REDUCTION OF THE CODIMENSION

VOL. 74 1997 NO. 2

REDUCTION OF THE CODIMENSION

OF A GENERIC MINIMAL SUBMANIFOLD IMMERSED IN A COMPLEX PROJECTIVE SPACE

BY

MASAHIRO Y A M A G A T A

MASAHIRO K O N (HIROSAKI)

1. Introduction. Let M be a Kaehlerian manifold with almost complex structure J and M be a Riemannian manifold isometrically immersed in M . We denote by T x (M ) and T x (M ) ^⊥ the tangent space and the normal space of M respectively at a point x of M . If J T x (M ) ^⊥ ⊂ T x (M ) for any point x of M , then we call M a generic submanifold of M . If J T x (M ) ^⊥ = T x (M ), then M is an anti-invariant (or totally real ) submanifold of M . If a generic submanifold is not anti-invariant, then we call it a proper generic subman- ifold . In [1] the second author proved that if the Ricci tensor S of a com- pact n-dimensional generic minimal submanifold M of a complex projec- tive space CP ^m satisfies S(X, X) ≥ (n − 1)g(X, X) + 2g(P X, P X), then M is a real projective space RP ⁿ , or M is the pseudo-Einstein real hy- persurface π(S ^(n+1)/2 (p1/2) × S ^(n+1)/2 (p1/2)), where P X is the tangen- tial part of J X and π denotes the projection with respect to the fibration S ¹ → S ^2m+1 → CP ^m , S ^k (r) being the k-dimensional Euclidean sphere with radius r. On the other hand, Maeda [2] studied an n-dimensional complete minimal real hypersurface M with (n − 1)g(X, X) ≤ S(X, X) ≤ (n + 1)g(X, X), and proved that M is congruent to π(S ^(n+1)/2 (p1/2) × S ^(n+1)/2 (p1/2)). The purpose of the present paper is to prove the follow- ing

Theorem 1. Let M be a compact n-dimensional proper generic minimal submanifold of a complex m-dimensional projective space CP ^m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X) for any vector field X tangent to M , then M is a real hypersurface of CP ^m , that is, 2m − n = 1.

2. Preliminaries. Let CP ^m denote the complex projective space of complex dimension m (real dimension 2m) equipped with the standard sym- metric space metric g normalized so that the maximum sectional curvature

1991 Mathematics Subject Classification: 53C55, 53C40.

[185]

is four. We denote by J the almost complex structure of CP ^m . Let M be a real n-dimensional Riemannian manifold isometrically immersed in CP ^m . We denote by the same g the Riemannian metric tensor field induced on M from that of CP ^m . Covariant differentiation with respect to the Levi-Civita connection in CP ^m (resp. M ) will be denoted by ∇ (resp. ∇). Then the Gauss and Weingarten formulas are respectively given by

∇ _X Y = ∇ X Y + B(X, Y ) and ∇ _X V = −A V X + D X V

for all vector fields X, Y tangent to M and every vector field V normal to M , where D denotes covariant differentiation with respect to the linear connection induced in the normal bundle T (M ) ^⊥ . A and B are both called the second fundamental forms of M , and are related by g(B(X, Y ), V ) = g(A V X, Y ). For the second fundamental form A we define its covariant derivative ∇ X A by

(∇ X A) V Y = ∇ X (A V Y ) − A D

V Y − A V ∇ _X Y.

If TrA V = 0 for any vector field V normal to M , then M is said to be minimal , where Tr denotes the trace of an operator. In the following, we assume that M is a generic submanifold of CP ^m . Then the tangent space T x (M ) is decomposed as follows:

T x (M ) = H x (M ) ⊕ J T x (M ) ^⊥

at each point x of M , where H x (M ) denotes the orthogonal complement of J T x (M ) ^⊥ in T x (M ). Then we see that H x (M ) is a holomorphic subspace of T x (M ). If M is a real hypersurface of CP ^m , then M is obviously a generic submanifold of CP ^m . In the following, we put 2m − n = p, which is the codimension of M . For a vector field X tangent to M , we put

J X = P X + F X,

where P X is the tangential part of J X and F X the normal part of J X.

