On the Chern-Type Problem in a Complex Projective Space

Hyang Sook Kim and Yong-Soo Pyo

Abstract

Chern pointed out that it is interesting to study the distribution of the values of the squared norm | α |_{2}=h_{2} of the second fundamental form α of the Kähler manifold. The purpose of this paper is to investigate the Chern-type problem in a complex projective space.

Mathematics Subject Classification: 53C50; 53C55, 53C56. Key words: Kalher manifold, second fundamental form, Chern problem, complex submanifolds

1 Introduction

Let M be an n-dimensional submanifold of an (n+p)-dimensional complex space form M^{n+p}(c) of constant holomorphic sectional curvature c. Chern pointed out that it is interesting to study the distribution of the values of the squared norm | α |_{2}=h_{2} of the second fundamental form α of M. The first value is of course 0 in the case where M is totally geodesic.

The purpose of this paper is to investigate the Chern-type problem in a complex space form. In this paper, the second fundamental form of complex submanifolds on a complex projective space CP^{n+p}(c) is treated. We prove the following Theorem. Let M be an n(=^{>} 3)-dimensional complete complex submanifold of an (n+ p)-dimensional complex projective space CP^{n+p}(c), p > 0, of constant holomorphic sectional curvature c. If the squared norm | α |_{2} of the second fundamental form α on M satisfies

| α |_{2}=^{<} \frac{c(n^{2} − 4)}{12n(n^{2} − 1)},

then M is totally geodesic.

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 69-81 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 Kähler manifolds

In this section, we shall consider M an n(=^{>} 2)-dimensional connected Kähler manifold. Then a local unitary frame field {E_{j}} = {E_{1}, ⋯, E_{n}} on a neighborhood of M can be chosen. This is a complex linear frame which is orthonormal with respect to the Kähler metric g of M, that is, g^{c}(E_{j}, Ē_{k}) = δ_{jk.} Its dual frame field {ω_{j}} = {ω_{1}, ⋯, ω_{n}} consists of complex valued 1-forms of (1, 0) on M such that ω_{j}(E_{k}) = δ_{jk} and ω_{1}, ⋯, ω_{n}, ω̄_{1}, ⋯, ω̄_{n} are linearly independent. Thus the natural extension g^{c} of the Kähler metric g of M can be expressed as g^{c} = 2 ∑_{j} ω_{j} ⊗ ω̄_{j.} Associated with the frame field {E_{j}}, there exist complex valued forms ω_{ik,} where the indices i and k run over the range 1, ⋯, n. They are usually called connection forms on M such that they satisfy the structure equations of M :

(2.1)  dω_{i} + ∑_{j} ω_{ij} ∧ ω_{j} = 0, ω_{ij} + ω̄_{ji} = 0,

(2.2)  dω_{ij} + ∑_{k} ω_{ik} ∧ ω_{kj} = Ω_{ij,}

(2.3)  Ω_{ij} = ∑_{k,l} R_{ījkl̄}ω_{k} ∧ ω̄_{l,}

where Ω = (Ω_{ij}) (resp. R_{ījkl̄}) denotes the curvature form (resp. the components of the Riemannian curvature tensor R) of M. The second formula of (2.1) means the skew-Hermitian symmetric of Ω_{ij,} which is equivalent to the symmetric condition

R_{ījkl̄} = R̄_{j̄ilk̄.}

Moreover, the first Bianchi identity implies the further symmetric relations

(2.4)  R_{ījkl̄} = R_{īkjl̄} = R_{l̄kjī} = R_{l̄jkī.}

Next, relative to the frame field chosen above, the Ricci tensor S of M can be expressed as follows

(2.5)  S = ∑_{i,j}(S_{ij̄}ω_{i} ⊗ ω̄_{j} + S_{īj}ω̄_{i} ⊗ ω_{j}),

where S_{ij̄} = ∑_{k} R_{k̄kij̄} = S_{j̄i} = S̄_{īj.} The scalar curvature K of M is also given by

(2.6)  K = 2 ∑_{j} S_{jj̄.}

Now, the components R_{ījkl̄m} and R_{ījkl̄m̄} (resp. S_{ij̄k} and S_{ij̄k̄}) of the covariant derivative of the Riemannian curvature tensor R (resp. the Ricci tensor S) are obtained by

