Symmetric Curvature-Like Tensors on a Semi-Definite Kähler Manifold

Yong-Soo Pyo and Hyang Sook Kim

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract The purpose of this paper is to investigate properties and relations of sym-

metric or locally symmetric semi-definite Kähler manifolds.

Mathematics Subject Classification: 53C50, 53C55, 53C56. Key words: Semi-definite Kähler manifold, concircular curvature tensor, projective curvature tensor, second curvature-like tensor, locally symmetric, symmetric

1 Introduction

As is well known, there exist some special tensors on the Kähler manifold extending the Riemannian curvature tensor on the Riemannian manifold, for example, the concircular, projective or conformal curvature tensor. In his paper [3], Bochner introduced three kinds of curvature-like tensors on the Kähler manifold which are closely related to the projective curvature tensor, the concircular curvature tensor and the conformal curvature tensor, and calculated the Betti numbers on a compact Kähler manifold under suitable conditions for the above curvature-like tensors. See Yano and Bochner [7]. The first tensor is called the Weyl curvature tensor and the third tensor is known as the Bochner curvature tensor. On the other hand, in their paper [4] Matsumoto and Tanno treated Kähler manifolds with parallel Bochner curvature tensor.

In this note, we investigate properties of curvature-like tensors H, Z and W on a semi-definite Kähler manifold.

Let (M, g) be an n(=^{>} 2)-dimensional semi-definite Kähler manifold with Kähler connection ∇. Let R or S be the Riemannian curvature tensor or the Ricci tensor with components K_{ījkl̄} or S_{ij̄} and let r be the scalar curvature. We denote by H 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})\},

Z the concircular curvature tensor on M with components Z_{ījkl̄} defined by

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 103-112 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

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

and W the projective curvature tensor on M with components W_{ījkl̄} defined by

W_{ījkl̄} = K_{ījkl̄} − \frac{1}{2(n + 1)}ϵ_{j}(δ_{ji}S_{kl̄} + δ_{jl}S_{kī}).

The Riemannian curvature tensor R (resp. the second curvature-like tensor H, the concircular curvature tensor Z or the projective curvature tensor W) is said to be parallel if it satisfies ∇R = 0 (resp. ∇H = 0, ∇Z = 0 or ∇W = 0). If the Riemannian curvature tensor R (resp. the second curvature-like tensor H, the concircular curvature tensor Z or the projective curvature tensor W) is parallel, then M is said to be locally symmetric (resp. H-symmetric, Z-symmetric or W -symmetric). Simply M is called to be symmetric when M is H-symmetric, Z-symmetric or W -symmetric. The class of semi-definite H-symmetric (resp. Z-symmetric or W -symmetric) Kähler manifolds contains semi-definite Kähler manifolds with vanishing second curvature-like tensor (resp. the concircular curvature tensor or the projective curvature tensor). From these facts it seems to be natural to consider properties of semi-definite symmetric Kähler manifolds, and to study relations of locally symmetric semi-definite Kähler manifolds and symmetric semi-definite Kähler manifolds.

The purpose of this paper is to investigate properties and relations of symmetric or locally symmetric semi-definite Kähler manifolds.

2 Semi-definite Kähler manifolds

This section is concerned with recalling basic formulas on a semi-definite Kähler manifold.

Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold with the semidefinite Kähler metric tensor g and the almost complex structure J. From the semidefinite 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(=^{>} 2)-dimensional connected semi-definite Kähler manifold 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 [5]. 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 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}).

The second Bianchi identity is given by the exterior derivative of the third equation of the structure equations (2.1), this is, we have

(2.10)  dΩ_{ij} = ∑_{k} ϵ_{k}(Ω_{ik} ∧ ω_{kj} − ω_{ik} ∧ Ω_{kj}).

