Notes on Kahler surfaces with distinct constant Ricci eigenvalues

NOTES ON K ¨ AHLER SURFACES WITH DISTINCT CONSTANT RICCI EIGENVALUES

Terumasa Nihonyanagi, Takashi Oguro and Kouei Sekigawa

Abstract. In this paper, we study a connected, simply connected homogeneous K¨ ahler surface with distinct constant Ricci eigenval- ues, and specify the local structure of them.

1. Introduction

Recently, V. Apostolov, T. Dr˘aghici and A. Moroianu studied com- pact K¨ahler manifolds whose Ricci tensor Ric (or rather the symmetric endomorphism of the tangent bundle corresponding to Ric via the met- ric) has two distinct constant eigenvalues and proved the followings.

Theorem A ([1]). Let M = (M, J, g) be a compact K¨ahler manifold whose Ricci tensor has two distinct non-negative eigenvalues λ and µ.

Then, the universal covering of M is the product of two simply connected K¨ ahler-Einstein manifolds of Ricci eigenvalues λ and µ respectively.

Theorem B ([1]). Let M = (M, J, g) be a compact K¨ahler surface whose Ricci tensor has two distinct constant eigenvalues. Then, one of the following alternatives holds:

(i) M is locally symmetric, i.e. M is locally the product of Riemannian surfaces of distinct constant Gaussian curvature;

(ii) if M is not as described in (i), then the eigenvalues of the Ricci tensor are both negative and (M, J) must be a minimal surface of general type with ample canonical bundle and with even and positive signature.

Moreover, in this case, reversing the orientations, the manifold would admit an Einstein, strictly almost K¨ ahler structure.

Received November 25, 2002.

2000 Mathematics Subject Classification: 53B20, 53C30, 53C55.

Key words and phrases: K¨ ahler manifold, almost Hermitian manifold, homoge-

neous manifold.

On one hand, H. Shima studied 2n-dimensional homogeneous K¨ahler manifolds with non-degenerate Ricci form of signature (2, 2(n − 1)) and obtained the following classification in dimension 4.

Theorem C ([8]). Let M = (M, J, g) be a connected, simply con- nected homogeneous K¨ ahler surface with non-degenerate Ricci form ρ.

Then, the signature of ρ is (4, 0) or (2, 2) or (0, 4). Moreover,

(i) if the signature is (4, 0), M is either P 1 (C) × P ¹ (C) or P 2 (C), where P n (C) is a complex projective space;

(ii) if the signature is (0, 4), M is either H 1 (C) × H ¹ (C) or H 2 (C), where H n (C) is a complex hyperbolic space;

(iii) if the signature is (2, 2), M is a holomorphic fiber bundle over H 1 (C) with fiber P 1 (C).

In this paper, we specify the local structure of a connected, simply connected homogeneous K¨ahler surface M = (M, J, g) with distinct con- stant Ricci eigenvalues (§ 3). From the argument of § 3, we will also find that if the signature of Ricci form of M is (2, 2), then M is locally the product of Riemannian surfaces of constant Gaussian curvature.

We remark that the problem of existence of K¨ahler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the Gold- berg conjecture (in the open case, i.e. the negative scalar curvature).

In fact, in [1], it was showed that the irreducible homogeneous K¨ahler manifolds whose Ricci tensor has two distinct constant negative eigenval- ues give rise to complete (non-compact) Einstein strictly almost K¨ahler manifolds of any even dimension grater than 4. For example, M ²ⁿ = SO(2, n)/(SO(2) × SO(n)), n ≥ 3.

2. Local structures

Let (M, J, g) be a K¨ahler surface of the constant Ricci eigenvalues λ, µ (λ < µ), and E λ , E µ the corresponding J-invariant eigenspaces at peach point of M . E _λ and E µ give rise to two smooth distributions on M . We denote by Ω the K¨ahler form of (J, g), which is given by Ω(·, ·) = g(J·, ·).

Now, we consider the 2-forms α, β defined respectively by (2.1) α(X, Y ) = Ω(Pr ^λ (X), Pr ^λ (Y )),

for vector fields X, Y ∈ T M, and

β = Ω − α

where Pr ^λ (resp. Pr ^µ ) denotes the orthogonal projection Pr ^λ : T M → E λ

(resp. Pr ^µ : T M → E µ ). The Ricci form ρ(·, ·) = Ric(J·, ·) is given by

(2.2) ρ = λα + µβ.

Thus, we have

(2.3) α = 1

λ − µ (ρ − µΩ), β = 1

λ − µ (λΩ − ρ).

Since ρ and Ω are both closed, so are α and β.

Proposition 1. The distributions E _λ and E _µ are both involutive.