∑_{m}(R_{ījkl̄m}ω_{m} + R_{ījkl̄m̄}ω̄_{m}) = dR_{ījkl̄}−

− ∑_{m}(R_{m̄jkl̄}ω̄_{mi} + R_{īmkl̄}ω_{mj} + R_{ījml̄}ω_{mk} + R_{ījkm̄}ω̄_{ml}),

∑_{k}(S_{ij̄k}ω_{k} + S_{ij̄k̄}ω̄_{k}) = dS_{ij̄} − ∑_{k}(S_{kj̄}ω_{ki} + S_{ik̄}ω̄_{kj}).

The second Bianchi formula is given by

R_{ījkl̄m} = R_{ījml̄k,}

and hence we have

S_{ij̄k} = S_{kj̄i} = ∑_{l} R_{j̄ikl̄l,} K_{i} = 2 ∑_{j} S_{ij̄j,}

where dK = ∑_{j}(K_{j}ω_{j} + K̄_{j}ω̄_{j}). The components S_{ij̄kl} and S_{ij̄kl̄} of the covariant derivative of S_{ij̄k} are expressed by

(2.7)  ∑_{l}(S_{ij̄kl}ω_{l} + S_{ij̄kl̄}ω̄_{l}) = dS_{ij̄k} − ∑_{l}(S_{lj̄k}ω_{li} + S_{il̄k}ω̄_{lj} + S_{ij̄l}ω_{lk}).

By the exterior differentiation of the definition of S_{ij̄k} and taking account of (2.7), the Ricci formula for the Ricci tensor S is given by

S_{ij̄kl̄} − S_{ij̄l̄k} = ∑_{m}(R_{l̄kim̄}S_{mj̄} − R_{l̄kmj̄}S_{im̄}).

The sectional curvature of the non-degenerate holomorphic plane P spanned by u and Ju is called the holomorphic sectional curvature, which is denoted by H(P) = H(u). The Kähler manifold M is said to be of constant holomorphic sectional curvature if its holomorphic sectional curvature H(P) is constant for all nondegenerate holomorphic planes P and for all points of M. Then M is called a complex space form, which is denoted by M_{s}^{n}(c′) provided that it is of constant holomorphic sectional curvature c′, of complex dimension n. The standard models of complex space forms are the following three kinds : the complex Euclidean space C^{n}_{,} the complex projective space CP^{n}(c′) or the complex hyperbolic space CH^{n}(c′)_{,} according as c′ = 0, c′ > 0 or c′ < 0. It is seen that they are only complete, simply connected and connected complex space forms of dimension n.

The Riemannian curvature tensor R_{ījkl̄} of M_{s}^{n}(c′) is given by

(2.8)  R_{ījkl̄} = \frac{c′}{2} (δ_{ij}δ_{kl} + δ_{ik}δ_{jl}).

3 Complex submanifolds

Let (M′, g′) be an (n + p)-dimensional connected Kähler manifold and let M be an ndimensional connected complex submanifold of M′. Then M is the Kähler manifold endowed with the induced metric tensor g. We choose a local unitary frame field {E_{A}} = {E_{1}, ⋯, E_{n+p}} on a neighborhood of M′ in such a way that restricted to M, E_{1}, ⋯, E_{n} are tangent to M and the others are normal to M. Here and in the sequel, the following convention on the range of indices is used throughout this paper, unless otherwise stated :

A, B, ⋯ = 1, ⋯, n, n + 1, ⋯, n + p,
i, j, ⋯ = 1, ⋯, n,
x, y, ⋯ = n + 1, ⋯, n + p.