In fact, we find

dΩ_{ij} = ∑_{k} ϵ_{k}d(ω_{ik} ∧ ω_{kj})
= ∑_{k} ϵ_{k}(dω_{ik} ∧ ω_{kj} − ω_{ik} ∧ dω_{kj})

= ∑_{j} ϵ_{k} {(Ω_{ik} − ∑_{l} ϵ_{l}ω_{il} ∧ ω_{lk}) ∧ ω_{kj} − ω_{ik} ∧ (Ω_{kj} − ∑_{l} ϵ_{l}ω_{kl} ∧ ω_{lj})}

= ∑_{j} ϵ_{k}(Ω_{ik} ∧ ω_{kj} − ω_{ik} ∧ Ω_{kj}),

where the first equality is derived from the property that d^{2} = 0, the second one follows from the fact that the complex connection form is an 1-form and the property of the exterior derivative and the third one is derived from the structure equations (2.1). We can regard Ω = (Ω_{ij}) and ω = (ω_{ij}) as complex matrices of order n. Then (2.10) can be reformed as

(2.11)  dΩ = Ω ∧ ω − ω ∧ Ω.

By the straightforward calculation we have

(2.12)  K_{ījkl̄m} = K_{ījml̄k,}

and hence

(2.13)  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

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

Putting i = j in (2.13) and summing up with respect to i, we have

(2.15)  ∑_{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

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

3 Curvature-like tensors

This section is concerned with curvature-like tensors H, Z and W on a semi-definite Kähler manifold.

Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s with a semi-definite Kähler metric g. 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 [7] 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}{n(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 eqivalent 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 [7]. It is said to be the concircular curvature tensor on M.

Moreover, a tensor W with components W_{ījkl̄} is defined by

(3.3)  W_{ījkl̄} = K_{ījkl̄} − \frac{1}{2(n + 1)}ϵ_{j}(δ_{ji}S_{kl̄} + δ_{jl}S_{kī}).

It is the formal analogue to the Weyl projective curvature tensor on the Riemannian manifold. This is introduced by Bochner. It is also called the projective curvature tensor on M. As is easily seen in definite case (cf. Yano and Bochner [7]), the semidefinite Kähler manifold M with vanishing projective curvature tensor is of constant holomorphic sectional curvature. Lemma 3.1. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the second curvature-like tensor H with components H_{ījkl̄} satisfies

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

(3.5)  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}.

Proof. Owing to the definition (3.1) of H, (3.4) is clear.

Since S_{ij̄} = ∑_{k} ϵ_{k}K_{k̄kij̄}, we have S_{ij̄} = S_{j̄i} = S̄_{jī} and hence we obtain H_{ij̄} =

H_{j̄i} = H̄_{jī}.

Moreover, putting i = j in (3.1) and summing up with respect to j, we get

H_{ij̄} = \frac{1}{4(n + 1)}\{2nS_{ij̄} − rϵ_{i}δ_{ij}\},

where we have used (2.6).

It completes the proof. □ Proposition 3.2. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the second curvature-like tensor H has vanishing t, where t is the scalar-like curvature of H defined by t=2 ∑_{r,s} ϵ_{rs} H_{r̄rss̄}.

Proof. Putting i = j in (3.5) and summing up j, we obtain

t = \frac{1}{2(n + 1)}(nr − nr) = 0.

It completes the proof. □ Lemma 3.3. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the concircular curvature tensor Z with components Z_{ījkl̄} satisfies

(3.6)  Z_{ījkl̄} = Z_{īkjl̄} = Z_{l̄jkī} = Z_{l̄kjī},

(3.7)  Z_{ij̄} = Z_{j̄i} = Z̄_{jī} = S_{ij̄} − \frac{r}{n}ϵ_{i}δ_{ij},

where Z_{ij̄} is the components of the Ricci-like tensor of Z defined by Z_{ij̄}=∑_{r} ϵ_{r} Z_{r̄rij̄}.

Proof. Owing to the definition (3.2) of Z, (3.6) is clear.

Since S_{ij̄} = ∑_{k} ϵ_{k}K_{k̄kij̄}, we have S_{ij̄} = S_{j̄i} = S̄_{jī} and hence we obtain Z_{ij̄} = Z_{j̄i} =

Z̄_{jī}. Putting k = l in (3.2) and summing up with respect to k, we find

Z_{ij̄} = S_{ij̄} − \frac{r}{n}ϵ_{i}δ_{ij}.

