to (4.3) (v-~17)* = vx.17* + G(~X*, 17,)~. We denote by/~ and/~ the Riemannian curvature tensors of N and 21I respectively. 74 ICE,TARO u - IV[ASAm-~O K0~: Generic submani]olds Then we have (~.~) (/~(X, Y)z)* =/~(x*, :Y*)z* + G(z*, pY*)px*- -- G(Z*, ~X*)~Y*-- 2G(Y*, q~X*)q~Z* for any vector fields X, Y and Z on M. We now study the relations between submanifolds of a Sasakian manifold and those of Ki~hlerian manifold. Let N be an (n + 1)-dimensional submanifold immersed in 37 and being tangent to the structure vector field ~ of _~ and M be an n-dimensional submanifold immersed in _~r. In the following we ussume that there exists a fibration ~: N-> M such that the diagram N i' :,~ (c) i commutes and the immersion i' is a diffeomorphism on the fibres. We denote by the same G and g the induced metric tensor fields of h r and M respectively. Let V' (resp. V) be the operator of eovariant differentiation with respect to G (resp. g). We denote by g (resp. B) the second fundamental form of the immersion i' (resp. i) and the associated second fundamental tensors will be denoted by H und A respec- tively. For nny vector fields X and Y on M, the Gauss formulas are given by ~.Y = V~Y + B(X, Y) and 9x,:Y* = V~.Y* + ~(X*, Y*). From (4.2) we have I , (4.5) (V~Y)* = -- q~ Vx, Y , (4.6) (B(X, ]~))* = ~(X*, ~*) . Let D' and D be the operators of covariant differentiation with respect to the linear connections induced in the normal bundles of /V and M respectively. For any tangent vector field X and any normal vector field V to M, we have the Wein- garten formulas ! (4.7) VxV:--ArX+DxV, Vx.V*-~--Hv, X*+Dx.V*. From (4.2) and (4.7) we have respectively (4.8) (AvX)* = -- ~o~Hv.X* , K~TARO u - MASA~rRO K0~: Generic submani]olds 75 and (4.9) (D x V)* = D'x. V* . L]~A 4.1. - M is minimal i] and only i] N is minimal. P~ooF. - Since the structure vector field ~ of 2r is tangent to N, we have o = ~o~ = ~ = v'~ + =(~, ~), from which (4.10) ~(~, ~) = o. We take an orthonormal frame {e~} for T~(M). Then {e*, ~} is an orthonormal frame for T~(N) (Jr(y)= x). Therefore (4.6) and (4.10) imply n (4.11) n --F 1 #* =/#' where/z and #' denote the mean curvature vectors of M and N respectively. From (4.11) we have our assertion. From (4.1) we see that the following lemmas are valid. LE~WA 4.2. - M is anti-invariant i] and only i] 2r is anti-invariant. L]~lVr~A 4.3. - M is invariant i] and only i] s is invariant. We say that a submanifold N of a Sasaki~n manifold N is generic if ~cT~(2r c c T~(N) for all point x of N. Thus we have LE~WA 4.4. - M is a generic submani]old o] a Kghlerian mani]old .M i] and only i] 2( is a generic submani]old of a Sasakian mani]old ~. From (4.9) and (4.11) we have LEnA 4.5. - I] the mean curvature vector #' of 2~ is parallel, then the mean curva- ture vector tt o/ M is also parallel. Let e~ ..., e~+~ be an orthonormal frame for N. Then we have (n-~ 1)# = n+l = ~ ~(e~, ei). On the other hand, for any vector field X tangent to N, we have i=1 ~ = v~ + ~(x, ~) = ~x. Thus we have (4.12) (~X) ~ = ~(X, ~), (?X) ~ = V'z~ , 76 KE~TARO YA~O - lVIASA~0 KO~: Generic submanifolds where (pX) -~ and (pX) ~" denote the normal and tangential parts of pX respectively. Using (4.12)~ we obtain LE~ 4.6. - Zet ~ be a submanifold tangent to the structure vector field ~ o] a Sasakian manifold N. Then we have t l 1)~oo~. - Since N is a Sasakian manifold, the second fundamental form ~ of N satisfies (V~)(X, 17) ~ (Vx~)(~, l~) for any vector fields X and Y tangent to IY Let {e~} be an orthonormal frame of N. Then we have (n + 1)G(D'j, v) = a((v~)(e~, e,), v) = ~ G((V, ~)(~, e,), ~) Z [~(D; ~(~, e~), v)- ~(~(V;/, e,), v)] i e(~e,(Vei) 2", V) : ~ Ee((VetV)ei, V) + e(VVeei, V)] i i Z e(~(e~, e~), v) = (n + 1)a(~#', v) , i where v is a normal vector field to N. We have denoted by the same e~ local, orthonormal vector fields on M which extend e~ of the orthonormul frame (e~} of N, and which are covariant constant with respect to V' at x E N. Thus we have (4.13). If N is a generic submanifold of N, then F#' is tangent to N and hence D'~#' -~ 0. On the other hand, (4.9) and (4.11) imply n = Dx.