With respect to the unitary frame field {E_{A}}, let {ω_{A}} = {ω_{i}, ω_{x}} be its dual frame field. Then the Kähler metric tensor g′ of M′ is given by g′ = 2 ∑_{A} ω_{A} ⊗ω̄_{A.} The canonical forms ω_{A} and the connection forms ω_{AB} of the ambient space satisfy the structure equations

(3.1)  \begin{matrix}dω_{A} + ∑_{B} ω_{AB} ∧ ω_{B} = 0, ω_{AB} + ω̄_{AB} = 0, \\ dω_{AB} + ∑_{C} ω_{AC} ∧ ω_{CB} = ΩAB,^{′} \\ ΩAB^{′} = ∑_{C,D} RĀBCD̄^{′}ω_{C} ∧ ω̄_{D,} \end{matrix}

where Ω′ = (ΩAB^{′}) (resp. RĀBCD̄^{′}) denotes the curvature form with respect to the unitary frame field {E_{A}} (resp. the components of the Riemannian curvature tensor R′) of M′. Restricting these forms to the submanifold M, we have

(3.2)  ω_{x} = 0,

and the induced Kähler metric tensor g of M is given by g = 2 ∑_{j} ω_{j} ⊗ ω̄_{j}. Then {E_{j}} is a local unitary frame field with respect to this metric and {ω_{j}} is a local dual frame field due to {E_{j}}, which consists of complex valued 1-forms of type (1.0) on M. Moreover, ω_{1}, ⋯, ω_{n}, ω̄_{1}, ⋯, ω̄_{n} are linearly independent. It follows from (3.2) and Cartan’s lemma that the exterior derivatives of (3.2) give rise to

(3.3)  ω_{xi} = ∑_{j} h_{ij}^{x}ω_{j,} h_{ij}^{x} = h_{ji}^{x}.

The quadratic form ∑_{i,j,x} h_{ij}^{x}ω_{i} ⊗ ω_{j} ⊗ E_{x} with values in the normal bundle is called the second fundamental form of the submanifold M. From the structure equations of M′, it follows that the structure equations for M are similarly given by

(3.4)  \begin{matrix}dω_{i} + ∑_{j} ω_{ij} ∧ ω_{j} = 0, ω_{ij} + ω̄_{ji} = 0, \\ dω_{ij} + ∑_{k} ω_{ik} ∧ ω_{kj} = Ω_{ij,} \\ Ω_{ij} = ∑_{k,l} R_{ījkl̄}ω_{k} ∧ ω̄_{l}, \end{matrix}

where Ω = (Ω_{ij}) (resp. R_{ījkl̄}) denotes the curvature form with respect to the unitary frame field {E_{A}} (resp. the component of the Riemannian curvature tensor R) of M.

Moreover, the following relationships are obtained :

(3.5)  \begin{matrix}dω_{xy} + ∑_{A} ω_{xA} ∧ ω_{Ay} = Ω_{xy,} \\ Ω_{xy} = ∑_{k,l} R_{x̄ykl̄}ω_{k} ∧ ω̄_{l,} \end{matrix}

where Ω_{xy} is called the normal curvature form of M. For the Riemannian curvature tensors R and R′ of M and M′, respectively, it follows from (3.1), (3.3) and (3.4) that we have the Gauss equation

(3.6)  R_{ījkl̄} = R′_{ījkl̄} − ∑_{x} h_{jk}^{x}h̄_{il,}^{x}

and by means of (3.1), (3.3) and (3.5), we have

R_{x̄ykl̄} = R′_{x̄ykl̄} + ∑_{j} h_{kj}^{x}h̄_{jl.}^{y}

Using (2.5), (2.6) and (3.6), the components of the Ricci tensor S and the scalar curvature K of M are given by

(3.7)  \begin{matrix}S_{ij̄} = ∑_{k} R′_{k̄kij̄} − h_{ij̄,}^{2} \\ K = 2(∑_{j,k} R′_{k̄kjj̄} − h_{2}), \end{matrix}

where h_{ij̄}^{2} = h_{j̄i}^{2} = ∑_{k,x} h_{ik}^{x}h̄_{kj}^{x} and h_{2} = ∑_{j} h_{jj̄.}^{2}

Now, the components h_{ijk}^{x} and h_{ijk̄}^{x} of the covariant derivative of the second fundamental form of M are given by

\begin{aligned}∑_{k}(h_{ijk}^{x}ω_{k} + h_{ijk̄}^{x}ω̄_{k}) = dh_{ij}^{x} & − & ∑_{k}(h_{kj}^{x}ω_{ki} + h_{ik}^{x}ω_{kj}) \\ & + & ∑_{y} h_{ij}^{y}ω_{xy}. \end{aligned}

