A Recurrent Curvature-Like Tensor on Semi-Definite Kähler Manifolds

Y-S Pyo and H.S. Kim

Abstract

The purpose of this paper is to investigate properties of semi-definite Kähler manifolds with the second curvature-like tensor H. In paticular, we study a semi-definite H-recurrent Kähler manifold.

Mathematics Subject Classification: 53C50, 53C55 Key words: Semi-definite Kähler manifold, curvature-like tensor, Ricci-like tensor, recurrent curvature tensor, complex space form

1 Introduction

As is well known, there exist some special tensors on the Kähler manifold which extend the Riemannian curvature tensor on the Riemannian manifold, for example, the concircular, projective or conformal curvature tensor. In [3], Bochner introduced three kinds of curvature-like tensors on the Kähler manifold which are closely related to the projective curvature tensor and the conformal curvature tensor, and calculated Betti numbers on a compact Kähler manifold under suitable conditions for these three curvature-like tensors. We can see detailed calculations with respect to these three kinds of curvature-like tensors in Yano and Bochner [8]. The first tensor W is called the Weyl curvature tensor and the third tensor B is known as the Bochner curvature tensor. In [4], Matsumoto and Tanno treated Kähler manifolds with parallel Bochner curvature tensor.

In this note, we investigate properties of semi-definite Kähler manifolds with the second curvature-like tensor H of the above three curvature-like tensors.

Let (M, g) be an n-dimensional semi-definite Kähler manifold n ≥ 2 with the Kähler connection ∇. Let R or S be the Riemannian curvature tensor or the Ricci tensor with components K_{ījkl̄} or S_{ij̄} and let H be the second curvature-like tensor on M with components H_{ījkl̄} defined by

H_{ījkl̄} = K_{ījkl̄} − \frac{1}{2(n + 1)}\{ϵ_{j}(δ_{ji}S_{kl̄} + δ_{jl}S_{kī}) + ϵ_{k}(δ_{ki}S_{jl̄} + δ_{kl}S_{jī})\}.

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 89-98 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

The Riemannian curvature tensor R (resp. the second curvature-like tensor H) is said to be recurrent, if there exists an 1-form α such that ∇R = α⊗R (resp. ∇H = α⊗H) and then M is said to be R-recurrent (resp. H-recurrent). The class of semi-definite Hrecurrent Kähler manifolds contains locally symmetric semi-definite Kähler manifolds, semi-definite Kähler manifolds with parallel second curvature-like tensor and semidefinite Kähler manifolds with vanishing second curvature-like tensor. From these facts it is natural to consider properties of semi-definite R-recurrent or H-recurrent Kähler manifolds.

The purpose of this paper is to prove the following Theorem. Let M be an n-dimensional semi-definite Kähler manifold (n ≥ 2) with non-null second curvature like tensor H. If it is H-recurrent, then it is R-recurrent.

2 Semi-definite Kähler manifolds

This section is concerned with recalling basic formulas on semi-Kähler manifolds.

Let M be an n-dimensional semi-definite Kähler manifold (n ≥ 2) with a semidefinite Kähler metric tensor g and almost complex structure J. From the semi-definite Kähler structure {g, J}, it follows that J is integrable and the index of g is even, say 2s(0 ≤ s ≤ n). In the case where the index 2s is contained in the range 0 < s < n, the structure {g, J} is said to be indefinite Kähler structure and, in particular, in the case where s = 0 or n, it is said to be definite Kähler structure.

In this section, we shall consider M an n-dimensional connected semi-definite Kähler manifold (n ≥ 2) of index 2s, 0 ≤ s ≤ n. Then a local unitary frame field {U_{j}} = {U_{1}, …, U_{n}} on a neighborhood of M can be chosen. This is a complex linear frame which is orthonormal with respect to the semi-definite Kähler metric g of M, that is, g(U_{j}, U_{k}) = ϵ_{j}δ_{jk,} where

ϵ_{j} = −1 or 1 according as 0 ≤ j ≤ s or s + 1 ≤ j ≤ n.