Proof. Let {e ¹ , e 2 = Je 1 } and {e ³ , e 4 = Je 3 } be any local unitary frame fields for E _λ and E _µ respectively. We set

∇ e

e _j =

4

X

k=1

Γ _ijk e _k . Then, we have

(2.4) Γ _ijk = −Γ ikj , Γ _{i¯ j¯ k} = Γ _ijk ,

where we adopt the notational convention e ¯ i = Je _i (i = 1, . . . , 4). From the definition of α and β, we see that

(2.5) α(e 1 , e 2 ) = −α(e ² , e 1 ) = 1, β(e 3 , e 4 ) = −β(e ⁴ , e 3 ) = 1, and the others are zero. We denote by A and B the linear endomor- phisms of T M corresponding to the 2-forms α and β respectively. Since α and β are closed, we have

X,Y,Z S g((∇ X A)Y, Z) = 0, S

X,Y,Z g((∇ X B)Y, Z) = 0,

where S X,Y,Z denotes the cyclic sum with respect to X, Y , Z. In par- ticular, from (2.4) and (2.5), we get

0 = S

e

,e

,e

g((∇ ^e

A)e 2 , e 3 ) = S

e

,e

,e

g(∇ ^e

(Ae 2 ) − A(∇ ^e

e 2 ), e 3 )

= −Γ ¹¹³ + Γ 232 = Γ 142 − Γ ²⁴¹ , and hence

(2.6) Γ 142 − Γ ²⁴¹ = 0.

Similarly, from S e

,e

,e

g((∇ e

A)e 2 , e 4 ) = 0, we get

(2.7) Γ 132 − Γ ²³¹ = 0.

Thus, E _λ is involutive. By the similar way, we obtain

(2.8) Γ 314 − Γ ⁴¹³ = Γ 324 − Γ ⁴²³ = 0.

Thus, we see also that E µ is involutive.

From Proposition 1, we may choose a local coordinate system (x 1 , y 1 , x 2 , y 2 ) such that {∂/∂x ¹ , ∂/∂y 1 } spans E λ and {∂/∂x ² , ∂/∂y 2 } spans E _µ . Since E _λ and E _µ are J-invariant, we may set

J ∂

∂x 1

= J ₁ ¹ ∂

∂x 1

+ J ₁ ^{¯ 1} ∂

∂y 1

, J ∂

∂y 1

= J ¯ 1 ¹

∂

∂x 1

+ J ¯ 1 ^{¯ 1}

∂

∂y 1

, J ∂

∂x 2

= J ₂ ² ∂

∂x 2

+ J ₂ ^{¯ 2} ∂

∂y 2

, J ∂

∂y 2

= J ¯ 2 ²

∂

∂x 2

+ J ¯ 2 ^{¯ 2}

∂

∂y 2

. Then, J ² = −1 implies

(2.9)

(J ₁ ¹ ) ² + J ¯ 1 ¹ J ₁ ^{¯ 1} = −1, (J ¯ 1 ^{¯ 1} ) ² + J ¯ 1 ¹ J ₁ ^{¯ 1} = −1, J ₁ ^{¯ 1} (J ₁ ¹ + J ¯ 1 ^{¯ 1} ) = 0, J ¯ 1 ¹ (J ₁ ¹ + J ¯ 1 ^{¯ 1} ) = 0, (J ₂ ² ) ² + J ¯ 2 ² J ₂ ^{¯ 2} = −1, (J ¯ 2 ^{¯ 2} ) ² + J ¯ 2 ² J ₂ ^{¯ 2} = −1,

J ₂ ^{¯ 2} (J ₂ ² + J ¯ 2 ^{¯ 2} ) = 0, J ¯ 2 ² (J ₂ ² + J ¯ 2 ^{¯ 2} ) = 0.

Thus, we have J ₁ ^{¯ 1} 6= 0, J ^¯ 1 ¹ 6= 0 and (2.10) J ¯ 1 ^{¯ 1} = −J 1 ¹ . Since g(J(∂/∂x 1 ), ∂/∂x 1 ) = 0, we have

J ₁ ¹ g 11 + J ₁ ^{¯ 1} g _{1¯ 1} = 0.

Thus, we may put

(2.11) J ₁ ¹ = −F g 1¯ 1 , J ₁ ^{¯ 1} = F g 11 .

From g(J(∂/∂x 1 ), ∂/∂y 1 ) + g(∂/∂x 1 , J(∂/∂y 1 )) = 0 and (2.10), we have J ₁ ^{¯ 1} g ¯ 1¯ 1 + J ¯ 1 ¹ g 11 = 0,

and hence,

(2.12) J ¯ 1 ¹ = −F g ^¯ 1¯ 1 . Taking account of (2.9)∼(2.12), we have

det µg 11 g _{1¯ 1} g _{1¯ 1} g ¯ 1¯ 1

¶

= 1 F ² .

We put e ^2σ

= 1/F ² . Summing up the above arguments, we obtain (2.13) µg 11 g _{1¯ 1}

g _{1¯ 1} g ¯ 1¯ 1

¶

= e ^−σ

µ J ₁ ^{¯ 1} −J 1 ¹

−J 1 ¹ −J ^¯ 1 ¹

¶

.

Similarly, we get (2.14)

µg 22 g _{2¯ 2} g _{2¯ 2} g ¯ 2¯ 2

¶

= e ^−σ

µ J ₂ ^{¯ 2} −J 2 ²

−J 2 ² −J ¯ 2 ²

¶

, where e ^2σ

= det µg 22 g _{2¯ 2} g _{2¯ 2} g ¯ 2¯ 2

¶ . From (2.13) and (2.14), we have

Ω = e ^−σ

dx 1 ∧ dy ¹ + e ^−σ

dx 2 ∧ dy ² .