Then, substituting dh_{ij}^{x} into the exterior derivative of (3.3), we have

h_{ijk}^{x} = h_{jik}^{x} = h_{ikj,}^{x} h_{ijk̄}^{x} = −R′_{x̄ijk̄.}

Similarly the components h_{ijkl}^{x} and h_{ijkl̄}^{x} of the covariant derivative of h_{ijk}^{x} can be defined by

\begin{aligned}∑_{l}(h_{ijkl}^{x}ω_{l} + h_{ijkl̄}^{x}ω̄_{l}) = dh_{ijk}^{x} & − & ∑_{l}(h_{ljk}^{x}ω_{li} + h_{ilk}^{x}ω_{lj} \\ & + & h_{ijl}^{x}ω_{lk}) + ∑_{y} h_{ijk}^{y}ω_{xy}, \end{aligned}

and by the simple calculation the Ricci formula for the second fundamental form are given by

\begin{matrix}h_{ijkl}^{x} = h_{ijlk,}^{x} \\ h_{ijkl̄}^{x} − h_{ijl̄k}^{x} = ∑_{r}(R_{l̄kir̄}h_{rj}^{x} + R_{l̄kjr̄}h_{ir}^{x}) − ∑_{y} R_{x̄ykl̄}h_{ij}^{y}. \end{matrix}

In particular, let the ambient space be an (n + p)-dimensional complex space form M^{n+p}(c) of constant holomorphic sectional curvature c. Then, from (2.8), (3.6) and (3.7), we get

(3.8)  R_{ījkl̄} = \frac{c}{2}(δ_{ij}δ_{kl} + δ_{ik}δ_{jl}) − ∑_{x} h_{jk}^{x}h̄_{il}^{x},

(3.9)  \begin{matrix}S_{ij̄} = \frac{(n + 1)c}{2}δ_{ij} − h_{ij̄}^{2}, \\ h_{ijk̄}^{x} = 0. \end{matrix}

And hence from (3.8) we obtain

(3.10)  \begin{aligned}h_{ijkl̄}^{x} & = & \frac{c}{2}(h_{ij}^{x}δ_{kl} + h_{jk}^{x}δ_{il} + h_{ki}^{x}δ_{jl}) \\ & − & ∑_{r,y}(h_{ri}^{x}h_{jk}^{y} + h_{rj}^{x}h_{ki}^{y} + h_{rk}^{x}h_{ij}^{y})h̄_{rl}^{y}. \end{aligned}

Let us denote by h_{4} = ∑_{i,j} h_{ij̄}^{2}h_{jī}^{2} and A = (A_{y}^{x}), where A_{y}^{x} = ∑_{i,j} h_{ij}^{x}h̄_{ij}^{y}. Then, by

means of (3.10), the Laplacian △h_{2} of the function h_{2} is given by

(3.11)  △h_{2} = \frac{c}{2}(n + 2)h_{2} − 2h_{4} − T rA^{2} + ∑_{i,j,k,x} h_{ijk}^{x}h̄_{ijk.}^{x}

4 Totally real bisectional curvature

Let (M, g) be an n-dimensional Kähler manifold with almost complex structure J. In this section, we consider the totally real bisectional curvature on M.

A plane section P in the tangent space T_{x}M of M at any point x in M is said to be totally real if P is orthogonal to JP. For the non-degenerate totally real plane P spanned by orthonormal vectors u and v, the totally real bisectional curvature B(u, v) is defined by

(4.1)  B(u, v) = \frac{g(R(u, Ju)Jv, v)}{g(u, u)g(v, v)}.

For a complex submanifold, using the first Bianchi identity to (4.1) and the fundamental properties of the Riemannian curvature tensor of a complex submanifold, we get

(4.2)  B(u, v) = g(R(u, v)v, u) + g(R(u, Jv)Jv, u) = K(u, v) + K(u, Jv),

where K(u, v) means the sectional curvature of the plane spanned by u and v. Example 4.1. Let Q^{n} be a complex quadric in a complex projective space CP^{n+1}(c) of constant holomorphic sectional curvature c. Then it is seen in Kobayashi and Nomizu [4] that its totally real bisectional curvature B satisfies

0 =^{<} B =^{<} \frac{c}{2}.