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}(U_{k}) = ϵ_{j}δ_{jk} and ω_{1}, …, ω_{n}, ω̄_{1}, …, ω̄_{n} are linearly independent. Thus the natural extension g^{c} of the semi-definite Kähler metric g of M can be expressed as g^{c} = 2 ∑_{j} ϵ_{j}ω_{j} ⊗ ω̄_{j.} Associated with the frame field {U_{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 :

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

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

Ω_{ij} = ∑_{k,l} ϵ_{kl}K_{ījkl̄}ω_{k} ∧ ω̄_{l,}

where ϵ_{k...l} = ϵ_{k} … ϵ_{l} and Ω = (Ω_{ij}) (resp. K_{ījkl̄}) denotes the curvature form (resp. components of the semi-definite Riemannian curvature tensor R) of M in Kobayashi and Nomizu [6]. The second formula of (2.1) means the skew-Hermitian symmetricity of Ω_{ij,} which is equivalent to the symmetric condition

(2.2)  K_{ījkl̄} = K̄_{j̄ilk̄.}

Moreover, substitution the third equation of (2.1) into the exterior differential of the first equation of (2.1), the first Bianchi identity

(2.3)  ∑_{j} ϵ_{j}Ω_{ij} ∧ ω_{j} = 0

is given. It implies further symmetric relations

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

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

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

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

(2.6)  r = 2 ∑_{j} ϵ_{j}S_{jj̄.}

An n-dimensional semi-definite Kähler manifold M is said to be Einstein, if the Ricci tensor S is given by

(2.7)  S_{ij̄} = \frac{r}{2n}ϵ_{i}δ_{ij,} S = \frac{r}{2n}g.

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

(2.8)  \begin{matrix}∑_{m} ϵ_{m}(K_{ījkl̄m}ω_{m} + K_{ījkl̄m̄}ω̄_{m}) = dK_{ījkl̄}− \\ − ∑_{m} ϵ_{m}(K_{m̄jkl̄}ω̄_{mi} + K_{īmkl̄}ω_{mj} + K_{ījml̄}ω_{mk} + K_{ījkm̄}ω̄_{ml}), \end{matrix}

and

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

By differentiating the third equation in (2.1) exteriorly, we have

(2.10)  K_{ījkl̄m} = K_{ījml̄k,}

and hence we have

(2.11)  S_{ij̄k} = S_{kj̄i} = ∑_{l} ϵ_{l}K_{j̄ikl̄l.}

On the other hand, the exterior differential dr of the scalar curvature r on M is given

by

(2.12)  dr = ∑_{m} ϵ_{m}(r_{m}ω_{m} + rm̄ω̄_{m}).

Putting i = j in (2.11) and summing up with respect to i, we have

(2.13)  ∑_{r} ϵ_{r}S_{rr̄k} = ∑_{r,s} ϵ_{rs}K_{r̄rks̄s} = ∑_{r,s} ϵ_{rs}K_{r̄rss̄k} = \frac{r_{k}}{2} .

Now, a semi-definite Kähler manifold M of constant holomorphic sectional curvature is called a semi-definite complex space form. An n-dimensional semi-definite complex space form of constant holomorphic sectional curvature c and of index 2s 0 ≤ s ≤ n, is denoted by M_{s}^{n}(c). The standard models of semi-definite complex space forms are the following three kinds : the semi-definite complex Euclidean space C_{s}^{n}_{,} the semi-definite complex projective space CP_{s}^{n}(c) or the semi-definite complex hyperbolic space CH_{s}^{n}(c)_{,} according as c = 0, c > 0 or c < 0. For any integer s (0 ≤ s ≤ n), it is seen that they are only complete and simply connected semi-definite complex space forms of dimension n and of index 2s. The Riemannian curvature tensor K_{ījkl̄} of M_{s}^{n}(c) is given by

(2.14)  K_{ījkl̄} = \frac{c}{2}ϵ_{jk}(δ_{ij}δ_{kl} + δ_{ik}δ_{jl}).

3 The second curvature-like tensor H

This section is concerned with the second curvature-like tensor H on a semi-definite Kähler manifold M.

Under the notation prepared in §2, the second curvature-like tensor H with components H_{ījkl̄} is defined by

(3.1)  H_{ījkl̄} = K_{ījkl̄} − \frac{1}{2(n + 1)}\{ϵ_{j}(δ_{ji}S_{kl̄} + δ_{jl}S_{kī}) + ϵ_{k}(δ_{ki}S_{jl̄} + δ_{kl}S_{jī})\}.

As is easily seen (cf. Yano and Bocher [8] in the definite case), the semi-definite Kähler manifold M with vanishing second curvature-like tensor H is of constant holomorphic sectional curvature.

