Some Examples of Almost Kähler 4-Manifolds

Takuji Sato

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

Abstract

In the framework of P. Nurowski and M. Przanowski [3], we construct 4dimensional examples of Ricci flat almost Kähler manifolds, almost Kähler manifolds of pointwise constant holomorphic sectional curvature, and weakly ∗-Einstein almost Kähler manifolds.

Mathematics Subject Classification: 53C25, 53C55 Key words: almost Kähler manifolds, pointwise constant holomorphic sectional curvature, weakly ∗-Einstein manifolds

1 Introduction

An almost Hermitian manifold M = (M, J, g) is called an almost Kähler manifold if its Kähler form Ω is closed. One of the interest problems on almost Kähler manifolds is so called Goldberg’s conjecture, which asserts the almost complex structure of a compact Einstein almost Kähler manifold is integrable (and the manifold is necessarily Kähler) [2]. In connection with this conjecture, P. Nurowski and M. Przanowski [3] recently constructed a non-compact example of a strictly almost Kähler, Ricci-flat manifold. This shows that the assumption about compactness of the Einstein manifold is essential for the Goldberg conjecture. This example is also a space of pointwise positive constant holomorphic sectional curvature and a weakly ∗-Einstein manifold (see also [4]).

By considering a real expression for the Nurowski-Przanowski example, the author [5] has costructed a new example of almost Kähler manifold of pointwise negative constant holomorphic sectional curvature. In the present paper, we shall mainly deal with 4-dimensional almost Kähler manifolds M(f, u, v, ϕ) which will be defined in §2, and show that there exists a family of Ricci flat almost Kähler manifolds which include the Nurowski-Przanowski example. In [5], we investigated almost Kähler manifolds M(f, u, v, ϕ) with u = v = 0 and ϕ = 0. For the case where u and v are any constants and ϕ is arbitrary, we can obtain the examples of almost Kähler manifolds with pointwise constant holomorphic sectional curvature which are generalizations of [5]. These examples are also weakly ∗-Einstein, but not Einstein. We also show that there are other examples of weakly ∗-Einstein almost Kähler manifolds. Our examples are 4-dimensional and non-compact. We do not know whether or not there exist arbitrary dimensional, compact almost Kähler manifolds which are of (pointwise) constant holomorphic sectional curvature, or which are (weakly) ∗-Einstein.

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 113-137 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

In §2, we recall a characterization for a 4-dimensional almost Kähler manifold of pointwise constant holomorphic sectional curvature by using the expressions A_{ij} introduced by J. T. Cho and K. Sekigawa [1]. We give a real version of the Nurowski-Przanowski’s construction, and define an almost Kähler manifold M(f, u, v, ϕ). §3 is devoted to the construction of Ricci-flat examples. By calculating Ricci tensor ρ_{ij}, we find functions u, v and f for ρ_{ij} vanishing. In §4, we show that the Ricci flat almost Kähler manifolds in §3 are also of pointwise constant holomorphic sectional curvature, by using the characterization of [1]. Moreover, when u, v are constants α, β, we derive conditions for M(f, α, β, ϕ) to be of pointwise constant holomorphic sectional curvature. Taking account of these conditions, we shall obtain examples (Theorem 4.4). In the last §5, we give other examples of weakly ∗-Einstein almost Kähler manifolds. Especially, the last example (Theorem 5.3) shows that it depends on the value of ϕ that M(f, α, β, ϕ) is weakly ∗-Einstein.

2 Preliminaries

Let M = (M, J, g) be a four-dimensional almost Hermitian manifold with an almost Hermitian structure (J, g). We denote by Ω and N the Kähler form and the Nijenhuis tensor of M defined respectively by Ω(X, Y) = g(X, JY) and N(X, Y) = [JX, JY] − [X, Y] − J[JX, Y] − J[X, JY] for X, Y ∈ X (M), where X (M) is the Lie algebra of all smooth vector fields on M. The Nijenhuis tensor N has the properties

N(JX, Y) = N(X, JY) = −JN(X, Y), X, Y ∈ X (M).