From now on, we suppose that u and v are orthonormal vectors in the non-

degenerate totally real plane P. If we put u′ = \frac{1}{\sqrt{2}}(u + v) and v′ = \frac{1}{\sqrt{2}}(u − v),

then it is easily seen that

g(u′, u′) = 1, g(v′, v′) = 1, g(u′, v′) = 0.

Thus we get

B(u′, v′) = g(R(u′, Ju′)Jv′, v′) = \frac{1}{4}\{H(u) + H(v) + 2B(u, v) − 4K(u, Jv)\},

where H(u) = K(u, Ju) means the holomorphic sectional curvature of the holomorphic plane spanned by u and Ju. Hence we have

(4.3)  4B(u′, v′) − 2B(u, v) = H(u) + H(v) − 4K(u, Jv).

If we put u′′ = \frac{1}{\sqrt{2}}(u + Jv) and v′′ = \frac{1}{\sqrt{2}}(Ju + v), then we get

g(u′′, u′′) = 1, g(v′′, v′′) = 1, g(u′′, v′′) = 0.

Using the similar method as (4.3), we have

(4.4)  4B(u′′, v′′) − 2B(u, v) = H(u) + H(v) − 4K(u, v).

Summing up (4.3) and (4.4) and taking account of (4.2), we obtain

(4.5)  2B(u′, v′) + 2B(u′′, v′′) = H(u) + H(v).

Next, let M be an n(=^{>} 3)-dimensional complex submanifold of M′ = M^{n+p}(c) and let b(M) or a(M) be the supremum or the infimum of the set B of the totally real bisectional curvatures on M. Suppose that the totally real bisectional curvature is bounded from above (resp. below) by a constant b (resp. a). From the assumption and (4.5), it follows that we have

(4.6)  H(u) + H(v) =^{<} 4b (resp. =^{>} 4a).

We can choose an unitary frame field {E_{1}, E_{2}, ⋯, E_{n}} on a neighborhood of

M. Let \{ω_{1}, ω_{2}, ⋯, ω_{n}\} be a dual frame field. If we put e_{j} = \frac{1}{\sqrt{2}}(E_{j} + Ē_{j}), then

{e_{1}, e_{2}, ⋯, e_{n}} is an orthonormal basis of T_{x}M. Thus the holomorphic sectional curvature H(e_{j}) of the holomorphic plane defined by E_{j} is given by

H(e_{j}) = g(R(e_{j}, Je_{j})Je_{j}, e_{j}) = R_{j̄jjj̄}.

On the other hand, it is easily seen that the plane spanned by e_{j} and e_{k} (j ≠ k) is totally real and the totally real bisectional curvature B(e_{j}, e_{k}) is given by

(4.7)  B(e_{j}, e_{k}) = g(R(e_{j}, Je_{j})Je_{k}, e_{k}) = R_{j̄jkk̄,} j ≠ k.

From the inequality (4.6) for u = e_{j} and v = e_{k,} we find

(4.8)  R_{j̄jjj̄} + R_{k̄kkk̄} =^{<} 4b (resp. =^{>} 4a), j ≠ k.

Thus we have

(4.9)  ∑_{j<k}(R_{j̄jjj̄} + R_{k̄kkk̄}) =^{<} 2bn(n − 1) (resp. =^{>} 2an(n − 1)),

which implies that

(4.10)  ∑_{j} R_{j̄jjj̄} =^{<} 2bn (resp. =^{>} 2an),

where the equality holds if and only if R_{j̄jjj̄} = 2b (resp. = 2a) for any index j. Since the scalar curvature K is given by

K = 2 ∑_{j,k} R_{j̄jkk̄} = 2(∑_{j} R_{j̄jjj̄} + ∑_{j≠k} R_{j̄jkk̄}),

we have by (4.7)