Since Ω is closed, it must follow that σ i = σ i (x i , y i ), (i = 1, 2). Thus, we conclude the following.

Proposition 2. There exists a local Darboux coordinate system (x 1 , y 1 , x 2 , y 2 ) such that {∂/∂x ¹ , ∂/∂y 1 } and {∂/∂x ² , ∂/∂y 2 } are local frame fields for E λ and E µ respectively.

We remark that Propositions 1 and 2 do not necessarily guarantee the existence of local complex coordinate system (z 1 = x 1 + √

−1 y ¹ , z 2 = x 2 + √

−1 y ² ) which is compatible with the complex structure J such that {∂/∂x ¹ , ∂/∂y 1 } and {∂/∂x ² , ∂/∂y 2 } are local bases for the distributions E λ and E µ respectively. In fact, we may easily show that, for a K¨ahler surface M = (M, J, g) of constant distinct Ricci eigenvalues λ and µ, if there exists a complex coordinate system (z 1 = x 1 + √

−1 y ¹ , z 2 = x 2 +

√ −1 y ² ) around any point of M which is compatible with the complex structure J, then M is locally the product of two Riemann surfaces with Gaussian curvature λ and µ (compare with the Example in § 4).

3. Homogeneous K¨ ahler surface with distinct constant Ricci eigenvalues

An almost Hermitian manifold M = (M, J, g) is homogeneous if there exists a Lie subgroup G of the automorphism group Aut(M, J, g) which acts transitively on M . The following result is known.

Theorem D ([5]). Let M = (M, J, g) be a homogeneous almost Hermitian manifold. Then, there exists a skew-symmetric tensor field T of type (1, 2) satisfying the following conditions for any X ∈ T M:

(1) ∇ X J = T (X)·J, (2) ∇ X R = T (X)·R, (3) ∇ X T = T (X)·T,

where T (X)Y = T (X, Y ) and the symbol “ T (X) · ” means that T (X)

acts as a derivation of the tensor algebra. The tensor field T is called a

homogeneous structure of M .

Now, let U (M ) be the unitary frame bundle over a connected, simply connected, complete almost Hermitian manifold M = (M, J, g) and T a skew-symmetric tensor field of type (1, 2) satisfying the conditions (1), (2) and (3) in Theorem D. Let u 0 be an arbitrary fixed point of U (M ) and ∇ ^∗ be the linear connection defined by ∇ ^∗ X = ∇ ^X − T (X) for X ∈ T M. Then, we may easily check that g and J are parallel with respect to ∇ ^∗ , i.e. ∇ ^∗ g = 0 and ∇ ^∗ J = 0. The holonomy bundle G of ∇ ^∗ through u 0 has a Lie group structure with identity u 0 and acts transitively on M as a group of automorphisms. This is the converse of Theorem D.

A homogeneous almost Hermitian manifold M = (M, J, g) satisfies the following condition H(m) for all non-negative integers m.

H(m): for any x, y ∈ M, there exists a linear isometry Φ : (T ^x M, g x ) → (T _y M, g _y ) such that

Φ ◦ J ^x = J y ◦ Φ, and Φ(∇ ^k R) x = (∇ ^k R) y for k = 0, 1, . . . , m.

In fact, Φ is given by putting Φ = dϕ x , the differential mapping of a holomorphic isometry ϕ of M with ϕ(x) = y. For a Riemannian manifold M = (M, g), we denote by P (m) the following condition:

P (m) : for x, y ∈ M, there exists a linear isometry Φ : (T ^x M, g x ) → (T _y M, g _y ) such that

Φ(∇ ^k R) _x = (∇ ^k R) _y for k = 0, 1, . . . , m.

A Riemannian manifold M satisfying the condition P (0) is called cur- vature homogeneous. In general, a curvature homogeneous Riemannian manifold is not homogeneous even if it is connected and simply con- nected ([4]), and the following result is known.

Theorem E ([7]). Let M = (M, g) be a 4-dimensional connected, simply connected, complete Riemannian manifold satisfying the condi- tion P (1). Then, M is homogeneous, i.e. M is a locally symmetric space or a group space.

Now, let M = (M, J, g) be a 2n-dimensional homogeneous almost Hermitian manifold with the automorphisms group ˆ G = Aut(M, J, g) which acts transitively on M . For a point u 0 = (x; e 1 , Je 1 , . . . , e _n , Je _n ) ∈ U (M ), we set G = © ˆγ(u 0 ) ¯

¯ γ ∈ ˆ ˆ G ª. Then, G is a principal subbundle of U (M ) and has a Lie group structure with the identity ε = u 0 which is isomorphic to ˆ G by the canonical correspondence ˆ γ 7→ γ = ˆγ(ε).

Moreover, G acts transitively on M as a group of automorphism by

γ(x) = π(γγ ^′ ), γ ∈ G, x ∈ M, where π : U(M) → M is the bundle

projection and π(γ ^′ ) = x. Then, relating to G (or ˆ G), we have a skew- symmetric tensor field T of type (1, 2) on M satisfying the conditions (1), (2) and (3) in Theorem D. We define functions f _h

_···h

;ijkl and T _ijk on U (M ) respectively by

f h

···h

;ijkl (u) = g((∇ ^s e

···e

R)(e i , e j )e k , e l ), T ijk (u) = g(T (e i , e j ), e k ), for u = (x; e 1 , Je 1 , . . . , e n , Je n ) ∈ U(M, J, g). Then, we have the follow- ing.