# n + l (D~#)* ' ' From this and Lemma 4.6 we have L]~nv~A 4.7. - Let M be a generic submani]old o] a Kiihlerian manifold M and N be a submani]old o1 a Sasal~ian manifold N satisfying (C). Then the mean curvature vector/~ of M is parallel if and only if the mean curvature vector #' o] N is parattlel. We denote in the sequel by S and S' the squares of the length of the second fundamental forms of M and N respectively. Then (4.1), (4.6) and (4.12) imply (4.14) ~= S ~- 2 ~g(Fe~, Fe~), i=1 KENTAI~0 YAN0 - ~ASAttIIr KON: Generic submani]olds 77 where {ei} denotes an orthonormal frame of M. If M is a generic submanifold of a K~hlerian manifold 2~ and if dim T~(M)z= p, then (4.14) reduces to (4.15) S'= S ~ 2p. On the other hand, (4.14) shows that, if S ---- S', then M is an invariant submanifold, and that if S ~ S'--2n, then M is an anti-invariant submanifold. We next study the relations between covariant derivatives of the second funda- mental forms B and a. First of all, from (4.6) and (4.9), we have (VxB(]5 Z))* = 2)~.(B(~, Z))* = 2)'~. a(~*, Z*), from which ((VxB)(Y, Z))* + (B(VxY, Z))* + (B(Y, VxZ))* = = (V~.~)(~*, Z*) + ~(V~.Y*, Z*) + ~(Y*, V~.Z*). Therefore, taking account of (4.5), we see that (4.16) ((VxB)(Y, Z))* = = (Vx.~)(Y*, Z*) - G(Y*, q~X*)a($, Z*) - a(Z*, ~X*)~($, 17'). From (4.12) and (4.16) we have (4.17) (Vx.a)(Y* , Z*) = ((VxB)(Y, Z) + g(:Y, ~PX)_~Z -]- g(Z, PX)2~Y)*. On the other hand, we have (V..~)(:Y*, ~) =/)~,~(I ~*, ~)- a(V..Y, ~) - ~(I~*, V.,~) = D'~.(~*)'-- (~ V'~.~*) • ~(]~*, (~X*)'). Moreover, we have, for any vector field V normal to N, G(D'x.(cyY*) • V)= Dx.a(q~Y*! , V)--G((plz*, Dx.Vl ) = G(cyVx.Y*l , V)-}- + a(~(x., ~.), v) - a(~,, 2)'~. v) - G(~*, HvX*) + G(q~*, D~.V), from which (4.18) G(D'~.(~*)5 V) = = e(vv~.P, v) + G(w(x*, ~*), v) - a(a(x*, (~*)~), v). 78 KEi~TAgO YA_No - MisAm~o Ko~: Generic submanifolds Consequently, we have (4.19) (Vx.a)(Y* , }) = -- g(Y*, (~X*) *) -- g(X*, (q~:y.)T) _j_ ((p~(X*, Y*))". But (4.6) shows that ~(P, (~x*) *) + ,(x*, (~o~*)*) = B(Y, _PX)* + B(X, f'r)*. From this and (4.19) we have (4.20) G((Vx.~)(Y*, ~), V*) = g(PAv Y, X)*-- g(Av2Y, X)* + g(JB(X, Y), V)* for any vector field V normal to M. L]~i 4.8. - Let M be a generic submanifold of a Kdhlerian manifold M and N be a submani]old o] a Sasakian manifold N satisfyin (C). Then the second fundamental form o: of N is parallel if and only if the second fundamental form B of M satisfies the conditions (4.21) (VxB)(Y, Z) : g(.PY, X)FZ -[- g(.PZ, X)FY and (4.22) AvP = I~Av for any vector fields X, Y and Z tangent to M and any vector field V normal to M. PR00~. - If ~ is parallel, then (4.17) implies (4.21). ~Ioreover, since we have g(JB(X, ~), V) = O, (4.20) implies (4.22). We next assume that the second funda- mental form B of M satisfies (4.21) and (4.22). We next assume that the second fundamental form B of M satisfies (4.21) and (4.22). Then, we see from (4.17) that (Vx.~)(:Y* , Z*)= 0 and (4.19) implies (Vx.~)(Y* , ~)= 0. Since N is a Sasakialn manifold, /s ~)Y* is tangent to N and hence (].3) implies that the second fun- damental form ~ satisfies (Vx,~)(Y* , ~)= (V~)(X*, 17"). On the other hand, we easily see that (V~.