Further we denote by ∇, R, ρ, τ, ρ^{∗} and τ^{∗} the Riemannian connection, the Riemannian curvature tensor, the Ricci tensor, the scalar curvature, the Ricci ∗-tensor and the ∗-scalar curvature of M, respectively. The Ricci ∗-tensor ρ^{∗} satisfies

ρ^{∗}(JX, JY) = ρ^{∗}(Y, X), X, Y ∈ X (M).

An almost Hermitian manifold M is called a weakly ∗-Einstein manifold if it satisfies ρ^{∗} = λ^{∗}g for some function λ^{∗} on M. In particular, if λ^{∗} is constant on M, then M is called a ∗-Einstein manifold .

The holomorphic sectional curvature H = H(x) = −R(x, Jx, x, Jx) (x ∈ T_{p}(M), ∥x∥ = 1) can be regarded as a differentiable function on the unit tangent bundle U(M) of M. If the function H is constant along each fiber, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole of U(M), then M is called a space of constant holomorphic sectional curvature.

Now we assume that M = (M, J, g) is a four-dimensional almost Kähler manifold. Then we have

(2.1)  2g((∇_{X}J)Y, Z) = g(JX, N(Y, Z)),

(2.2)  τ − τ^{∗} = −\frac{1}{2}∥∇J∥^{2} = −\frac{1}{8}∥N∥^{2}.

In the sequel, we adopt the following notational convention: for an orthonormal basis {e_{i}} of a tangent space T_{p}M, we put

\begin{matrix}J_{ij} = g(Je_{i}, e_{j}), & ∇_{i}J_{jk} = g((∇_{e_{i}}J)e_{j}, e_{k}), \\ N_{ijk} = g(e_{i}, N(e_{j}, e_{k})), & R_{ijkl} = R(e_{i}, e_{j}, e_{k}, e_{l}), \\ ∇īJ_{jk} = g((∇_{Je_{i}}J)e_{j}, e_{k}), & N_{ījk} = g(Je_{i}, N(e_{j}, e_{k})), \\ N_{īj̄k} = g(Je_{i}, N(Je_{j}, e_{k})), & R_{ijk̄l̄} = R(e_{i}, e_{j}, Je_{k}, Je_{l}), etc. \end{matrix}

Then it is easy to see that

∇_{i}J_{jk} + ∇īJ_{j̄k} = 0,
∇īJ_{jk} = ∇_{i}J_{j̄k} = ∇_{i}J_{jk̄},
N_{ijk} = −2∇īJ_{jk}, 2∇_{i}J_{jk} = N_{ījk}.

We set

(2.3)  A_{ij} = g(e_{i}, (∇_{e_{j}}N)(e_{1}, e_{3})) = ∇_{j}N_{i13},

for a unitary basis {e_{i}} = {e_{1}, e_{2} = Je_{1}, e_{3}, e_{4} = Je_{3}} of T_{p}M, p ∈ M. We note that

A_{ij} − A_{ji} = −2(R_{ij13} − R_{ij24}).

By using these A_{ij}, J. T. Cho and K. Sekigawa obtained the following characterization of almost Kähler manifolds of pointwise constant holomorphic sectional curvature:

Proposition 2.1 ([1]). Let M be a 4-dimensional almost Kähler manifold of pointwise constant holomorphic sectional curvature c = c(p)(p ∈ M). Then

R_{1212} = R_{3434} = −c(p),

R_{1234} = −\frac{c(p)}{2} − \frac{1}{16}(τ^{∗} − τ),

R_{1324} = −\frac{c(p)}{4} + \frac{1}{32}(τ^{∗} − τ) + \frac{1}{8}(A_{13} − A_{31} − A_{24} + A_{42}),

R_{1423} = \frac{c(p)}{4} + \frac{3}{32}(τ^{∗} − τ) + \frac{1}{8}(A_{13} − A_{31} − A_{24} + A_{42}),