K =^{<} 2 ∑_{j} R_{j̄jjj̄} + 2bn(n − 1) (resp. =^{>} 2 ∑_{j} R_{j̄jjj̄} + 2an(n − 1)),

from which it follows that

(4.11)  ∑_{j} R_{j̄jjj̄} =^{>} \frac{K}{2} − bn(n − 1) (resp. =^{<} \frac{K}{2} − an(n − 1)),

where the equality holds if and only if R_{j̄jkk̄} = b (resp. = a) for any distinct indices j and k. In this case, M is locally congruent to M^{n}(2b) (resp. M^{n}(2a)) due to Houh [1]. Also (4.8) gives us ∑_{j≠k}(R_{j̄jjj̄} + R_{k̄kkk̄}) =^{<} 4b(n − 1) (resp. =^{>} 4a(n − 1)), so that

(n − 2)R_{j̄jjj̄} + ∑_{k} R_{k̄kkk̄} =^{<} 4b(n − 1) (resp. =^{>} 4a(n − 1)).

Combining this with (4.11), we have

(4.12)  \begin{matrix}(n − 2)R_{j̄jjj̄} =^{<} b(n − 1)(n + 4) − \frac{K}{2} \\ (resp. =^{>} a(n − 1)(n + 4) − \frac{K}{2} ), \end{matrix}

for any index j, so that the holomorphic sectional curvature R_{j̄jjj̄} is bounded from above (resp. below) for n =^{>} 3. Moreover, the equality holds for some index j if and only if M is locally congruent to M^{n}(2b) (resp. M^{n}(2a)).

On the Chern-Type Problem in a Complex Projective Space 77

Since the Ricci curvature S_{jj̄} is given by

S_{jj̄} = R_{j̄jjj̄} + ∑_{k≠j} R_{j̄jkk̄,}

we have by (4.7)

S_{jj̄} =^{<} R_{j̄jjj̄} + b(n − 1) (resp. =^{>} R_{j̄jjj̄} + a(n − 1))

and hence, from (4.12), we get

(4.13)  \begin{matrix}S_{jj̄} =^{<} \frac{1}{2(n − 2)}\{4b(n − 1)(n + 1) − K\} \\ (resp. =^{>} \frac{1}{2(n − 2)}\{4a(n − 1)(n + 1) − K\}). \end{matrix}

On the other hand, for the scalar curvature K, we see by (4.13)

K  = 2S_{jj̄} + 2 ∑_{k≠j} S_{kk̄}
=^{<} 2S_{jj̄} + \frac{1}{n − 2}(n − 1)\{4b(n − 1)(n + 1) − K\}
(resp. =^{>} 2S_{jj̄} + \frac{1}{n − 2}(n − 1)\{4a(n − 1)(n + 1) − K\}),

and hence we see

(4.14)  \begin{matrix}S_{jj̄} =^{>} \frac{1}{2(n − 2)}\{(2n − 3)K − 4b(n − 1)^{2}(n + 1)\} \\ (resp. =^{<} \frac{1}{2(n − 2)}\{(2n − 3)K − 4a(n − 1)^{2}(n + 1)\}). \end{matrix}

This together with (4.12) and R_{j̄jiī} =^{<} b implies

(4.15)  \begin{matrix}R_{j̄jkk̄} =^{>} \frac{1}{n − 2}\{(n − 1)K − (2n^{3} − 3n + 2)b\} \\ (resp. =^{<} \frac{1}{n − 2}\{(n − 1)K − (2n^{3} − 3n + 2)a\} \end{matrix}

for any distinct indices j and k.

5 Complex submanifolds of a complex projective

space

Let M′ = CP^{n+p}(c) be an (n + p)-dimensional complex projective space and let M be an n(=^{>} 3)-dimensional complex submanifold of CP^{n+p}(c). Then by (3.8), we have

R_{j̄jkk̄} = \frac{c}{2} − ∑_{x} h_{jk}^{x}h̄_{jk}^{x} =^{<} \frac{c}{2}, j ≠ k.