~)(~, ~)= (V~)(X*, ~)= 0. Consequently, the second funda- mental form ~ of N is parallel. In the following we assume that M is an n-dimensional submanifold of a complex projective space C/~ with constant holomorphic sectional curvature 4 and that N is an (n -}- 1)-dimensional submanifold of a unit sphere S ~+~ such that the diagram KE~TAIr YAI~o - ~ASA~0 ]~0N: Generic submani]o~ds 79 commutes. From Lemma 4.4, we see that if M is a generic submanifold of CP ~, then N is also a generic submanifold od S ~+1. Thus we have LE~:y~A 4.9. - Zet M be an n-dimensional generic submani]old of C_P~ with ]lat normal connection and dim H~(M) > 4. I] (VxB)(Y, Z) -~ g(PY, X)FZ ~- g(t'Z, X)~Y , then the second ]undamental ]orm ~ o/N is parallel. To prove Lemma 4.9, we need the following lcmma. LE~ 4.10. - Let M be a generic submani]old o] a K~ihlerian mani]old M and N be a submani]old o/ a Sasakian mani]old N whivh satis]ies the condition (C). Then the normal connection o/ M is ]lat i] and only if the normal connection o] N is fiat. P~ooF. - Let X and I r be vector fields tangent to M and U and V be vector fields normal to M. Then, from (4.9) we have (DxD r V)* = Dx,; DI,,; V:g , (DrD x V)* = Dr.Dx.! ; V* Since [X, ]7]* ~- IX*, ;Y*] q- 2G(~X*, :Y*)~, we find (D~x,r I V)* = Dtx,,r, ~V + 2G(~oX*, ~'*)D~ V* From these equations we see that (4.23) g(R• Y) V, U)* = G(K• *, Y*) V*, U*) -- 2g(t)X, Y)*g(JV, U)*, where K ~ denotes the normal curvature tensor of N. Since M is a generic submuni- fold of .M, g(JV, U)~ 0 and hence (4.23) reduces to (4.24) g(R• Y) V, U)* : G(K• *, ~*) V*, U*) . On the other hand, from the equation of l~icci, we have (4.25) G(K~(X *, ~) V*, V*) = G(EHv., Hv.]X*, ~) . We now compute the right hand side of (4.25). First of all, we have (4.26) Hv. ~ -~ -- (?V*) ~ . 8O KE~T~0 YA~o - M).SAmar Ko~: Generic sq~bmani]olds Therefore we have, using (4.8), G([Hv., H,~X*, ~) = G(~ V*, Hv.X*)- a(q~V*, H.X*) = g(JV, AvX)*-- g(JU, AvX)* = g(AvJV, X)*-- g(AvJU, X)* ~ 0. In the last step we have used Lemma 2.1. Consequently, from (4.25) we see that G(K• *, ~)V*, U*)----0. From this and (4.24) we have our assertion. P~o0F oF LE~A 4.9. - By Lemma 4.10 the normal connection of N is fiat and, by the assumption and (4.17), (Vx~)(Y, Z) ---- 0 for any vectors X, :Y and Z tangent to N and orthogonal to $. First of all, we have (v: v:~)(z, w) = G(Z, +X)(V:~)(~, W) + G(W, ~X)(V:~)(Z, ~), (~.3s) (V~ V~)(Z, W) = G(Z, ~];)(V~)(~, W) + a(w, ~y)(v~)(z, ~), (V:~,~)(Z, W) --= 3~(X, ~Y)(V~)(Z, W), because of G(V~Z, ~) ~ -- G(Z, q~X) and G([X, Y], ~) : 2G(X, ?Y). Since the normal connection of N is flat, (4.37) and (4.28) give (4.39) ~(K(X, ~) Z, W) + ~(Z, K(X, Y) W) = = 2G(X, ~Y)(V~)(Z, W) -- G(Z, ~X)(V~)(~, W) -- G(W, ~X)(V~)(Z, ~) + + G(Z, ~')(V~)(~, W) + G(W, ~)(V~)(Z, ~). Since the ambient manifold of N is a unit sphere 2 ~+1, the equation of Gauss and (4.29) imply G(Y, Z)~(X, W)- G(X, Z)~(:Y, W) + ~(H~,(7,z)X, W)- ~(tt~,(x,~)~Y, W) -t- § G(Y, W)~(Z, X)- G(X, W)~(Y, Z) § ~(H~,(~.,~)X, Z)- o:(H~,(~,,~):Y, Z) =- = 2G(X, ~')(V~)(Z, W) -- G(Z, ~X)(V~)(~, W) -- G(W, ~oX)(V~)(Z, ~) § + G(Z, ~Y)(V~)(~, W) § ~(W, q~Y)(V~)(Z, ~). Putting W ~ Z in this equation, we find G(:Y, Z)~(X, Z)- G(X, Z)~(:Y, Z) -t- ~(H~,(~,z)X, Z)- ~(tt~,(x,z) :Y, Z) ---- = G(X, ~]~)(v~)(z, z) - G(Z, ~X)(V~)(~, Z) + G(Z, ~]~)(V~)(~, Z), KES;TAR0 u - ~ASAItlI~0 KObT: Generic submani]olds 81 from which, putting X ---- ~Y~ T~(27) (~ ~T~(27), we have (~.3o) G(Y, Z)~(~, Z)- G(~Y, Z)~(Y, Z) § § ~[~(e~, Z)G(e,, Ho~]~)G(Ho];, Z)- ~(e~, Z)G(e~, H~]:)G(Ho~]:, Z)] = i,a = G(]z, ~z)(V~)(Z, Z) § G(Z, Y)(V~)(~, Z) + G(Z, ~)(V~)(~, Z), where {e,} denotes a frame for 27 and (v~} a normal frume. Replacing Z by v'Z in (4.30), we obtain (4.3~) G(:Y, g)~(~:Y, Z)- G(~]5, Z)~(Y, Z)- -- ~ [~(e~, ~Z)a(e~, H~]:)G(]~, H~Z) -- ~(e~, ~Z)G(c~, ~ Y)G(~Y_, HoZ)] = i,a = G(Y, Y)(V~)(Z, Z) + G(Z, ~)(V~)(~, Z) + G(Z, ~)(V~)(~, Z) for any vector field Z and any Y ~ T~(N)n q~T~(N). Since the normal connection of N is fiat, we see that H~Ho = HbH~ for ~11 a, b. Thus we can choose an orthonormal frame {e~) for T~(N) such that Ha e~-= ~ e, for any a. Thus (4.31) reduces to (4.32) G(Y, e~)a(~Y, e~)- G(?Y, G)a(Y, e~)- -- ~ ~a(e~, ?