On the other hand, let Z be a tensor with components Z_{ījkl̄} such that

(3.2)  Z_{ījkl̄} = K_{ījkl̄} − \frac{r}{2n(n + 1)}ϵ_{jk}(δ_{ji}δ_{kl} + δ_{jl}δ_{ki}).

In the case where the semi-definite Kähler manifold M is Einstein, the tensor Z is equivalent to the second curvature-like tensor H. It is trivial that the semi-definite Kähler manifold M with Z = 0 is of constant holomorphic sectional curvature. The tensor Z is the formal analogue to the concircular curvature tensor on the Riemannian manifold [8]. Lemma 3.1([5]). The second curvature-like tensor H with components H_{ījkl̄} satisfies

(3.3)  H_{ījkl̄} = H_{īkjl̄} = H_{l̄jkī} = H_{l̄kjī},

(3.4)  H_{ij̄} = H_{j̄i} = H̄_{jī} = \frac{1}{4(n + 1)}\{2nS_{ij̄} − rϵ_{i}δ_{ij}\}.

where H_{ij̄} is the components of the Ricci-like tensor of H defined by H_{ij̄} = ∑_{r} ϵ_{r}

H_{r̄rij̄}. Moreover, ∑_{i} H_{iī} = 0.

Theorem 3.2. Let M be an n-dimensional semi-(definite) Kähler manifold (n ≥ 2). If the Ricci tensor S is parallel, then the second fundamental tensor H is parallel to the curvature tensor R. Proof. By the definition of H, we have

(3.5)  \begin{matrix}H_{ījkl̄m} = K_{ījkl̄m}− \\ −\frac{1}{2(n + 1)}\{ϵ_{j}(δ_{ji}S_{kl̄m} + δ_{jl}S_{kīm}) + ϵ_{k}(δ_{kl}S_{jīm} + δ_{ki}S_{jl̄m})\}. \end{matrix}

Putting i = m in (3.5) and summing up with respect to i, we obtain

∑_{r} ϵ_{r}H_{r̄jkl̄r} = S_{jl̄k} − \frac{1}{4(n + 1)}(4S_{jl̄k} + r_{j}ϵ_{k}δ_{kl} + r_{k}ϵ_{j}δ_{jl}) = 0,

where we have used (2.10), (2.11), (2.13) and the relation

H_{ij̄k} = \frac{1}{4(n + 1)}(2nS_{ij̄k} − ϵ_{i}δ_{ij}r_{k}).

From this it follows that

(3.6)  4nS_{jl̄k} = r_{j}ϵ_{k}δ_{kl} + r_{k}ϵ_{j}δ_{jl}.

Putting k = l in (3.6) and summing up with respect to k, we have (n−1)r_{j} = 0. Since n ≥ 2, we get r_{j} = 0. It means that the scalar curvature r is constant, and hence by the above equation (3.6), we have S_{jl̄k} = 0, from which together with (3.5) we have H_{ījkl̄m} = K_{ījkl̄m}. Thus we obtain ∇H = ∇R.

It completes the proof. □ In [5], the authors of the present paper proved the following : Let M be an n-dimensional semi-definite Kähler manifold (n ≥ 2). Then its second

curvature-like tensor H is parallel if and only if its Riemannian curvature tensor R is parallel.

4 Semi-definite H-recurrent Kähler manifolds

Now, we assume that the curvature-like tensor H is recurrent, namely, there exists an 1-form α such that ∇H = α ⊗ H. Then M is said to be H-recurrent. In terms of components, we assume that

(4.1)  H_{ījkl̄m} = α_{m}H_{ījkl̄}.

Substituting (3.1) and (3.5) into (4.1), we get

(4.2)  \begin{matrix}K_{ījkl̄m} = α_{m}K_{ījkl̄}+ \\ +\frac{1}{2(n + 1)}[ϵ_{j}\{δ_{ji}(S_{kl̄m} − α_{m}S_{kl̄}) + δ_{jl}(S_{kīm} − α_{m}S_{kī})\} \\ +ϵ_{k}\{δ_{kl}(S_{jīm} − α_{m}S_{jī}) + δ_{ki}(S_{jl̄m} − α_{m}S_{jl̄})\}]. \end{matrix}