Proposition 3. On G, the functions f _h

_···h

;ijkl and T _ijk are con- stant.

For each point x ∈ M, we define a Lie group G ^x s by G ^x _s = © a ∈ U(T x M, J _x , g _x ) ¯

¯ t _i ◦ a = t i for i = 0, . . . , s ª, where t i and t i ◦ a are defined respectively by

t i (X i , . . . , X 1 , Y, Z) = (∇ ⁱ X

···X

R)(Y, Z),

(t _i ◦ a)(X i , . . . , X 1 , Y, Z) = a ⁻¹ (∇ ⁱ aX

···aX

R)(aY, aZ)a,

for X _i , . . . , X 1 , Y , Z ∈ T x M . We denote by g ^x _s the Lie algebra of G ^x _s and put k = g ^x ₁ . We note that g ^x ₁ and g ^y ₁ are isomorphic for any x, y ∈ M.

In the sequel, we assume that (M, J, g) is a connected, simply con- nected homogeneous K¨ahler surface with distinct constant Ricci eigen- values λ, µ (λ < µ). Then, the following cases are possible.

Case I: k = {0},

Case II: k = u(1) ⊕ {0} or Case II ^′ : k = {0} ⊕ u(1), Case III: k = u(1) ⊕ u(1).

Case I

In this case, the corresponding Lie group G of automorphisms acts

simply transitively on M . We denote by ∇ ^∗ the linear connection on

the principal subbundle G of U (M, J, g) through u 0 ∈ U(M, J, g). The

tensor field T of type (1,2) defined by T = ∇ ^∗ −∇ satisfies the condition

(1), (2) and (3) in Theorem D. If γ = (x; e 1 , e 2 = Je 1 , e 3 , e 4 = Je 3 ) ∈ G,

then we see that ∇ ^∗ e

e j = 0 and hence T _ijk = Γ _ijk are regarded as

constant-valued functions on M . Then, taking account of (2.4) and

(2.6)∼(2.8), we have

(3.1)

R 1313 = Γ 341 (2Γ 134 − Γ ¹¹² ) − Γ ¹³² (2Γ 312 − Γ ³³⁴ ) + Γ ² ₁₃₂ + Γ ² ₁₄₂ + Γ ² ₃₄₁ + Γ ² ₃₄₂ ,

R 1414 = −Γ ³⁴¹ (2Γ 134 − Γ ¹¹² ) − Γ ¹⁴² (2Γ 412 − Γ ⁴³⁴ ) + Γ ² ₁₃₂ + Γ ² ₁₄₂ + Γ ² ₃₄₁ + Γ ² ₃₄₂ ,

R 2424 = Γ 142 (2Γ 412 − Γ ⁴³⁴ ) − Γ ³⁴² (2Γ 234 − Γ ²¹² ) + Γ ² ₁₃₂ + Γ ² ₁₄₂ + Γ ² ₃₄₁ + Γ ² ₃₄₂ ,

R 2323 = Γ 132 (2Γ 312 − Γ ³³⁴ ) + Γ 342 (2Γ 234 − Γ ²¹² ) + Γ ² ₁₃₂ + Γ ² ₁₄₂ + Γ ² ₃₄₁ + Γ ² ₃₄₂ .

Since R 1313 = R 2424 and R 1414 = R 2323 , we have Γ 341 (2Γ 134 − Γ ¹¹² ) − Γ ¹³² (2Γ 312 − Γ ³³⁴ )

= Γ 142 (2Γ 412 − Γ ⁴³⁴ ) − Γ ³⁴² (2Γ 234 − Γ ²¹² ),

− Γ ³⁴¹ (2Γ 134 − Γ ¹¹² ) − Γ ¹⁴² (2Γ 412 − Γ ⁴³⁴ )

= Γ 132 (2Γ 312 − Γ ³³⁴ ) + Γ 342 (2Γ 234 − Γ ²¹² ), and hence,

(3.2) Γ 132 (2Γ 312 − Γ ³³⁴ ) + Γ 142 (2Γ 412 − Γ ⁴³⁴ ) = 0, Γ 341 (2Γ 134 − Γ ¹¹² ) + Γ 342 (2Γ 234 − Γ ²¹² ) = 0.

On one hand, we have

R 1314 = −Γ ¹⁴² (2Γ 312 − Γ ³³⁴ ) − Γ ³⁴² (2Γ 134 − Γ ¹¹² ), R 1413 = −Γ ¹³² (2Γ 412 − Γ ⁴³⁴ ) − Γ ³⁴² (2Γ 134 − Γ ¹¹² ).

Since R 1314 = R 1413 , we have

(3.3) Γ 132 (2Γ 412 − Γ ⁴³⁴ ) − Γ ¹⁴² (2Γ 312 − Γ ³³⁴ ) = 0.