~ G)G(e~, ~a~ ~) ~ G( ~, e]c) -- ~(ei, ~02 e/c)G(ci, Ha ~) ~ G(q~ ~, ek) ] : i,a = G(Y, Y)(V~a)(ek, G) -t- G(G, Y)(Vra)(~, ek) + G(ek, ?:Y)(V~r~)(~, ek) Since dim H~(M)>4, we see that dim T~(N)C~ ~T~(N)>4 and hence we can choose :Y and ~Yin 2~(5V) n ~oT~(N) such that Y and TY are orthogonM to ek. Thus (4.32) implies (V~)(e~, e~) = o for all k. Since g is symmetric, we have (V~)(X, Y)= 0 for any X, Y e T~(27). Consequently, the second fundamental form ~ of N is parallel. We now put T(X, Y, Z) ---- (VxB)(Y, Z) q- g(Y, PX)FZ q- g(Z, t)X)FY. Then T(X, Y, Z) = 0 if and only if (VxB)(Y, Z) = g(PY, X)HZ Jr g(PZ, X)HY. In the following we compute the square of the length of T. Let {e~} be an orthonormal frame for T~(M). Then we have g(T, T) ---- g(VB, VB) if- ~ [g(ej, Pe,)g(e~, PeJg(FG, Fek) q- i,j,lc q- g(G, -Pe~)g(G, Pe~)g(Fej, Fej) q- 2g((V~B)(ej, G), FG)g(ej, Pe~) -}- -}- 2g(ej, Pe~)g(G, Pe~)g(FG, Hej) q- 2g(e~, _Pe~)g((V~B)(ej, e~), Fe~)], 6 - Annali di Matematica 82 KE~o Y~o - M~s~m~o Ko~: Generic submani]olds where g(T, T) and g(VB, VB) denote the squares of the length of T and VB respec- tively. Using the fact that FP = O, we find g(T, T) : g(VB, VB) -~ 2hp 4- 4g((V~B)(Pe~, e~), Fe~) , where we h~ve put h:dimH~(M), p:dimT~(M) -~, hq-p:n:dimM. We now assume that the ambient manifold ;]~ is u complex projective space CP ~ of complex dimension m and with constant holomorphic sectional curvature 4. Then from the equntion o2 Codazzi, we have (VxB)(Y, Z) ----(V~B)(X, Z) + g(PY, Z)FX-- g(PX, Z)FY ~- 2g(X, PY)FZ , from which Zg((Ve~)(~ei, Gj), ~flej) = Z [g((VeJ ~)(ei~ Pei), ~ej) ~- ~,~ i,~ g(Pe~. Pei)g( Fei, Eel)- g(Pei~ Pei)g(Fe~, Eel) -~ 2g(ei, Pe~)g(Ft)ei~ Fe~)] . Since FP = O, we have g((V~B)(Pei, ej), Fe~) = ~ g((V~B)(e~, Pe~), Fej) -- hp . Moreover, since P is skew-symmetric, (2.13) and (2.15) imply Zg((V ,B) (Pei, Fe ) = - Z g(A o ei, (V P)ei)- hp = i,i,k i,i ~j,k i,i : ~ g(AF~j ek, J,.~ ej) -- .~.g(AF~j e~, AF,, ej) -- hp : -- hp. ~,Ir ~,~ Consequently, the square of the length of T is given by g(T, T) = G(VB, VB)- 2hp . LEPTA 4.11. - Let M be an n-dimensionat generic submani]old o/ CP% Then we have (4.33) g(VB, VB) >2hp and the equality here holds i] and only i] (VxB)(Y, Z) = g(PY, X)FZ ~- g(PZ, X)FY. KEI~T•162 YA:N0 - M:ASA:HI~0 K0~: Generic submani]olds 83 Rn~A~K. -- Let M be a re~l hypersurface of a complex projective space CP ~. Then OKU~A [8] proved that, if AP ~ PA, then the second fundamental form of iV satisfying (C) is parallel and determines M as will be shown by using the results of RYA~ [10]. Furthermore, MAEDA [7] proved that (4.21) and (4.22) are equivalent to each other for real hypersurface M of CP% 5. - Generic submanifolds. In this section we study ~n n-dimensional generic submanifold M of a complex m-dimensional projective space CP ~ with constant holomorphic sectional curvature 4. In the following, we put h -~ dim H~(M) and p = dim JT~(M) • = dim T~(M) • , h d- P = n. T]~EOR]~ 5.1. - Let M be a complete n-dimensional generic submani]old o] 0t) m with fiat normal connection. I] h>~4 and g(VB, VB)~ 2hp. then M is k k ~(s~'(r~) • • s~(r~)), n + 1 = Zm~, 2<~ P~ooP. - First of all, we consider the following commutative diagram: i' > S2m+ 1 (5.1) M. i > Cp ~ Since the normal connection of M is fiat, by Lemma 4.10, the normal connection of N is also flat. Therefore Lemmas 4.9 and 4.11 imply that the second fundamental form of N is parallel. On the other hand, if M admits a geodesic section, that is, if there exists a normal vector v such that g(B(X, Y), v) ---- 0 for all vector fields X and :Y tangent to M, then (2.6) implies that the holomorphic sectional curvature of the ambient manifold CP ,~ is zero. This is a contradiction. Consequently, the immersion i is full. Thus the immersion i' is also full by (4.6). Therefore The- orem 1.2 and Example 5 prove our assertion. T~fEO~S.