Putting k = l in (4.2) and summing up with respect to k, we have

(4.3)  S_{jīm} − α_{m}S_{jī} = \frac{1}{2n}\{(r_{m} − rα_{m})ϵ_{j}δ_{ji}\},

where we have used (2.5) and (2.11). Theorem 4.1. Let M be an n(≥ 2)-dimensional semi-definite Kähler manifold. Put M′ = {x ∈ M| < H, H >= 0, H ≠ 0 at x}, and assume that the subset M − M′ is dense in M. If it is H-recurrent, then it is R-recurrent. Proof. If H = 0 on an open subset U in M, then the holomorphic sectional curvature is constant and hence ∇H = ∇R = 0 on U. So it is sufficient to consider on an open set V = {x ∈ M|⟨H, H⟩ ≠ 0 at x}. The assumption ∇H = α ⊗ H implies that α_{mp̄} = α_{p̄m} on V. Thus

(4.4)  H_{ījkl̄mp̄} = (α_{mp̄} + α_{m}ᾱ_{p})H_{ījkl̄} = H_{ījkl̄p̄m}.

It follows from d^{2} H_{k̄lmp̄} = 0 combined with (4.6) that

(4.5)  ∑_{r} ϵ_{r}(−K_{ījrk̄}H_{r̄lmp̄} + K_{ījlr̄}H_{k̄rmp̄} + K_{ījmr̄}H_{k̄lrp̄} − K_{ījrp̄}H_{k̄lmr̄}) = 0.

Putting i = j in (4.5) and summing up with respect to i, we have

(4.6)  ∑_{r} ϵ_{r}(−S_{rk̄}H_{r̄lmp̄} + S_{lr̄}H_{k̄rmp̄} + S_{mr̄}H_{k̄lrp̄} − S_{rp̄}H_{k̄lmr̄}) = 0.

Moreover, differentiating (4.5) covariantly, we get

(4.7) ∑_{r} ϵ_{r}(−K_{ījrk̄q}H_{r̄lmp̄} + K_{ījlr̄q}H_{k̄rmp̄} + K_{ījmr̄q}H_{k̄lrp̄} − K_{ījrp̄q}H_{k̄lmr̄}) = 0,

where the equality is derived from (4.1) and (4.5).

Differentiating (4,6) covariantly and then, from (4.1) and (4.8), we get

(4.8)  ∑_{r} ϵ_{r}(−S_{rk̄q}H_{r̄lmp̄} + S_{lr̄q}H_{k̄rmp̄} + S_{mr̄q}H_{k̄lrp̄} − S_{rp̄q}H_{k̄lmr̄}) = 0.

A Recurrent Curvature-Like Tensor on Semi-Definite Kähler Manifolds 95 Subtraction α_{q} times (4.6) from (4.8), we have

(4.9)  \begin{matrix}∑_{r} ϵ_{r}\{−(S_{rk̄q} − α_{q}S_{rk̄})H_{r̄lmp̄} + (S_{lr̄q} − α_{q}S_{lr̄})H_{k̄rmp̄}+ \\ +(S_{mr̄q} − α_{q}S_{mr̄})H_{k̄lrp̄} − (S_{rp̄q} − α_{q}S_{rp̄})H_{k̄lmr̄}\} = 0. \end{matrix}

Substituting (4.2) into (4.7), we have

∑_{r} ϵ_{rj}[−{δ_{ji}(S_{rk̄q} − α_{q}S_{rk̄}) + δ_{jk}(S_{rīq} − α_{q}S_{rī})}H_{r̄lmp̄}+

+{δ_{ji}(S_{lr̄q} − α_{q}S_{lr̄}) + δ_{jr}(S_{līq} − α_{q}S_{lī})}H_{k̄rmp̄}+ +{δ_{ji}(S_{mr̄q} − α_{q}S_{mr̄}) + δ_{jr}(S_{mīq} − α_{q}S_{mī})}H_{k̄lrp̄}− −{δ_{ji}(S_{rp̄q} − α_{q}S_{rp̄}) + δ_{jp}(S_{rīq} − α_{q}S_{rī})}H_{k̄lmr̄} ]+

+ ∑_{r} ϵ_{r}[−ϵ_{r}{δ_{rk}(S_{jīq} − α_{q}S_{jī}) + δ_{ri}(S_{jk̄q} − α_{q}S_{jk̄})}H_{r̄lmp̄}+

+ϵ_{l}{δ_{lr}((S_{jīq} − α_{q}S_{jī}) + δ_{li}(S_{jr̄q} − α_{q}S_{jr̄})}H_{k̄rmp̄}+ +ϵ_{m}{δ_{mr}(S_{jīq} − α_{q}S_{jī}) + δ_{mi}(S_{jr̄q} − α_{q}S_{jr̄})}H_{k̄lrp̄}− −ϵ_{r}{δ_{rp}(S_{jīq} − α_{q}S_{jī}) + δ_{ri}(S_{jp̄q} − α_{q}S_{jp̄})}H_{k̄lmr̄} ] = 0.