R_{1313} = −\frac{c(p)}{4} + \frac{1}{32}(τ^{∗} − τ) − \frac{3}{8}(A_{13} − A_{31}) − \frac{1}{8}(A_{24} − A_{42}),

R_{1414} = −\frac{c(p)}{4} + \frac{5}{32}(τ^{∗} − τ) − \frac{1}{8}(A_{13} + A_{42}) − \frac{3}{8}(A_{31} + A_{24}),

R_{2323} = −\frac{c(p)}{4} + \frac{5}{32}(τ^{∗} − τ) + \frac{1}{8}(A_{31} + A_{24}) + \frac{3}{8}(A_{13} + A_{42}),

R_{2424} = −\frac{c(p)}{4} + \frac{1}{32}(τ^{∗} − τ) + \frac{3}{8}(A_{24} − A_{42}) + \frac{1}{8}(A_{13} − A_{31}),

R_{1334} = −R_{2434} = −\frac{1}{4}(A_{34} − A_{43}),

R_{1213} = −R_{1224} = −\frac{1}{4}(A_{12} − A_{21}),

R_{1434} = R_{2334} = −\frac{1}{4}(A_{33} + A_{44}),

R_{1214} = R_{1223} = −\frac{1}{4}(A_{11} + A_{22}),

R_{1323} = \frac{1}{8}(A_{14} + A_{41} + A_{32} − 3A_{23}),

R_{2324} = \frac{1}{8}(A_{14} + A_{41} + A_{23} − 3A_{32}),

R_{1314} = −\frac{1}{8}(A_{23} + A_{32} + A_{14} − 3A_{41}),

R_{1424} = −\frac{1}{8}(A_{23} + A_{32} + A_{41} − 3A_{14}),

for any unitary basis {e_{i}} of T_{p}M at each point p ∈ M.

Let M be an open set of R^{4}, and let (x_{1}, x_{2}, x_{3}, x_{4}) be the Euclidean coordinates on M. We put

z_{1} = x_{1} + \sqrt{−1}x_{2}, z_{2} = x_{3} + \sqrt{−1}x_{4}.

Let f be a non-zero real function and h be a complex function on M. Then P. Nurowski and M. Przanowski proved the following Lemma 2.2 ([3]). Let (z_{1}, z̄_{1}, z_{2}, z̄_{2}) be coordinates on M. Then for each value of the real constant ϕ ∈ [0, 2π), the metric

g = 2f^{2}(dz_{1} + hdz_{2})(dz̄_{1} + h̄dz̄_{2}) + \frac{2}{f^{2}} dz_{2}dz̄_{2}

and the almost complex structure

J_{e^{\sqrt{−1}ϕ}}^{+} = 2Re [\sqrt{−1} e^{\sqrt{−1}ϕ}\{f^{2}(dz_{1} + hdz_{2}) ⊗ (\frac{∂}{∂z̄_{2}} − h̄\frac{∂}{∂z̄_{1}} ) − \frac{1}{f^{2}} dz_{2} ⊗ \frac{∂}{∂z̄_{1}} \}]

define an almost Kähler structure on M.

The Riemannian metric g = (g_{ij}) and the almost complex structure J_{e^{\sqrt{−1}ϕ}}^{+} =

(J_{i}^{j}) in the above Lemma 2.2 are given with respect to the real coordinates (x_{1}, x_{2}, x_{3}, x_{4}) by

(2.4)  \begin{matrix}g_{11} = g_{22} = 2f^{2}, \\ g_{12} = g_{21} = g_{34} = g_{43} = 0, \\ g_{13} = g_{31} = g_{24} = g_{42} = f^{2}u, \\ −g_{14} = −g_{41} = g_{23} = g_{32} = f^{2}v, \\ g_{33} = g_{44} = 2f^{2}(u^{2} + v^{2} + \frac{1}{f^{4}} ), \end{matrix}

and

(2.5)