Similarly, from R 1323 = R 2313 , we obtain

(3.4) Γ 341 (2Γ 234 − Γ ²¹² ) − Γ ³⁴² (2Γ 134 − Γ ¹¹² ) = 0.

From the equalities (3.2), (3.3) and (3.4), we see that the following cases are possible.

Case(I-1): (Γ 132 , Γ 142 ) 6= (0, 0) and (Γ ³⁴¹ , Γ 342 ) 6= (0, 0),

Case(I-2): (Γ 132 , Γ 142 ) = (0, 0) and (Γ 341 , Γ 342 ) 6= (0, 0),

Case(I-3): (Γ 132 , Γ 142 ) 6= (0, 0) and (Γ ³⁴¹ , Γ 342 ) = (0, 0),

Case(I-4): (Γ 132 , Γ 142 ) = (0, 0) and (Γ 341 , Γ 342 ) = (0, 0).

Case (I-1). In this case, (3.2), (3.3) and (3.4) are reduced to (3.5)

2Γ 134 − Γ ¹¹² = 0, 2Γ 234 − Γ ²¹² = 0, 2Γ 312 − Γ ³³⁴ = 0, 2Γ 412 − Γ ⁴³⁴ = 0.

Taking account of (3.5), we have

R 1213 = −Γ ¹³² (2Γ 212 − Γ ²³⁴ ) − Γ ¹⁴² (2Γ 112 − Γ ¹³⁴ )

= −3(Γ ¹³² Γ 234 + Γ 142 Γ 134 ),

R 3413 = −Γ ³⁴¹ (2Γ 434 − Γ ⁴¹² ) − Γ ³⁴² (2Γ 334 − Γ ³¹² )

= −3(Γ ³⁴¹ Γ 412 + Γ 342 Γ 312 ).

Since R 1213 + R 3413 = 0, we have

(3.6) Γ 132 Γ 234 + Γ 142 Γ 134 + Γ 341 Γ 412 + Γ 342 Γ 312 = 0.

Similarly, from R 1214 + R 3414 = 0, we obtain

(3.7) Γ 132 Γ 134 − Γ ¹⁴² Γ 234 + Γ 342 Γ 412 − Γ ³⁴¹ Γ 312 = 0.

On one hand, we get

R 1412 = −2Γ ³⁴¹ Γ 132 + 2Γ 342 Γ 142 + Γ 132 Γ 112 − Γ ¹⁴² Γ 212

+ Γ 134 Γ 312 + Γ 412 Γ 212 − Γ ³⁴¹ Γ 312 + Γ 342 Γ 412 , R 1434 = 2Γ 341 Γ 132 − 2Γ ³⁴² Γ 142 + Γ 132 Γ 134 − Γ ¹⁴² Γ 234

+ Γ 134 Γ 334 + Γ 412 Γ 234 − Γ ³⁴¹ Γ 334 + Γ 342 Γ 434 . Since R 1412 + R 1434 = 0, we have

(3.8) Γ 132 Γ 134 − Γ ¹⁴² Γ 234 + Γ 312 (Γ 134 − Γ ³⁴¹ ) + Γ 412 (Γ 234 + Γ 342 ) = 0.

Similarly, from R 1312 + R 1334 = 0, we obtain

(3.9) Γ 132 Γ 234 + Γ 142 Γ 134 − Γ ³¹² (Γ 234 − Γ ³⁴² ) + Γ 412 (Γ 134 + Γ 341 ) = 0.

Thus, from (3.6)∼(3.9), we have

(3.10) Γ 312 Γ 134 + Γ 412 Γ 234 = 0, Γ 412 Γ 134 − Γ ³¹² Γ 234 = 0.

By direct calculation, we get

R 1212 = Γ ² ₁₁₂ + Γ ² ₂₁₂ − 2(Γ ² 132 + Γ ² ₁₄₂ ) = 4(Γ ² ₁₃₄ + Γ ² ₂₃₄ ) − 2(Γ ² 132 + Γ ² ₁₄₂ ), R 1234 = 2(Γ ² ₁₃₂ + Γ ² ₁₄₂ + Γ ² ₁₃₄ + Γ ² ₂₃₄ ).

Thus, we have

(3.11) −λ = R ¹²¹² + R 1234 = 6(Γ ² ₁₃₄ + Γ ² ₂₃₄ ).

Similarly, we have

(3.12) −µ = R ³⁴¹² + R 3434 = 6(Γ ² ₃₁₂ + Γ ² ₄₁₂ )(≥ 0).

Since λ 6= 0, we see that (Γ ¹³⁴ , Γ 234 ) 6= (0, 0) by virtue of (3.11). Thus, (3.10) implies (Γ 312 , Γ 412 ) = (0, 0). Then, (3.8) and (3.9) are reduced to

Γ 142 Γ 134 + Γ 132 Γ 234 = 0, Γ 132 Γ 134 − Γ ¹⁴² Γ 234 = 0.

Since (Γ 134 , Γ 234 ) 6= (0, 0), it must follow that (Γ ¹³² , Γ 142 ) = (0, 0). But, this contradicts to the assumption of case (I-1). Thus, case (I-1) cannot occur.