It completes the proof. □ Proposition 3.4. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the concircular curvature tensor Z satisfies u = −r, where u is the scalar-like curvature of Z defined by u=2 ∑_{r,s} ϵ_{rs} Z_{r̄rss̄}.

Proof. Putting i = j in (3.7) and summing up j, we find

u = r − 2r = −r.

It completes the proof. □ Lemma 3.5. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the projective curvature tensor W with components W_{ījkl̄} satisfies

(3.8)  W_{ījkl̄} = W_{l̄jkī},

(3.9)  W_{ij̄} = \frac{1}{2}S_{ij̄},

where W_{ij̄} are the components of the Ricci-like tensor of W defined by W_{ij̄}=∑_{r} ϵ_{r}

W_{r̄rij̄}. Proof. Owing to the definition (3.3) of W, (3.8) is clear.

Since S_{ij̄} = ∑_{k} ϵ_{k}K_{k̄kij̄}, if we put i = j in (3.3) and summing up with respect to

j, we obtain

W_{ij̄} = \frac{1}{2}S_{ij̄}.

It completes the proof. □ Remark 3.1. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the projective curvature tensor W with components W_{ījkl̄} has the following properties:

W_{ījkl̄} ≠ W_{īkjl̄},

W_{ījkk̄} = \frac{1}{4(n + 1)}\{2(2n + 1)S_{īj} − rϵ_{j}δ_{ji}\}.

Proposition 3.6. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s. Then the projective curvature tensor W satisfies 2v = r, where v is the scalar-like curvature of W defined by v=2 ∑_{r,s} ϵ_{rs} W_{r̄rss̄}.

Proof. Putting i = j in (3.9) and summing up j, we deduce

v = \frac{1}{2}r.

It completes the proof. □

4 Symmetric semi-definite Kähler manifolds

Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold of index 2s with the Kähler connection ∇. We choose a local frame field {U_{j}} of unitary frames on a neighborhood of M. Let H be the second curvature-like tensor 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_{kl}S_{jī} + δ_{ki}S_{jl̄})\}.

And let Z be the concircular curvature tensor with components Z_{ījkl̄} such that

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

Moreover, the projective curvature tensor W with components W_{ījkl̄} is defined by

W_{ījkl̄} = K_{ījkl̄} − \frac{1}{2(n + 1)}ϵ_{j}(δ_{ji}S_{kl̄} + δ_{jl}S_{kī}).

Now, we suppose that the curvature-like tensor Z vanishes identically. It is then easily seen that M is of constant holomorphic sectional curvature. The curvature-like tensor Z (resp. H or W) is said to be parallel if it satisfies ∇Z = 0 (resp. ∇H = 0 or ∇W = 0), namely, Z_{ījkm̄n} = 0 (resp. H_{ījkm̄n} = 0 or W_{ījkm̄n} = 0). If Z (resp. H or W) is parallel, then M is said to be Z(resp. H or W)-symmetric. Simply M is said to be symmetric when M is H-symmetric, Z-symmetric or W -symmetric.

Then we can prove Proposition 4.1. Let M be an n(=^{>} 2)-dimensional semidefinite Kähler manifold. Then it is H-symmetric if and only if it is locally symmetric. Proof. By the definition (3.1) of H and the assumption ∇H = 0, we have

(4.1) 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})\} = 0.

Putting i = m in (4.1), and summing up with respect to i, we obtain

S_{jl̄k} − \frac{1}{4(n + 1)}\{(2ϵ_{j} + 2ϵ_{k})S_{jl̄k} + r_{j}ϵ_{k}δ_{kl} + r_{k}ϵ_{j}δ_{jl}\} = 0,

where we have used (2.12), (2.13) and (2.15). From this it follows that

(4.2)  4nS_{jl̄k} = r_{j}ϵ_{k}δ_{kl} + r_{k}ϵ_{j}δ_{jl},

and hence 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 by the above equation (4.2) we have S_{jl̄k} = 0, from which, together with (4.1), we have K_{ījkl̄m} = 0. Thus we get ∇R = 0, that is, M is locally symmetric.