Then P is an endomorphism on the tangent bundle T (M ), and F is a normal bundle valued 1-form on the tangent bundle T (M ). Then we see that F P X = 0 and P ² X = −X − J F X. Moreover, we have

(∇ X P )Y = J B(X, Y ) + A F Y X, (∇ X F )Y = −B(X, P Y ),

where we have put (∇ X P )Y = ∇ X (P Y )−P ∇ X Y and (∇ X F )Y = D X (F Y )

− F ∇ X Y . For any vector field U normal to M , we also have

∇ _X J U = −P A U X + J D X U, B(X, J U ) = −F A U X.

For all vector fields U and V normal to M , we obtain

A U J V = A V J U.

Let R denote the Riemannian curvature tensor of M . Then we have the Gauss equation

R(X, Y )Z = g(Y, Z)X − g(X, Z)Y + g(P Y, Z)P X − g(P X, Z)P Y + 2g(X, P Y )P Z + A B(Y,Z) X − A B(X,Z) Y.

The Codazzi equation of M is given by

(∇ X A) V Y − (∇ Y A) V X = g(F X, V )P Y − g(F Y, V )P X − 2g(X, P Y )J V.

We now define the curvature tensor R ^⊥ of the normal bundle of M by R ^⊥ (X, Y ) = [D X , D Y ] − D _{[X,Y ]} . Then we have the Ricci equation

g(R ^⊥ (X, Y )U, V )

= g([A U , A V ]X, Y ) + g(F Y, U )g(F X, V ) − g(F X, U )g(F Y, V ).

If R ^⊥ vanishes identically, the normal connection of M is said to be flat . 3. Proof of the theorem. From the Gauss equation the Ricci tensor S of M is given by

S(X, Y ) = (n − 1)g(X, Y ) + 3g(P X, P Y ) − X

a

g(A a X, A a Y ) for all vector fields X and Y tangent to M , where we have put A a = A v

, {v a } being an orthonormal basis of the normal space of M . By assumption we have

S(X, X) − (n − 1)g(X, X) = 3g(P X, P X) − X

a

g(A a X, A a X) ≥ 0.

Hence we obtain, for any vector field V normal to M , A a J V = 0

for all a. This means that A U J V = 0 for all vector fields U and V normal to M . Using the equation above, we find

(∇ X A) U J V + A U ∇ X J V = (∇ X A) U J V − A U P A V X = 0, from which

g((∇ X A) U J V, Y ) = g((∇ X A) U Y, J V ) = g(A U P A V X, Y ).

Thus we have, by the Codazzi equation,

2g(P X, Y )g(V, U ) = g(A U P A V X, Y ) + g(A V P A U X, Y ).

In particular, we obtain

A V P A V X = P X

for any vector field X tangent to M and any vector field V normal to M . On the other hand, we have

S(P X, P X) = (n + 2)g(P X, P X) − X

a

g(A a P X, A a P X), from which

X

a

g(A a P X, A a P X) = (n + 2)g(P X, P X) − S(P X, P X)

= X

a

g(A a P A a X, P X) + (n + 2 − p)g(P X, P X) − S(P X, P X), where we have put p = 2m − n, which is the codimension of M . Therefore we obtain

1 2

X

a

|[P, A a ]| ² = (n + 2 − p)(n − p) − X

i

S(P e i , P e i )

= (n + 2 − p)(n − p) − (n + 2)(n − p) + X

a

Tr A ² _a

= −(n − p)p + X

a

Tr A ² _a ,

where {e i } denotes an orthonormal basis of the tangent space of M . By assumption we see

0 ≤ X

i

S(P e i , P e i ) − (n − 1)(n − p) = 3(n − p) − X

a

Tr A ² _a . Hence we have

X

a

Tr A ² _a ≤ 3(n − p).

Consequently, we conclude that 1

2 X

a

|[P, A _a ]| ² ≤ −(n − p)p + 3(n − p) = (n − p)(3 − p).