J_{1}^{1} = −J_{2}^{2} = −v cos ϕ + u sin ϕ,
J_{1}^{2} = J_{2}^{1} = u cos ϕ + v sin ϕ,
J_{1}^{3} = −J_{2}^{4} = − sin ϕ,
J_{1}^{4} = J_{2}^{3} = − cos ϕ,
J_{3}^{1} = −J_{4}^{2} = (u^{2} + v^{2} + \frac{1}{f^{4}} ) sin ϕ,
J_{3}^{2} = J_{4}^{1} = (u^{2} + v^{2} + \frac{1}{f^{4}} ) cos ϕ,
−J_{3}^{3} = J_{4}^{4} = v cos ϕ + u sin ϕ,
J_{3}^{4} = J_{4}^{3} = −u cos ϕ + v sin ϕ,

where u and v are the real and imaginary part of the complex function h, respectively. It is easy to see that the Kähler form Ω is given by

Ω = 2 sin ϕ dx_{1} ∧ dx_{3} + 2 cos ϕ dx_{1} ∧ dx_{4} + 2 cos ϕ dx_{2} ∧ dx_{3} − 2 sin ϕ dx_{2} ∧ dx_{4},

and (J_{e^{\sqrt{−1}ϕ}}^{+}, g) is an almost Kähler structure. In the present paper, we shall mainly

deal with this almost Kähler manifold (M, J_{e^{\sqrt{−1}ϕ}}^{+}, g). Since the almost Kähler struc-

ture (J_{e^{\sqrt{−1}ϕ}}^{+}, g) is determined by functions f, u, v and a real constant ϕ, we denote this almost Kähler manifold by M(f, u, v, ϕ).

We define a unitary frame field {e_{1}, e_{2} = Je_{1}, e_{3}, e_{4} = Je_{3}} on M(f, u, v, ϕ) by

(2.6)

e_{1} = \frac{1}{\sqrt{2}f} \frac{∂}{∂x_{1}} ,

e_{2} = \frac{f}{\sqrt{2}} \{(−v cos ϕ + u sin ϕ)\frac{∂}{∂x_{1}} + (u cos ϕ + v sin ϕ)\frac{∂}{∂x_{2}}

− sin ϕ\frac{∂}{∂x_{3}} − cos ϕ\frac{∂}{∂x_{4}} \} ,

e_{3} = \frac{1}{\sqrt{2}f} \frac{∂}{∂x_{2}} ,

e_{4} = \frac{f}{\sqrt{2}} \{(u cos ϕ + v sin ϕ)\frac{∂}{∂x_{1}} + (v cos ϕ − u sin ϕ)\frac{∂}{∂x_{2}}

− cos ϕ\frac{∂}{∂x_{3}} + sin ϕ\frac{∂}{∂x_{4}} \} .

118 T. Sato With respect to this unitary frame {e_{i}}_{i=1,2,3,4} we set

∇_{e_{i}}e_{j} = ∑ Γ_{ijk}e_{k}.

Then we have

(2.7)

Γ_{ijj} = 0,

Γ_{112} = −Γ_{121} = \frac{1}{\sqrt{2}}\{cos ϕ(∂_{4}f + v∂_{1}f − u∂_{2}f + f∂_{1}v)

+ sin ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f − f∂_{1}u)} ,

Γ_{113} = −Γ_{131} = −\frac{∂_{2}f}{\sqrt{2}f^{2}} ,

Γ_{114} = −Γ_{141} = \frac{1}{\sqrt{2}}\{cos ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f − f∂_{1}u)

− sin ϕ(∂_{4}f + v∂_{1}f − u∂_{2}f + f∂_{1}v)} ,

Γ_{123} = −Γ_{132} = \frac{f}{2\sqrt{2}}\{cos ϕ(∂_{1}u − ∂_{2}v) + sin ϕ(∂_{2}u + ∂_{1}v)\},