~ 5.2. - Let M be a complete n-dimensional generic minimal submani]old o] Ct )~. I] g(VB, VB) ~ 2hp, S~(n-- 1)p and i] the second ]undamental ]orm A o] M satis]ies (4.22), then M is Ir ~(S~(rl)•215 rt-~V/mt/(n-d-1) (t --~ l, ... , k), rid-1 = ~ m~, i=l where ml, ..., m~ are odd numbers such that m~, ..., m~>l, p --- k-- 1. 8~ K]~TARO u - ~AShm~O Ko~: Generic submanifolds P~OOF. - By the assumption~ ~ is a minimal submanifold of S ~+1 in (5.1). On the other h~nd~ (4.15) implies that S'~>(n-- 1)p -F 2p -= (n --F 1)p , where S' denotes the square of the length of the second fundamental form of 2f. Thus our assertion follows from Theorem 1.1, Example 2 and Lemmas 4.8, 4.11. THEO~E~ 5.3. -- Let M be an n-dimensional complete generic minimal submanifold of CP ~ with flat normal connection. If S-= n- p~ then M is '--V- l where k is an odd number and p = 1. P~ooF. - We h~ve seen in the proof of Theorem 5.1 that the immersion i ~nd i' are full. Therefore our assertion follows from Theorem C and Lemmas 4.1, 4.10. Let /~j and K~ denote the sectional curvature of M and N with respect to the section spanned by {e,, e~} und {e*, e*} respectively. Then (1.2), (4.4) and (4.6) imply thut (5.2) Kij-= R,-- 3g(Pei, ej) 2 . Thus, if R~j>~3g(Pei~ e~) 2, then the sectional curvature K~j of s is non-negative. Therefore Theorem B and Lemmas ~.7, 4.10 give the following THEO~E~ 5.4. - Let M be an n-dimensional complete generic submanifold of CP ~ with parallel mean curvature vector and with flat normal connection. I] S is constant and if .Rij~3g(Pei, ej) 2 for all i, j~ then M is x... x n + 1 = 2< where ml~ ...~ m~ are odd numbers and m-= k--1). Tm~o~[ 5.5. - Let M be an n-dimensional complete generic submanifold of CP ~ with flat normal connection. If the second fundamental form o/ M is parallel~ then M is an anti-invariant submanifold of CP ~ and is n+l ~(Sl(r~) x... x Sl(r.+l)), ~r~ = 1. ~=1 PROOF. - By (4.33), if the second fundamental form B of M is parallel, then h = 0. Therefore M is an ~nti-invariant submanifold und m = n. On the other KE~TA]r TAN0 -MASAHn~O KO~: Generic submanifoIds 85 hand, M is flat if and only if the normal connection of M is flat (cf. [19], [21]). Moreover (5.2) becomes K,j ~ R,j-:-- 0, and by the second equation of (4.12) we see that the structure vector field ~ is parallel on 5 r. Therefore 2V is also flat. Since the second fundamental form of M is parallel, the mean curvature vector of M is obviously parallel. From these considerations and Theorem 5.4, we have our asser- tion. I~E~A~K. -- In Theorem 5.5, we also see, by Proposition A, that M is an anti- invariant submanifold. 6. - Commutative second fundamental form. Let M be an n-dimensionM generic submanifold of a complex m-dimensional Ki~hlerian manifold 217. If the second fundamental form of M is commutative, then the equation of l~icci (1.4) becomes g(R(X, Y) U, V) = g(R• Y) U, V) . Moreover, if the ambient manifold 217 is of constant holomorphic sectional curvature c, then (2:6) implies (6.1) lc[g(~'Y, U)g(FX, V)- g(FX, U)g(FY, V)] : g(R• Y) U, V). If the normal bundle T(M) • of M admits a parallel section, say U, then we have U)g(FX, V)- g(FX, U)g(FY, V)] = 0. Thus, if dim T~(M)• p>~2, then c = 0. Therefore we have LEPTA 6.1. - Let M be an n-dimensional generic submani]old o] a complex space ]orm Mm(c) with commutative second ]undamental ]orm. If the codimension p of M saris]its p >~2 and i] the normal bundle o] M admits a non-zero parallel section, then c ~-- 0 and the normal connection of M is flat. We now choose an orthonormal frame {e~} of M in such a way that el,..., e._.~ form an orthonormal frame for H~(M) and e~_~+l, ..., e~ an orthonormal frame for JT~(M) J- such that Je~_~+~ ---- Fe~_~+~ ~ e~+~, ..., Je~ ~-- Fe, ~ e2~ form an orthonormal frame for T~(M) • Unless otherwise stated, we use the following convention on the ranges of indices: i, j, k, .... 