Combining (4.9) with the above equation, we can obtain

(4.10)  \begin{matrix}∑_{r} ϵ_{rj}\{−δ_{jk}(S_{rīq} − α_{p}S_{rī})H_{r̄lmp̄} + δ_{jr}(S_{līq} − α_{q}S_{lī})H_{k̄rmp̄}+ \\ +δ_{jr}(S_{mīq} − α_{q}S_{mī})H_{k̄lrp̄} − δ_{jp}(S_{rīq} − α_{q}S_{rī})H_{k̄lmr̄}\}− \\ −(S_{jk̄q} − α_{q}S_{jk̄})H_{īlmp̄} − (S_{jp̄q} − α_{q}S_{jp̄})H_{k̄lmī}+ \\ + ∑_{r} ϵ_{r}\{ϵ_{l}δ_{li}(S_{jr̄q} − α_{q}S_{jr̄})H_{k̄rmp̄} + ϵ_{m}δ_{mi}(S_{jr̄q} − α_{q}S_{jr̄})H_{k̄lrp̄}\} = 0. \end{matrix}

Putting j = k in (4.10) and summing up with respect to j, we have

(n + 1) ∑_{r} ϵ_{r}(S_{rīq} − α_{q}S_{rī})H_{r̄lmp̄} − (S_{līq} − α_{q}S_{lī})H_{mp̄}−

−(S_{mīq} − α_{q}S_{mī})H_{lp̄} + (r_{q} − rα_{q})H_{īlmp̄} + ∑_{r} ϵ_{r}(S_{rp̄q} − α_{q}S_{rp̄})H_{r̄lmī}−

− ∑_{r,s} ϵ_{rs}{ϵ_{l}δ_{li}(S_{rs̄q} − α_{q}S_{rs̄})H_{r̄smp̄} + ϵ_{m}δ_{mi}(S_{rs̄q} − α_{q}S_{rs̄})H_{r̄slp̄}} = 0.

Taking account of (4.3), the last equation turns out to be

(4.11)  (r_{q} − rα_{q}){(2n + 1)H_{īlmp̄} − 2ϵ_{m}δ_{mi}H_{lp̄} − 2ϵ_{l}δ_{li}H_{mp̄}} = 0.

Putting m = p in (4.11) and summing up with respect to the index p, we have 3n(r_{q} − rα_{q})H_{lī} = 0 for any indices q, l and i. Hence (4.11) implies that

(4.12)  (r_{q} − r α_{q})H_{īlmp̄} = 0

for any indices q, i, l, m and p.

Since H ≠ 0 on V, (4.12) implies that ∇r = rα on V. So, from (4.5), we obtain that ∇S = α ⊗ S.

Moreover, it follows from (4.4) that ∇R = α ⊗ R.

It completes the proof. □ Remark 4.1. In the last section, an example of an n(≥ 2)-dimensional indefinite H-recurrent Kähler manifolds which satisfies H ≠ 0 and ∥H∥ = 0 is given.

In particular, we consider the case that M is positive definite. Then the condition ∥H∥ = 0 is equivalent to H = 0. Thus we have the following Corollary 4.2. Let M be an n(≥ 2)-dimensional Kähler manifold. If it is H-recurrent, then it is R-recurrent.

5 Examples

This section is devoted to some examples of H-recurrent indefinite Kähler manifold. For any integer p(≥ 3) an indefinite complex hypersurface M(p, c) of a (2n + 1)- dimensional indefinite complex Euclidean space C_{n}^{2n+1} of index 2n defined as follows ([1], [6]) :

Let

{z^{A}} = {z^{j}, z^{j^{∗}}, z^{2n+1}} = {z^{1}, …, z^{2n+1}}

be a complex coordinate of C_{n}^{2n+1} and let c be a complex number such that |c| = 1. Then M(p, c) is an indefinite complete complex hypersurface of index 2n defined by

z^{2n+1} = ∑_{j} h_{j}(z^{j} + cz^{j^{∗}}), j^{∗} = n + j, h_{j}(z) = z^{p}.