Case (I-2). In this case, from (3.2) and (3.4), we have (3.13) 2Γ 134 − Γ ¹¹² = 0, 2Γ 234 − Γ ²¹² = 0.

Further, we get (3.14)

R 1212 = Γ ² ₁₁₂ + Γ ² ₂₁₂ , R 1313 = Γ ² ₃₄₁ + Γ ² ₃₄₂ , R 1414 = Γ ² ₃₄₁ + Γ ² ₃₄₂ . Thus, we have

(3.15) −λ = R ¹²¹² + R 1313 + R 1414 = Γ ² ₁₁₂ + Γ ² ₂₁₂ + 2(Γ ² ₃₄₁ + Γ ² ₃₄₂ ), and hence, λ < 0. By direct calculation, we get also

R 1213 = 0, R 3413 = −Γ ³⁴¹ (2Γ 434 − Γ ⁴¹² ) − Γ ³⁴² (2Γ 334 − Γ ³¹² ), R 1214 = 0, R 3414 = Γ 342 (2Γ 434 − Γ ⁴¹² ) − Γ ³⁴¹ (2Γ 334 − Γ ³¹² ).

Thus, form R 1213 + R 3413 = 0 and R 1214 + R 3414 = 0, we have

−Γ ³⁴¹ (2Γ 434 − Γ ⁴¹² ) − Γ ³⁴² (2Γ 334 − Γ ³¹² ) = 0, Γ 342 (2Γ 434 − Γ ⁴¹² ) − Γ ³⁴¹ (2Γ 334 − Γ ³¹² ) = 0.

Since (Γ 341 , Γ 342 ) 6= (0, 0), we get

(3.16) 2Γ 334 − Γ ³¹² = 0, 2Γ 434 − Γ ⁴¹² = 0.

Thus, we have also

R 3413 = R 3414 = 0.

Further, we have

(3.17) R 1234 = 2(Γ ² ₁₃₄ +Γ ² ₂₃₄ ), R 3434 = −2(Γ ² 341 +Γ ² ₃₄₂ )+Γ ² ₃₃₄ +Γ ² ₄₃₄ , and hence,

(3.18) −µ = 2(Γ ² 134 + Γ ² ₂₃₄ ) − 2(Γ ² 341 + Γ ² ₃₄₂ ) + Γ ² ₃₃₄ + Γ ² ₄₃₄ . On one hand, we have

R 1314 = 0, R 3412 = 2(Γ ² ₃₄₁ + Γ ² ₃₄₂ + Γ ² ₃₃₄ + Γ ² ₄₃₄ ).

Since R 1234 = R 3412 , we have

(3.19) Γ ² ₁₃₄ + Γ ² ₂₃₄ = Γ ² ₃₄₁ + Γ ² ₃₄₂ + Γ ² ₃₃₄ + Γ ² ₄₃₄ .

Substituting this equality into (3.18), we get (3.20) −µ = 3(Γ ² 334 + Γ ² ₄₃₄ ), and hence, µ ≤ 0. Further, we have

R 1212 = Γ ² ₁₁₂ + Γ ² ₂₁₂ = 4(Γ ² ₁₃₄ + Γ ² ₂₃₄ ) = 2R 1234 , and hence,

−λ = R ¹²¹² + R 1234 = 3R 1234 . Therefore, we obtain

(3.21) R 1212 = − 2

3 λ, R 1234 = − λ

3 , R 1313 = R 1414 = − λ 6 . Since R 1412 = R 1214 = 0 and R 1312 = R 1213 = 0, we have

Γ 312 (Γ 134 − Γ ³⁴¹ ) + Γ 412 (Γ 212 + Γ 342 ) = 0, Γ 312 (Γ 212 − Γ ³⁴² ) − Γ ⁴¹² (Γ 134 + Γ 341 ) = 0.

Thus, we see that (Γ 312 , Γ 412 ) 6= (0, 0) or (Γ ³¹² , Γ 412 ) = (0, 0). First, we assume that (Γ 312 , Γ 412 ) 6= (0, 0). Then, the equality

Γ ² ₃₄₁ + Γ ² ₃₄₂ = Γ ² ₁₃₄ + Γ ² ₂₁₂

holds. On one hand, from R 2412 = R 1213 = 0 and R 2312 = −R ¹²¹⁴ = 0, we have

(Γ 234 − Γ ³⁴² )Γ 312 − (Γ ³⁴¹ + Γ 112 )Γ 412 = 0, (Γ 341 − Γ ¹¹² )Γ 312 − (Γ ²³⁴ + Γ 342 )Γ 412 = 0.

Since (Γ 312 , Γ 412 ) 6= (0, 0), we have

Γ ² ₃₄₁ + Γ ² ₃₄₂ = Γ ² ₁₁₂ + Γ ² ₂₃₄ = 4Γ ² ₁₃₄ + 1 4 Γ ² ₂₁₂ .

Thus, we have Γ ² ₁₃₄ = Γ ² ₂₁₂ /4, and hence Γ ² ₁₁₂ = Γ ² ₂₁₂ . If we put γ = Γ ² ₁₁₂ , then

R 1212 = 2γ, R 1234 = γ, R 1313 = R 1414 = 5 4 γ.