Γ_{124} = −Γ_{142} = \frac{f^{3}}{2\sqrt{2}}\{(∂_{3}v − u∂_{1}v − v∂_{2}v) + (∂_{4}u + v∂_{1}u − u∂_{2}u)\},

Γ_{134} = −Γ_{143} = −\frac{f}{2\sqrt{2}}\{cos ϕ(∂_{2}u + ∂_{1}v) − sin ϕ(∂_{1}u − ∂_{2}v)\},

Γ_{212} = −Γ_{221} = −\frac{∂_{1}f}{\sqrt{2}f^{2}} ,

Γ_{213} = −Γ_{231} = −\frac{f}{2\sqrt{2}}\{cos ϕ(∂_{1}u + ∂_{2}v) − sin ϕ(∂_{2}u − ∂_{1}v)\},

Γ_{214} = −Γ_{241} = \frac{f^{3}}{2\sqrt{2}}\{(∂_{3}v − u∂_{1}v − v∂_{2}v) + (∂_{4}u + v∂_{1}u − u∂_{2}u)\},

Γ_{223} = −Γ_{232} = \frac{∂_{2}f}{\sqrt{2}f^{2}} ,

Γ_{224} = −Γ_{242} = −\frac{1}{\sqrt{2}}\{cos ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f)

− sin ϕ(∂_{4}f + v∂_{1}f − u∂_{2}f)} ,

Γ_{234} = −Γ_{243} = −\frac{f^{3}}{2\sqrt{2}}\{(∂_{3}u − u∂_{1}u − v∂_{2}u) − (∂_{4}v + v∂_{1}v − u∂_{2}v)\},

Γ_{312} = −Γ_{321} = −\frac{f}{2\sqrt{2}}\{cos ϕ(∂_{1}u − ∂_{2}v) + sin ϕ(∂_{2}u + ∂_{1}v)\},

Γ_{313} = −Γ_{331} = \frac{∂_{1}f}{\sqrt{2}f^{2}} ,

Γ_{314} = −Γ_{341} = −\frac{f}{2\sqrt{2}}\{cos ϕ(∂_{2}u + ∂_{1}v) − sin ϕ(∂_{1}u − ∂_{2}v)\},

Γ_{323} = −Γ_{332} = −\frac{1}{\sqrt{2}}\{cos ϕ(∂_{4}f + v∂_{1}f − u∂_{2}f − f∂_{2}u)

+ sin ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f − f∂_{2}v)} ,

Γ_{324} = −Γ_{342} = −\frac{f^{3}}{2\sqrt{2}}\{(∂_{3}u − u∂_{1}u − v∂_{2}u) − (∂_{4}v + v∂_{1}v − u∂_{2}v)\},

Γ_{334} = −Γ_{343} = \frac{1}{\sqrt{2}}\{cos ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f − f∂_{2}v)

− sin ϕ(∂_{4}f + v∂_{1}f − u∂_{2}f − f∂_{2}u)},

Γ_{412} = −Γ_{421} = −\frac{f^{3}}{2\sqrt{2}}\{(∂_{3}v − u∂_{1}v − v∂_{2}v) + (∂_{4}u + v∂_{1}u − u∂_{2}u)\},

Γ_{413} = −Γ_{431} = \frac{f}{2\sqrt{2}}\{cos ϕ(∂_{2}u − ∂_{1}v) + sin ϕ(∂_{1}u + ∂_{2}v)\},

Γ_{414} = −Γ_{441} = −\frac{∂_{1}f}{\sqrt{2}f^{2}} ,

Γ_{423} = −Γ_{432} = −\frac{f^{3}}{2\sqrt{2}}\{(∂_{3}u − u∂_{1}u − v∂_{2}u) − (∂_{4}v + v∂_{1}v − u∂_{2}v)\},

Γ_{424} = −Γ_{442} = \frac{1}{\sqrt{2}}\{cos ϕ(∂_{4}f + v∂_{1}f − u∂_{2}f)

+ sin ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f)},

Γ_{434} = −Γ_{443} = −\frac{∂_{2}f}{\sqrt{2}f^{2}} ,

where we denote

∂_{i}f = \frac{∂f}{∂x_{i}} .