Conversely, differentiating (3.1) covariantly and using the assumption ∇R = 0, we find

\begin{aligned}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})\} = 0. \end{aligned}

This implies ∇H = 0.

It completes the proof. □ Proposition 4.2. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold. Then it is Z-symmetric if and only if it is locally symmetric. Proof. By the definition (3.2) of Z and the assumption ∇Z = 0, we have

(4.3)  K_{ījkl̄m} − \frac{r_{m}}{n(n + 1)}ϵ_{jk}(δ_{ji}δ_{kl} + δ_{jl}δ_{ki}) = 0.

Putting i = j in (4.3), and summing up with respect to i, we obtain

S_{kl̄m} = \frac{r_{m}}{n(n + 1)}(n + 1)ϵ_{k}δ_{kl}.

From this it follows that

S_{kl̄m} = \frac{r_{m}}{n} ϵ_{k}δ_{kl},

and hence we have r_{m} = 0. It means that the scalar curvature r is constant and by the above equation (4.3) we have K_{ījkl̄m} = 0. Thus we get ∇R = 0, that is, M is locally symmetric.

Conversely, differentiating (3.2) covariantly, we have

Z_{ījkl̄m} = K_{ījkl̄m} − \frac{r_{m}}{n(n + 1)}ϵ_{jk}(δ_{ji}δ_{kl} + δ_{jl}δ_{ki}).

Using the assumption ∇R = 0, we have r_{m} = 0, and hence the above equation yields ∇Z = 0.

It completes the proof. □ Proposition 4.3. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold. Then it is W -symmetric if and only if it is locally symmetric. Proof. By the definition (3.3) of W and the assumption ∇W = 0, we have

(4.4)  K_{ījkl̄m} − \frac{1}{2(n + 1)}ϵ_{j}(δ_{ji}S_{kl̄m} + δ_{jl}S_{kīm}) = 0.

Putting i = m in (4.4), and summing up with respect to i, we obtain

S_{jl̄k} − \frac{1}{2(n + 1)}(S_{jl̄k} + r_{k}ϵ_{j}δ_{jl}) = 0,

where we have used (2.12), (2.13) and (2.15). From this it follows that

(4.5)  S_{jl̄k} = \frac{1}{2(2n + 1)}r_{k}ϵ_{j}δ_{jl},

and hence we have r_{j} = 0. It means that the scalar curvature r is constant and by the above equation (4.5) we have S_{jl̄k} = 0, from which together with (4.4) we have K_{ījkl̄m} = 0. Thus we get ∇R = 0, that is, M is locally symmetric.

Conversely, differentiating (3.3) covariantly and using the assumption ∇R = 0, we have

W_{ījkl̄m} = K_{ījkl̄m} − \frac{1}{2(n + 1)}ϵ_{j}(δ_{ji}S_{kl̄m} + δ_{jl}S_{kīm}) = 0.

Thus this means ∇H = 0.

It completes the proof. □ Remark 4.1. From the proof of Propositions 4.1∼ 4.3, we can see that an n(=^{>} 2)- dimensional symmetric semi-definite Kähler manifolds have the costant scalar curvature, parallel Ricci tensor S and parallel Riemannian curvature tensor R.

Owing to Propositions 4.1 ∼ 4.3, we have the following theorem. Theorem 4.4. Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold. Then the following are equivalent:

(1) H-symmetric,

(2) Z-symmetric,

(3) W -symmetric,

(4) locally symmetric.

Consequently, from Theorem 4.4, we get the following fact:

Let M be an n(=^{>} 2)-dimensional semi-definite Kähler manifold. Then it is symmetric if and only if it is locally symmetric. Aknowledgements. The first author was supported by Research Foundation of Pukyong National University in 2000.

References

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

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