Since M is proper, we must have n > p. Hence we have p ≤ 3. Suppose p = 3. Then P A a = A a P for all a, and hence

A a P A a X = A ² _a P X = P X for all a. This implies that P

a TrA ² _a = n − p. Moreover, we have S(P X, P X) = (n + 2)g(P X, P X) − X

a

g(A a P X, A a P X)

= (n + 2 − p)g(P X, P X) = (n − 1)g(P X, P X),

S(J V, J V ) = (n − 1)g(V, V ).

Therefore, M is Einstein. Since we have

g(A a P A b X, Y ) + g(A b P A a X, Y ) = 2g(P X, Y )g(v a , v b ), it follows that

g(A a P X, A b P X) = 0

for a 6= b. Suppose A a X = kX for X ∈ P T x (M ). Then A ² _a X = k ² X = X.

Hence we have k = ±1 6= 0. Moreover, we obtain 0 = g(A a X, A b X) = kg(X, A b X),

from which g(A b X, X) = 0 for b 6= a. This is a contradiction to the fact A ² _a X = X for all a and for X ∈ P T x (M ). Thus we must have p 6= 3. We next suppose that p = 2. Then

X

a,i,j

g(∇ j J v a , e i )g(e j , ∇ i J v a )

= X

a,i,j

[g(P A a e j , e i )g(e j , P A a e i ) − g(P A a e j , e i )g(e j , J D i v a )

− g(J D _j v a , e i )g(e j , P A a e i ) + g(J D j v a , e i )g(e j , J D i v a )]

= − X

a,j

g(P A a e j , A a P e j ) + X

a,i,j

g(D j v a , J e i )g(J e j , D i v a )

= X

a

Tr(P A a ) ² + X

a,b,c

g(D J b v a , v c )g(v b , D J c v a )

= X

a

Tr(P A a ) ² + X

a,b

g(D J b v a , v b ) ² ,

where we have put ∇ j , D j , D J b as ∇ e

, D e

, D J v

to simplify notation, and a, b, c = 1, 2. We also have

X

a

(div J v a ) ² = X

a,i,j

g(∇ i J v a , e i )g(∇ j J v a , e j )

= X

a,i,j

g(J D i v a , e i )g(J D j v a , e j )

= X

a,i,j

g(D i v a , J e i )g(D j v a , J e j ) = X

a,b

g(D J b v a , v b ) ² . Generally, we have (cf. Yano [3])

div(∇ X X) − div((div X)X)

= S(X, X) + X

i,j

g(∇ j X, e i )g(e j , ∇ i X) − (div X) ² .

Using the equations above, we obtain X

a

div(∇ J a J v a ) − X

a

div((div J v a )J v a )

= X

a

S(J v a , J v a ) + X

a

Tr(P A a ) ²

= X

a

(n − 1) + X

a

Tr(P A a ) ²

= 2(n − 1) + 1 2

X

a

|[P, A a ]| ² + X

a

Tr(P ² A ² _a )

= 2(n − 1) − 2(n − 2) + X

a

Tr A ² _a + X

a

Tr(P ² A ² _a ) ≥ 2.

If M is compact, the equation above gives a contradiction. Thus we have p 6= 2. Therefore, we must have p = 1, and hence M is a real hypersurface of CP ^m . This proves Theorem 1.

From Theorem 1 and a theorem of Maeda [2] we have

Theorem 2. Let M be a compact n-dimensional proper generic minimal submanifold of a complex m-dimensional projective space CP ^m . If the Ricci tensor S of M satisfies (n − 1)g(X, X) ≤ S(X, X) ≤ (n + 1)g(X, X) for any vector field X tangent to M , then M is a pseudo-Einstein real hypersurface π(S ^(n+1)/2 (p1/2) × S ^(n+1)/2 (p1/2)).

REFERENCES

[1] M. K o n, Generic minimal submanifolds of a complex projective space, Bull. London Math. Soc. 12 (1980), 355–360.

[2] S. M a e d a, Real hypersurfaces of a complex projective space II , Bull. Austral. Math.

Soc. 29 (1984), 123–127.

[3] K. Y a n o, On harmonic and Killing vector fields, Ann. of Math. 55 (1952), 38–45.

Department of Mathematics Faculty of Education Hirosaki University Hirosaki, 036 Japan

E-mail: yamagata@fed.hirosaki-u.ac.jp

Received 28 October 1996