1,...,n; x, y, z, .... n-- p ~- l, ..., n; ~, fi, ~, .... 1, .., n-- p. 86 K:E~-TA~0 Y~-~o - ~ASA~0 Ko~: Generic submani]olds If the second fund~mentM form B of M is of the form (6.2) B(X, Y) = ag(X, Y)~ -~ ~b~g(X, ee)g(Y, e~)Fe~, where ~ is a unit vector normal to M and a and be are functions, then M is said to be pseudo-umbilical. Then we see that g(B(X, Y), Fee) : g(AeX, Y) : ag(X, Y)g(~, Fez) ~ beg(X, ee)g(Y, ee), where we write Ae instead of A~ to simplify the notation. Consequently, the second fundamental form Ae is represented by a matrix form a x O' '"u 0 0 a~ (6.3) A~ : a~ O x=n--p~ l,...,n, 0 a~be... X "". 0 ax where we have put ae = ag(~, FG). On the other hand, from Lemma 2.1, we have seen that Aee~ = A~e~. Therefore, if x-va y, then (6.3) implies ax-~ g(A~%, e~) : g(A~ee, e~) -~ 0, which means that, if p>2, we have ae~-0 for all x. Thus (6.3) reduces to (6.4) A. --~ (i 0011b... x, x ~ n--p-~ 1, ..., n. If p:l, then (6.3) becomes (6.5) Ae~- (:~...... t a-k b} KE~TAI~O u - MAsAn:rno Ko~: Generic submanilolds 87 where a ---- a~ and b ~ b~. In this case, M is an ~-umbilicul real hypersurface of a Ki~hlerian manifold /~ (see [5], [12]): When p>~2, the second fundamental form B of a pseudo-umbilical generic sub- m~nifold M satisfies (6.6) B(X, Y)= ag(X, ~)~ for any vector X ~ H~(M) and any vector !~ tangent to M. Since we have a~---- =ag(~,Fe~)-~O for all x, and ~=/:0, we have a----0 and ag(X, Y)~=O and hence (6.6) becomes (6.7) B(X, Y) -~ 0 for any vector X e H~(M) and any vector Iz tangent to M. We now consider a distribution Lf: x -~ ~f~ ~ {X ~ T~(M) : FX = 0}. Then (2.14) shows that (Vx/7) 1~ ~ 0 for ~ny vector fields X and Y tangent to M. Consequently, we obtain F VzY = DxFY'-- (VxF) I ~ -----0 for any Y ~ H(M) and for any X ~ T(M). Therefore the distribution ~q~ is parallel and the maximal integral submanifold Mx is totally geodesic in M. Moreover M~ is totally geodesic in .~r and is a complex submanifold of M. Consequently, we have P~oeosi:r!o~ 6.1. - Let M be an n-dimensional pseudo-umbilical generic submanilold o] a complex m-dimensional Kiihlerian manilold M with codimension p >~2. Then the distribution ~ is completely integrable and its maximal integral submanilold M~ is totally geodesic in M and is a totally geodesic complex submanifold o] .M. From Propositions 2.1 and 6.1, we have PROPOSITIO~ 6.2. - _Let M be an n-dimensional pseudo-umbilical generic submani- /old o1 a complex m-dimensional K~ihlerian manilold M with codimension p ~ 2. Then M is locally a Riemannian direct product o1 the 1orm M1 • M~, where MI is a real (n- p)-dimensional totally geodesic complex submani]old of M and M2 a real p-dimen- sional anti-invariant submani]old o/ _M. Moreover~ M1 is totally geodesic in M. In the sequel, we assume that M is an n-dimensional pseudo-umbilical generic submanifold of a complex space from/17~(e). If the mean curvature vector # of M is parallel, then its length is constant. Moreover, if M is non-minimal, then Lemma 6.1 implies that c = 0 when p>~2. Combining this with Proposition 6.2, we have LEnA 6.2. - .Let M be an n-dimensional pseudo-umbilical generic submani]old oI a complex space 1orm M~(c) with non-zero parallel mean curvature vector. I1 p>2~ 88 Ka~TAI~O YA~O - MASAI~mO Ko~: Generic submani]olds then c = 0 and M is locally a Riemannian direct product M~ • Ms, where M~ is a flat totally geodesic complex submanijold of M~(O) and M2 is an anti-invariant submaniJold of M~(O) with non-zero parallel mean curvature vector. Now we use the following theorem [20]. TJgEORE~ D (YAN0-KoN [20]). - Let M be an n-dimensional complete anti-invar- iant submanifold oJ C ~ (n > 1) and M be with parallel mean curvature vector and com- mutative second fundamental