Thus, we have γ = R 1234 = R 1313 + R 1414 = 5γ/2, and hence γ = 0.

Then, we have λ = 0, but this is a contradiction. Next, we consider the case (Γ 312 , Γ 412 ) = (0, 0). Then, from (3.16), we have Γ 334 = Γ 434 = 0. Thus, from (3.20), we have µ = 0. Further, from (3.19), we have Γ ² ₁₃₄ + Γ ² ₂₃₄ = Γ ² ₃₄₁ + Γ ² ₃₄₂ . Thus, from (3.14), (3.17) and (3.21), we have (3.22) R 3434 = −2(Γ ² 341 + Γ ² ₃₄₂ ) = −2R ¹³¹³ = λ

3 .

Case (I-3). This case is essentially same as Case (I-2).

Case (I-4). In this case, M is locally the product of Riemannian surfaces of constant Gaussian curvature λ and µ.

Case II

In this case, taking account of Proposition 3, by direct calculation, we see that

(3.23)

R 1213 = 0, R 1214 = 0, R 1224 = 0, R 1223 = 0, R 3413 = 0, R 3414 = 0, R 3424 = 0, R 3423 = 0, R 1314 = 0, R 2324 = 0, R 1323 = 0, R 1424 = 0,

and R 1212 , R 1313 = R 1414 , R 2424 = R 2323 , R 3434 are constant. Further, we see that T 312 , T 412 , T 334 , T 434 , T 132 , T 231 , T 142 , T 241 are constant, and (3.24)

T 132 + T 231 = 0, T 142 + T 241 = 0, T _{i¯ j¯ k} = T _ijk (1 ≤ i, j, k ≤ 4).

Thus, from direct calculation, we have

∇ ¹ R 1213 = −Γ ¹³² (R 1212 − 4R ¹³¹³ ), ∇ ¹ R 1213 = −T ¹³² (R 1212 − 4R ¹³¹³ ),

∇ ¹ R 1214 = −Γ ¹⁴² (R 1212 − 4R ¹³¹³ ), ∇ ¹ R 1214 = −T ¹⁴² (R 1212 − 4R ¹³¹³ ),

∇ ² R 1213 = Γ 241 (R 1212 − 4R ¹³¹³ ), ∇ ² R 1213 = T 241 (R 1212 − 4R ¹³¹³ ),

∇ ² R 1214 = −Γ ²³¹ (R 1212 − 4R ¹³¹³ ), ∇ ² R 1214 = −T ²³¹ (R 1212 − 4R ¹³¹³ ), and hence,

(3.25)

(T 132 − Γ ¹³² )(R 1212 − 4R ¹³¹³ ) = 0, (T 142 − Γ ¹⁴² )(R 1212 − 4R ¹³¹³ ) = 0, (T 241 − Γ ²⁴¹ )(R 1212 − 4R ¹³¹³ ) = 0, (T 231 − Γ ²³¹ )(R 1212 − 4R ¹³¹³ ) = 0.

Further, we get

∇ ¹ R 3413 = Γ 132 (R 3434 − 4R ¹³¹³ ), ∇ ¹ R 3413 = T 132 (R 3434 − 4R ¹³¹³ ),

∇ ¹ R 3414 = Γ 142 (R 3434 − 4R ¹³¹³ ), ∇ ¹ R 3414 = T 142 (R 3434 − 4R ¹³¹³ ),

∇ ² R 3413 = −Γ ²⁴¹ (R 3434 − 4R ¹³¹³ ), ∇ ² R 3413 = −T ²⁴¹ (R 3434 − 4R ¹³¹³ ),

∇ ² R 3414 = Γ 231 (R 3434 − 4R ¹³¹³ ), ∇ ² R 3414 = T 231 (R 3434 − 4R ¹³¹³ ), and hence,

(3.26)

(T 132 − Γ ¹³² )(R 3434 − 4R ¹³¹³ ) = 0,

(T 142 − Γ ¹⁴² )(R 3434 − 4R ¹³¹³ ) = 0,

(T 241 − Γ ²⁴¹ )(R 3434 − 4R ¹³¹³ ) = 0,

(T 231 − Γ ²³¹ )(R 3434 − 4R ¹³¹³ ) = 0.

Thus, from (3.25) and (3.26), we see that (R 1212 − 4R ¹³¹³ , R 3434 − 4R 1313 ) = (0, 0) or (R 1212 − 4R ¹³¹³ , R 3434 − 4R ¹³¹³ ) 6= (0, 0). First, we assume that (R 1212 − 4R ¹³¹³ , R 3434 − 4R ¹³¹³ ) = (0, 0). Then, we have

λ = −R ¹²¹² − R ¹³¹³ − R ¹⁴¹⁴ = −6R ¹³¹³ = −R ³⁴³⁴ − 2R ¹³¹³ = µ.

But, this is a contradiction, and hence (R 1212 −4R ¹³¹³ , R 3434 −4R ¹³¹³ ) 6=

(0, 0). In this case, we have

Γ 132 = T 132 , Γ 231 = T 231 , Γ 142 = T 142 , Γ 241 = T 241 , and hence, from (2.6), (2.7) and (3.24), we have

Γ 132 = Γ 231 = Γ 142 = Γ 241 = 0.