Then the components h_{AB} and the components h_{ABC} of the second fundamental form and its covariant derivatives of M we have

h_{ij} = p(p − 1)δ_{ij}z^{p−2}, h_{i^{∗}j} = p(p − 1)cδ_{ij}z^{p−2}, h_{i^{∗}j^{∗}} = p(p − 1)c^{2}δ_{ij}z^{p−2},
h_{ijk} = p(p − 1)(p − 2)δ_{ij}δ_{ik}z^{p−3}, h_{i^{∗}jk} = p(p − 1)(p − 2)cδ_{ij}δ_{ik}z^{p−3},
h_{i^{∗}j^{∗}k} = p(p − 1)(p − 2)c^{2}δ_{ij}δ_{ik}z^{p−3}, h_{i^{∗}j^{∗}k^{∗}} = p(p − 1)(p − 2)c^{3}δ_{ij}δ_{ik}z^{p−3}.

Thus under the assumption of p the second fundamental form is not parallel.

Let K_{ĀBCD̄} (resp. K_{ĀBCD̄E}) the components of the Riemannian curvature tensor R (resp. the covariant derivative ∇R ) of the indefinite complex hypersurface M(p, c). In the theory of indefinite complex hypersurfaces in the indefinite complex space form the components are given by

K_{ĀBCD̄} = −h_{BC} h̄_{AD},
K_{ĀBCD̄E} = −h_{BCE}h̄_{AD},

which yield that R ≠ 0 and ∇R ≠ 0. Namely, M is not flat and not locally symmetric.

Now, by the property of the second fundamental form, they give to us

K_{ĀBCD̄E} = −\frac{δ_{AE}(p − 2)K_{ĀBCD̄}}{z} .

Thus we see that M is R-recurrent. Furthermore, in the theory of indefinite submanifolds of an indefinite complex space form, we have

\begin{aligned}S_{ij̄} & = & − ∑_{r} h_{ir}h̄_{rj} + ∑_{r} h_{ir^{∗}} h̄_{r^{∗}j} = \\ & = & p^{2}(p − 1)^{2}(−1 + |c|^{2})δ_{ij}z^{2(p−2)} = 0. \end{aligned}

Similarly we have

S_{AB̄} = − ∑_{r} h_{Ar}h̄_{rB} + ∑_{r} h_{Ar^{∗}} h̄_{r^{∗}B} = 0,

and from this we see that the Ricci tensor S is flat and hence the scalar curvature r vanishes. Hence this implies

H_{ĀBCD̄} = K_{ĀBCD̄}.

Furthermore it is easily seen that

\begin{aligned}∑_{R} ϵ_{R}K_{ĀBCR̄}K̄_{ĒF GR̄} & = & − ∑_{r} K_{ĀBCr̄}K̄_{ĒF Gr̄}+ \\ & + & ∑_{r} ϵ_{r}K_{ĀBCr̄^{∗}} K̄_{ĒF Gr̄^{∗}} = 0, \end{aligned}

and hence we obtain

∥H∥ = ∥R∥ = 0.

References

[1] R. Aiyama, T. Ikawa, J.H. Kwon and H. Nakagawa, Complex hypersurfaces of an indefinite complex space form, Tokyo J. Math. 10(1987), 349-361.

[2] A. Besse, Einstein manifolds, Springer-Verlag, 1987.

[3] S. Bochner, Curvatures and Betti numbers, Ann. of Math., 50(1949), 77-93.

[4] M. Matsumoto and S. Tanno, Kaehler spaces with parallel or vanishing Bochner curvature tensor, Tensor N. S., 27(1973), 21-30.

[5] Y. S. Pyo and H. S. Kim, Symmetric curvature-like tensors on a semi-definite Kähler manifold, Balkan J. Geom. Appl., 5-2(2000), 103-112.

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

[7] A. Romero, Some examples of indefinite complex Einstein hypersurfaces not locally symmetric, Proc. Amer. Math. J., 98(1986), 283-286.

[8] K. Yano and S. Bochner, Betti numbers and curvature, Studies of Prinston Univ. Press., no 32, 1953.

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

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