form. Then M is an n-dimensional plane R 5 or a pytha- gorean product of the form .or a pythagorean product o] the ]orm S~(r~) x Sl(r~) x ... x S*(r,) x/e ~ , where R ~-~ is an (n--s)-dimensional plane and n> s ) l. T~IEOlC]~ 6.1. - Let M be an n-dimensional complete pseudo-umbilical generic submani]old of a simply connected complete complex space form _M~(c) with non-zero parallel mean curvature vector. If p>2, then M is a pythagorean product of the form C(~-~)/2 • S~(rl) • • S~(%_~) • ~ in C ~ , l < s < p , or C (~-~)i~ • S~(r~) • • S~(%) in C "~ . PROOF. - By Lemma 6.2, M is Mr• and c~-0. Hence M~(c)=C ~. Since M~ is a totally geodesic complex submanifold of C5 we have M1 = C (~-~)/2. More- over, M2 is an anti-invariant submanifold of C ~-(~-~)/2 with parMle] me~m curva- ture vector and with commutative second fundamental form. Therefor% Theorem D proves our assertion. TI-IEO~E~ 6.2. -- Under +&e same assumptions as those o] Theorem 6.1, i/ M is compact, then M .is an anti-invariant submani]old and n = m and M is 81(rl) • • in (jn. ~ExA~LE 5. - We consider the following commutative diagram: M2 ~ i' > S2.+i M 2n-1 ~ > CP" KE~TA~0 YA~0 - MASA~O KO~: Generic submani]olds 89 Let M 2~ = S~'-~(r~) • S~(r2) where r~ + r~2 --~ 1. Then ~(M ~') ~- M ~-~ is called ~ geo- desic hypersphere of Ct)~, which is an ~-umbflicM real hypersurface of C/)~. We now use the following TI~:EORW~ E (TAKAGI[12]). -- If M is a complete v-umbilical real hypersur]ace in CP ~ (n>2)~ then M is a geodesic hypersphere. Tt~EOnE~ 6.3. - Let M be an n-dimensional complete pseudo-umbilical generic submani]old o] Ct )~ (m>2) with non-zero parallel mean curvature vector. Then m ~ (n + 1)/2 and M is a geodesic hypersphere o/ Ct)% P]cooF. - By the assumption and Lemma 6.1, if p>2, the holomorphie SeCtional curvature of CP ~ is zero. This is a contradiction. Thus we have p ---- 1 and M is an ~-umbilical real hypersurface of CP '~. From this and Theorem E, we have our assertion. RV.~A~K. -- A geodesic hypersphere of CP n has two constant principal curvatures. TA~GI[12] proved that if M is a connected complete real hypersurface in CP ~ with two constant principal curvatures, then M is a geodesic hypersphere. More- over, TA~AGI [13] determined real hypersurfaees of CP ~ with constant three prin- cipal curvatures. See also Ko~ [5]. The Bochner curvature tensor 0 of a complex m-dimensional K~ehlerian mani- fold 7~ is defined to be C(X, Y) Z = R(X, Y) Z 1 [g( y, Z) O~X -- g(QX, Z) Y + (2m + 4) + g(JY, Z)QJX - g(QJX, z)Jy + g(QY, Z) X - g(X, z)QY + + g(QJY, z)JX - g(JX, z)QJY - 2g(JX, QY)JZ - - 2g(JX, Y)QJZ] + (2m + 2)(2m + 4) [g(Y' Z)X - g(X, Z) Y + + g(JY, z) JX - g(JX, z) JY - 2g(JX, Y) JZ], for any vector fields X, Y and Z on/~, where R, Q and ~ are the Riemannian curva- ture tensor, the Ricei operator and the scalar curvature of M respectively. We now consider a pseudo-umbilical generic submanifold M of a Ki~hlerian manifold/~ with vanishing Boehner curvature tensor. We need the following Lemmas. Lw~L~A A (Ko~ [4]). - _Let M be a complex m-dimensional K~hlerian mani]otd with vanishing Bochner curvature tensor and M be a complex n-dimensional invariant submanilold o] M. If M is totally geodesic, then the Bochner curvature tensor o] M vanishes. 90 K:E:NTARO YANO !