Therefore, (M, J, g) is locally the product of Riemannian surfaces of constant Gaussian curvature λ and µ.

Case II ^′

This case is essentially same as Case II.

Case III

In this case, M is locally the product of Riemannian surfaces of con- stant Gaussian curvature λ and µ.

4. Example

In this section, we introduce an Example established by O. Kowal- ski ([3]). Let M = R ⁴ ₊ = © (x 1 , x 2 , x 3 , x 4 ) ∈ R ⁴ ¯

¯ x 1 > 0 ª and put e 1 = −x ¹ ∂

∂x 1

, e 2 = − 1 x 1

∂

∂y 1

, e 3 = √

x 1 y 1

∂

∂y 2

+ √ x 1

∂

∂x 2

, e 4 = 1

√ x 1

∂

∂y 2

.

We define an almost Hermitian structure (J, g) by Je 1 = e 2 , Je 3 = e 4 , and g(e i , e j ) = δ ij (1 ≤ i, j ≤ 4). Let {e ⁱ } be the dual basis of {e ⁱ }.

Then, we have e ¹ = − 1

x 1

dx 1 , e ² = −x ¹ dy 1 , e ³ = 1

√ x 1

dx 2 , e ⁴ = √

x 1 dy 2 − √

x 1 y 1 dx 2 , and

Ω = e ¹ ∧ e ² + e ³ ∧ e ⁴ = dx 1 ∧ dy ¹ + dx 2 ∧ dy ² .

Namely, (x 1 , y 1 , x 2 , y 2 ) is a Darboux coordinate system of M . With re- spect to the coordinate system (x 1 , y 1 , x 2 , y 2 ), the almost complex struc- ture J is expressed by

J ∂

∂x 1

= 1 x ² ₁

∂

∂y 1

, J ∂

∂y 1 = −x ² 1

∂

∂x 1

, J ∂

∂x 2

= x 1 y 1

∂

∂x 2

+ µ

x 1 y ² ₁ + 1 x 1

¶ ∂

∂y 2

, J ∂

∂y 2 = −x ¹ y 1

∂

∂y 2 − x ¹ ∂

∂x 2

. By direct calculation, we get

[e 1 , e 2 ] = e 2 , [e 1 , e 3 ] = − 1

2 e 3 , [e 1 , e 4 ] = 1 2 e 4 , [e 2 , e 3 ] = −e ⁴ , [e 2 , e 4 ] = 0, [e 3 , e 4 ] = 0, and

Γ

= 0, Γ

= 0, Γ

= 0, Γ

= 0, Γ

= 0, Γ

= 0, Γ

= −1, Γ

= 0, Γ

= 0, Γ

= 0, Γ

= 0, Γ

= − 1

2 , Γ

= 0, Γ

= 1

2 , Γ

= 0, Γ

= 0, Γ

= 1

2 , Γ

= 0, Γ

= 0, Γ

= 0, Γ

= − 1

2 , Γ

= 1

2 , Γ

= 0, Γ

= 0.

We have also

R 1212 = 1, R 1234 = 1

2 , R 1313 = 1

4 , R 1324 = 1 4 , R 1414 = 1

4 , R 1423 = − 1

4 , R 2323 = 1

4 , R 2424 = 1

4 , R 3434 = − 1 2 , and otherwise being zero up to sign. Thus, the Ricci eigenvalues are λ = −3/2 and µ = 0. We may easily check that (M, J, g) is a K¨ahler manifold. This example corresponds to the case (I-2).

References

[1] V. Apostolov, T. Dr˘ aghici and A. Moroianu, A splitting theorem for K¨ ahler man- ifolds whose Ricci tensors have constant eigenvalues, Int. J. of Math. 12 (2001), 769–789.

[2] S. I. Goldberg, Integrability of almost K¨ ahler manifolds, Proc. Amer. Math. Soc.

21 (1969), 96–100.

[3] O. Kowalski, Generalized Symmetric Spaces, Lecture Notes in Math. 805, Springer-Verlag, 1980.

[4] K. Sekigawa, On the Riemannian manifolds of the form B ×

F , Kodai Math.

Sem. Rep. 26 (1975), 343–347.

[5] , Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J.

7 (1978), 206–213.

[6] , On some compact Einstein almost K¨ ahler manifolds, J. Math. Soc. Japan 39 (1987), 677–684.

[7] K. Sekigawa, H. Suga and L. Vanhecke, Curvature homogeneity for four- dimensional manifolds, J. Korean Math. Soc. 32 (1995), no. 1, 93–101.

[8] H. Shima, On homogeneous K¨ ahler manifolds with non-degenerate canonical Her- mitian form of signature (2, 2(n − 1)), Osaka J. Math. 10 (1973), 477–493.

[9] I. M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685–697.

Terumasa Nihonyanagi

Graduate School of Science and Technology Niigata University

Niigata 950-2181, Japan Takashi Oguro

Department of Mathematical Sciences School of Science and Engineering Tokyo Denki University

Saitama 350-0394, Japan E-mail : [email protected] Kouei Sekigawa

Department of Mathematics Faculty of Science

Niigata University Niigata 950-2181, Japan

E-mail : [email protected]