By (2.7), we find (2.8)

∇_{1}J_{13} = −∇_{1}J_{31} = −∇_{1}J_{24} = ∇_{1}J_{42} = −∇_{2}J_{14} = ∇_{2}J_{41} = −∇_{2}J_{23} = ∇_{2}J_{32}
= \frac{1}{2\sqrt{2}} \{2 cos ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f) − 2 sin ϕ(∂_{4}f + v∂_{1}f − u∂_{1}f)
− f cos ϕ(∂_{1}u + ∂_{2}v) + f sin ϕ(∂_{2}u − ∂_{1}v)} ,
∇_{1}J_{14} = −∇_{1}J_{41} = ∇_{1}J_{23} = −∇_{1}J_{32} = ∇_{2}J_{13} = −∇_{2}J_{31} = −∇_{2}J_{24} = ∇_{2}J_{42}
= \frac{1}{2\sqrt{2}f^{2}} \{2∂_{2}f − f^{5}u(∂_{2}u + ∂_{1}v) + f^{5}v(∂_{1}u − ∂_{2}v) + f^{5}(∂_{4}u + ∂_{3}v)\} ,
∇_{3}J_{13} = −∇_{3}J_{31} = −∇_{3}J_{24} = ∇_{3}J_{42} = −∇_{4}J_{14} = ∇_{4}J_{41} = −∇_{4}J_{23} = ∇_{4}J_{32}
= −\frac{1}{2\sqrt{2}} \{2 sin ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f) + 2 cos ϕ(∂_{4}f + v∂_{1}f − u∂_{1}f)
− f sin ϕ(∂_{1}u + ∂_{2}v) − f cos ϕ(∂_{2}u − ∂_{1}v)} ,
∇_{3}J_{14} = −∇_{3}J_{41} = ∇_{3}J_{23} = −∇_{3}J_{32} = ∇_{4}J_{13} = −∇_{4}J_{31} = −∇_{4}J_{24} = ∇_{4}J_{42}
= \frac{1}{2\sqrt{2}f^{2}} \{−2∂_{1}f + f^{5}u(∂_{1}u − ∂_{2}v) + f^{5}v(∂_{2}u + ∂_{1}v) − f^{5}(∂_{3}u − ∂_{4}v)\} ,
∇_{i}J_{jk} = 0  (otherwise).

By (2.1) and (2.8), we then have (2.9)

N_{113} = −N_{131} = −N_{214} = N_{241} = −N_{223} = N_{232} = −N_{124} = N_{142}
= −\frac{1}{\sqrt{2}f^{2}} \{2∂_{2}f − f^{5}u(∂_{2}u + ∂_{1}v) + f^{5}v(∂_{1}u − ∂_{2}v) + f^{5}(∂_{4}u + ∂_{3}v)\} ,
N_{213} = −N_{231} = N_{114} = −N_{141} = N_{123} = −N_{132} = −N_{224} = N_{242}
= \frac{1}{\sqrt{2}} \{2 cos ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f) − 2 sin ϕ(∂_{4}f + v∂_{1}f − u∂_{1}f)
− f cos ϕ(∂_{1}u + ∂_{2}v) + f sin ϕ(∂_{2}u − ∂_{1}v)} ,
N_{313} = −N_{331} = −N_{414} = N_{441} = −N_{423} = N_{432} = −N_{324} = N_{342}
= −\frac{1}{\sqrt{2}f^{2}} \{−2∂_{1}f + f^{5}u(∂_{1}u − ∂_{2}v) + f^{5}v(∂_{2}u + ∂_{1}v) − f^{5}(∂_{3}u − ∂_{4}v)\} ,
N_{413} = −N_{431} = N_{314} = −N_{341} = N_{323} = −N_{332} = −N_{424} = N_{442}
= −\frac{1}{\sqrt{2}} \{2 sin ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f) + 2 cos ϕ(∂_{4}f + v∂_{1}f − u∂_{1}f)
− f sin ϕ(∂_{1}u + ∂_{2}v) − f cos ϕ(∂_{2}u − ∂_{1}v)} ,
N_{ijk} = 0 (otherwise).