~/[ASA:KI~0Ko~: Generic submani]olds LE~V[A B (YA~o-Ko~[21], p. 74). - Let M be an n-dimensional (n~>4) anti- invariant submani]old o] a complex m-dimensional K~ihlerian mani]old M with vanishing Bochner curvature tensor. I] the second ]undamental ]orm B o] M satis]ies g(B(X, W), B(Y, Z)) -- g(B(X, Z), B(Y, W)) = ~[g(X, W)g(Y, Z) -- g(X, Z)g(Y, W)] ]or some scalar ]unction ~ on M, then M is eon]ormally fiat. Tn~EO~)~ 6.4. - .Let M be an n-dimensional pseudo-umbilical generic submani]old o] a complex .m-dimensional Ki~hlerian mani]old M with vanishing Bochner curvature tensor. I] the codimension p satis]ies p~4, then M is locally a ttiemannian direct product o] the ]orm M~ • where M~ is a real (n- p)-dimensionat totally geodesic complex submani]old o] 1~ and M~ has vanishing Bochner curvature tensor and M~ is a real p-dimensional con]ormally fiat, anti-invariant submani]oZd o] M. Pl~ooF. - From Proposition 6.2, M is of the form M~ • where M~ is a totally geodesic complex submanifold of _M, and M~ is an anti-invariant submanifold of 7~. Since M~ is totally geodesic, from Lemma A, the Bochner curvature tensor of M~ vanishes. Moreover, since M2 is totally geodesic in M, the second fundamental form of M~ in M is given by B. On the other hand, from (6.7), we see that AvX -~ 0 for any vector field X e H~(M). Therefore AvY e JT~(M) • for any vector Y e JT~(M) • Let X, Y, Z and W be vectors in JT~(M) • Then (2.13) and (2.17) imply g(B(X, W), B(Y, Z)) -- g(B(X, Z), B(Y, W)) = g(AFx W, A~zZ)- g(AFzZ, A~ W) ~- = g(AExA~zZ~ W)- g(AFyAF~:Z, W) = 0. Therefore Lemma B proves our assertion. RE~K. - For anti-invariant submanifolds of a K~hlerian manifold with van- ishing Bochner curvature tensor, see YA~-o [17]. 7. - Anti-invm'iant submanifolds. In this section we study a nti-invariant submanifotds of a complex space form and give a generalization of a theorem proved in Yano-Kon [19]. Let M be an n.dimensional (n > 1) compact anti-invariant submanifold of a simply connected complete complex space form _M~(c) with parallel mean curvature vector and with commutative second fundamental form. If the mean curvature vector of M is non-zero, from Lemma 6.1, we have c ~-0. Consequently, the ambient manifold Ms(c) is C ~. Therefore Theorem 6.2 implies M --~ Sl(rl) • ... • Sl(r~) in C ~ . KEXTAI~O YA~O - MASAmlr Kox: Generic submani]olds 91 If the mean curvature vector of M vanishes identically, that is, if M is a mini- mal submanifold, then M is totally geodesic (cf. Corollary 1 of YA~o-Ko~ [19]). Consequently, we have Tn]~o%v,~ 7.1. - Let M be an n-dimensional (n > 1) compact anti-invariant sub- mani]old o] a simply connected complex space ]orm _~(c) with parallel mean curvature vector. I] the second ]undamental ]orm o/ M is commutative and i] M is not totally geodesic, then M = Sl(rl) x... X S~(r~) in C '~ . This theorem was proved in [19] under the assumption that the second funda- mental form of M is parallel. If the second fundamental form of M is parallel, then the mean curvature vector of M is obviously parallel. We next give an application of a theorem of Wells [15]. In [15] Wells proved that any compact anti-invariant submanifold M of dimension n imbedded in a com- plex number space C ~ has z(M) ---= 0, where z(M) denotes the Euler-Poincar6 charac- teristic of M. We now assume that n-----4 and M is compact. Then the Gauss- Bonnet formula is given by fEiiRil~- 411QII~ + rq*l = 32~x(2~) : o, M where r denotes the scalar curvature of M and *1 is the volume element and [IRH~, IIQII ~ are respectively given by IIRII = = g(R, R) = Z a(R(e~, c~) e~, R(e~, c~.)e~), i,j,,~ NQ]I ~ =- g(Q, Q) -= Zg(Qe~, Qe,) , for an orthonormal frame {e~} of M, where Q is the l~icci operator of M. Suppose that M is Einstein, then 4]fQI]~= r ~ and hence fllRIl~*l - 0. M Therefore IIRII~--- O, which means that M is flat. Thus we have Tm~Ol~E~ 7.2. - Zet M be a 4-dimensional compact anti-invariant submani]old imbedded in C 4. If M is an Einstein maul]old, then M is ]lat. 92 ~E~NTARO YA~'O - ~:~/[ASAttII~O KOST: Generic submani]olds BIBLIOGRAPHY [1] B. Y. C~EN - K. OGIUE, On totally real submani/olds, Trans. Amer. Math. Sot., 19$ (1974), pp. 257-266. [2] S. ISI~IHAI~A - K. YANO, On integrability conditions of a structure f satisfying ]~ + f = O, Quart. J. of Math., Oxford, (2), 15 (1964), pp. 217-222. [3] K. 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