Thus we see that any totally real plane section satisfies B(u, v) =^{<} \frac{c}{2}_{.}

Now, let a(M) be the infimum of the set B of totally real bisectional curvatures of M. Though the set B is bounded from above, we have no information on a(M). In their paper [3], Ki and Suh proved the following theorem. Theorem 5.1. Let M be an n(=^{>} 3)-dimensional complete complex submanifold of an (n + p)-dimensional complex projective space CP^{n+p}(c). If the totally real bisectional curvatures of M are bounded from below, then there exists a constant a_{1} = \frac{c}{2n(n^{2} + 2n + 3)}(n^{3} + 2n^{2} + 2n − 2) depending only upon n and c so that if

a(M) > a_{1,} then M is congruent to a complex projective space CP^{n}(c).

In the following theorem, the above estimate is improved Theorem 5.2. Let M be an n(=^{>} 3)-dimensional complete complex submanifold of a complex projective space CP^{n+p}(c). If the totally real bisectional curvatures of M are bounded from below, then there exists a constant a_{2}(< a_{1}) such that depending only upon n and c so that if a(M) > a_{2,} then M is congruent to a complex projective space CP^{n}(c), where a_{1} > a_{2.} Proof. Assume that the set B is bounded from below by a constant a. Since the matrix H = (h_{jk̄}^{2}) defined by h_{jk̄}^{2} = ∑_{x,r} h_{jr}^{x}h̄_{rk}^{x} and the matrix A = (A_{y}^{x}) defined by

A_{y}^{x} = ∑_{j,k} h_{jk}^{x}h̄_{jk}^{y} are both positive semi-definite Hermitian ones whose all eigenvalues

μ_{j} and μ_{x} are non-negative real valued functions on M. Thus we have

h_{2}^{2} =^{>} h_{4} = ∑_{j} μ_{j,}^{2} h_{2}^{2} =^{>} T rA^{2} = ∑_{x} μ_{x}^{2}.

By (3.11), we have

Δh_{2} =^{>} \frac{c}{2}(n + 2)h_{2} − 2h_{4} − T rA^{2},

from which together with the above properties about eigenvalues, it follows that

Δh_{2} =^{>} \frac{c}{2}(n + 2)h_{2} − 3h_{2}^{2}.

A non-negative function f is defined by h_{2.} Then the above inequality is reduced to

(5.1)  Δf =^{>} −3f^{2} + \frac{c}{2}(n + 2)f.

On the other hand, since the totally real bisectional curvatures are bounded from below by a constant a, we get

R_{j̄jkk̄} =^{>} a for any j, k(j ≠ k).

Hence, by (2.8), (3.7), (4.10) and (4.11), we have

2an =^{<} ∑_{j} R_{j̄jjj̄} =^{<} \frac{c}{2}n(n + 1) − h_{2} − an(n − 1).

Thus we get

2h_{2} =^{<} (c − 2a)n(n + 1).

Therefore we have

(5.2)  f = ∑_{j} μ_{j} =^{<} \frac{1}{2}(c − 2a)n(n + 1), μ_{j} =^{>} 0,

where the first equality holds if and only if R_{j̄jjj̄} = 2a and R_{j̄jkk̄} = a for any indices j ≠ k. This means that each eigenvalue μ_{j} is bounded. On the other hand, since the Ricci curvature S_{jj̄} of M is given by

S_{jj̄} = \frac{c}{2}(n + 1) − μ_{j,}

it is also bounded. Applying the generalized maximum principle due to Omori [6] and Yau [10] to the bounded function f, we see that for any sequence ϵ_{m} of positive numbers which converges to 0 as m tends to infinity, there exists a point sequence p_{m} such that

|∇f(p_{m})| < ϵ_{m,} Δf(p_{m}) < ϵ_{m,} sup f − ϵ_{m} < f(p_{m}).

Thus, we have

(5.3)  lim_{m→∞} Δf(p_{m}) =^{<} lim_{m→∞} ϵ_{m} = 0, lim_{m→∞} f(p_{m}) = sup f.

By (5.1) and (5.3), we see

sup f(sup f − \frac{c}{6}(n + 2)) =^{>} 0,

which means

sup f = 0 or sup f =^{>} \frac{c}{6}(n + 2).