Now, let u = α, v = β, where α, β are constants. In this case, we introduce a new coordinates system (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) on M(f, α, β, ϕ) by

(2.10)  ξ_{1} = x_{1} + αx_{3} − βx_{4}, ξ_{2} = x_{2} + βx_{3} + αx_{4}, ξ_{3} = x_{3}, ξ_{4} = x_{4}.

With respect to this coordinates system (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}), we adopt the notation ˜∂_{i}f =

\frac{∂f}{∂ξ_{i}}. Then, (2.9) can be written as

(2.11)

Γ_{112} = −Γ_{121} = −Γ_{323} = Γ_{332} = Γ_{424} = −Γ_{442}
= \frac{1}{\sqrt{2}}\{cos ϕ ˜∂_{4}f + sin ϕ ˜∂_{3}f\},

Γ_{113} = −Γ_{131} = −Γ_{223} = Γ_{232} = Γ_{434} = −Γ_{443} = −\frac{˜∂_{2}f}{\sqrt{2}f^{2}} ,
Γ_{114} = −Γ_{141} = −Γ_{224} = Γ_{242} = Γ_{334} = −Γ_{343}
= \frac{1}{\sqrt{2}}\{cos ϕ ˜∂_{3}f − sin ϕ ˜∂_{4}f\},

Γ_{212} = −Γ_{221} = −Γ_{313} = Γ_{331} = Γ_{414} = −Γ_{441} = −\frac{˜∂_{1}f}{\sqrt{2}f^{2}} ,
Γ_{ijk} = 0 (otherwise).

3 Ricci flat examples

In this section, we shall find out a family of Ricci flat metrics in the framework of Nurowski and Przanowski [3]. Let M(f, u, v, ϕ) be the almost Kähler 4-manifold defined in §2. Since the Nurowski-Przanowski’s Ricci flat example is given by

u = −2x_{3}, v = 2x_{4}, f = \frac{1}{\sqrt{2}} \{2x_{1} − 2(x_{3}^{2} + x_{4}^{2})\}^{− \frac{1}{4}} ,

we suppose that u = u(x_{3}, x_{4}), v = v(x_{3}, x_{4}). Then, by a straightforward computation, we obtain, with respect to the unitary basis {e_{i}} in (2.6),

(3.1)  \begin{aligned}ρ_{11} & = & \frac{1}{4 f^{4}} f^{10} (∂_{4}u + ∂_{3}v)^{2} − 10 (∂_{1}f)^{2} + 6 (∂_{2}f)^{2} \\ & − & 2 f^{4} (∂_{4}f + v ∂_{1}f − u ∂_{2}f)^{2} + (∂_{3}f − u ∂_{1}f − v ∂_{2}f)^{2} \\ & + & 2 f (∂_{1}^{2}f − ∂_{2}^{2}f) \\ & − & 2 f^{5} \{(∂_{4} + v∂_{1} − u∂_{2})^{2}f + (∂_{3} − u∂_{1} − v∂_{2})^{2}f)^{2}\}], \end{aligned}