If sup f = 0, then f vanishes identically because f is non-negative. Then M is totally geodesic. Suppose that M is not totally geodesic. So, f satisfies sup f =^{>} \frac{c}{6}(n + 2). On the other hand, by (5.2), we have

sup f =^{<} \frac{1}{2}(c − 2a)n(n + 1).

Thus, we see that

a =^{<} \frac{c}{6n(n + 1)}(3n^{2} + 2n − 2).

We denote the right hand side of the above inequality by a_{2,} which is the constant depending on the dimension n and c. In this case, we can easily prove that a_{1} > a_{2}.

It completes the proof. □

About the value of the squared norm h_{2} of the second fundamental form of M, we assert the following theorem. Theorem 5.3. Let M be an n(=^{>} 3)-dimensional complete complex submanifold of a complex projective space CP^{n+p}(c). If the squared norm h_{2} of the second fundamental form on M satisfies

h_{2} =^{<} \frac{c}{12n(n^{2} − 1)}(n^{2} − 4),

then M is totally geodesic. Proof. Suppose that M is not totally geodesic. Then, by Theorem 5.2, there exists a constant a_{2} = \frac{c}{6n(n + 1)}(3n^{2} + 2n − 2) so that the infimum a(M) of the totally real

bisectional curvatures of M satisfies a(M) =^{<} a_{2.} On the other hand, it is seen that we have

R_{j̄jkk̄} = \frac{c}{2} − ∑_{x} h_{jk}^{x}h̄_{jk}^{x} =^{<} \frac{c}{2}

for any distinct indices j and k, and hence it turns out to be

b(M) =^{<} \frac{c}{2},

where the equality holds if and only if M is totally geodesic. Since M is not totally geodesic, we have b(M) < \frac{c}{2}. By (2.8), (3.7) and (4.15), we see

R_{j̄jkk̄} =^{>} \frac{1}{n − 2}\{cn(n^{2} − 1) − 2(n − 1)h_{2} − b(M)(2n^{3} − 3n + 2)\}.

By the definition of a(M), we get

a(M) =^{>} \frac{1}{n − 2}\{cn(n^{2} − 1) − 2(n − 1)h_{2} − b(M)(2n^{3} − 3n + 2)\},

from the fact that b(M) < \frac{c}{2}, it follows that we have

h_{2} > \frac{1}{4(n − 1)}(c − 2a(M))(n − 2).

Since a(M) =^{<} a_{2}, we get

h_{2} > \frac{c}{12n(n^{2} − 1)}(n^{2} − 4).

It completes the proof. □

References

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[2] T. Ikawa, H. Nakagawa and A. Romero, Product complex submanifolds of indefinite complex space forms, Rocky Mountain J. Math., 18 (1988), 601-615.

[3] U-H, Ki and Y. J. Suh, On semi-Kaehler manifolds whose totally real bisectional curvature is bounded from below, J. Korean Math. Soc., 33 (1996), 1009-1038.

[4] S. Kobayashi and K. Nomizu, Foundations of differential geometry, I and II, Interscience Publ., 1963 and 1969.

[5] S. Montiel and A. Romero, Complex Einstein hypersurfaces of indefinite complex space forms, Math. Proc. Camb. Phil. Soc. 94 (1983), 495-508.

[6] H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19(1967), 205-211.

[7] B. O’Neill, Semi-Riemannian geometry with application to relativity, Academic Press, 1983.

[8] A. Romero, Differential geometry of complex hypersurfaces in an indefinite complex space form, Univ. de Granada, 1986.

[9] A. Romero, On a certain class of complex Einstein hypersurfaces in indefinite complex space forms, Math. Z., 192 (1986), 627-635.

[10] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure and Appl. Math., 28 (1975), 201-228.

Hyang Sook Kim^{1} Department of Mathematics School of Computer Aided Science Inje University Kimhae 621-749, Korea E-mail : mathkim@ijnc.inje.ac.kr

Yong-Soo Pyo Division of Mathematical Sciences Pukyong National University Pusan 608-737, Korea E-mail : yspyo@dolphin.pknu.ac.kr

1The first author was supported by 1999 Inje University Research Grant and the second by NULWON Culture Foundation.