122 T. Sato (3.2)

ρ_{22} = −\frac{1}{4 f^{4}} [6 \{(∂_{1}f)^{2} + (∂_{2}f)^{2}\} − 2f\{∂_{1}^{2}f + ∂_{2}^{2}f\}
+ f^{10} {(∂_{3}u − ∂_{4}v)^{2} + (∂_{4}u + ∂_{3}v)^{2}}
+ 2 f^{4} { (1 + 2 cos 2ϕ) (∂_{4}f + v∂_{1}f − u∂_{2}f)^{2}
+ (1 − 2 cos 2ϕ) (∂_{3}f − u∂_{1}f − v∂_{2}f)^{2}
+ 4 sin 2ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f)(∂_{4}f + v∂_{1}f − u∂_{2}f) }
+ 2 f^{5} { cos 2ϕ((∂_{4} + v∂_{1} − u∂_{2})^{2}f − (∂_{3} − u∂_{1} − v∂_{2})^{2}f)
+ sin 2ϕ((∂_{3} − u∂_{1} − v∂_{2})(∂_{4} + v∂_{1} − u∂_{2})f
+ (∂_{4} + v∂_{1} − u∂_{2})(∂_{3} − u∂_{1} − v∂_{2})f) } ],
(3.3)  \begin{aligned}ρ_{33} & = & \frac{1}{4 f^{4}} [f^{10} (∂_{3}u − ∂_{4}v)^{2} + 6 (∂_{1}f)^{2} − 10 (∂_{2}f)^{2} \\ & − & 2 f^{4} \{ (∂_{4}f + v ∂_{1}f − u ∂_{2}f)^{2} + (∂_{3}f − u ∂_{1}f − v ∂_{2}f)^{2} \} \\ & − & 2 f (∂_{1}^{2}f − ∂_{2}^{2}f) \\ & − & 2 f^{5} \{ (∂_{4} + v∂_{1} − u∂_{2})^{2}f + (∂_{3} − u∂_{1} − v∂_{2})^{2}f \} ], \end{aligned}

(3.4)

ρ_{44} = \frac{1}{4 f^{4}} [ − 6 \{(∂_{1}f)^{2} + (∂_{2}f)^{2}\} + 2f\{∂_{1}^{2}f + ∂_{2}^{2}f\}
− f^{10} {(∂_{3}u − ∂_{4}v)^{2} + (∂_{4}u + ∂_{3}v)^{2}}
− 2 f^{4} { (1 − 2 cos 2ϕ) (∂_{4}f + v∂_{1}f − u∂_{2}f)^{2}
+ (1 + 2 cos 2ϕ) (∂_{3}f − u∂_{1}f − v∂_{2}f)^{2}
− 4 sin 2ϕ(∂_{3}f − u∂_{1}f − v∂_{2}f)(∂_{4}f + v∂_{1}f − u∂_{2}f) }
+ 2 f^{5} { cos 2ϕ((∂_{4} + v∂_{1} − u∂_{2})^{2}f − (∂_{3} − u∂_{1} − v∂_{2})^{2}f)
+ sin 2ϕ((∂_{3} − u∂_{1} − v∂_{2})(∂_{4} + v∂_{1} − u∂_{2})f
+ (∂_{4} + v∂_{1} − u∂_{2})(∂_{3} − u∂_{1} − v∂_{2})f) } ],

(3.5)

ρ_{12} = \frac{1}{4 f^{2}} [f^{6}\{sin ϕ ∂_{4}(∂_{4}u + ∂_{3}v) − cos ϕ ∂_{3}(∂_{4}u + ∂_{3}v)\}
+ 4 ∂_{1}f {cos ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f) + sin ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f)}
− 6 f^{5} (∂_{4}u + ∂_{3}v){cos ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f) − sin ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f)}],
(3.6)  ρ_{13} = −\frac{1}{4 f^{4}} [f^{10} (∂_{3}u − ∂_{4}v)(∂_{4}u + ∂_{3}v) + 16 ∂_{1}f ∂_{2}f − 4f∂_{1}∂_{2}f],

Some Examples of Almost Kähler 4-Manifolds 123 (3.7)

ρ_{14} = \frac{1}{4 f^{2}} [f^{6}\{sin ϕ ∂_{3}(∂_{4}u + ∂_{3}v) + cos ϕ ∂_{4}(∂_{4}u + ∂_{3}v)\}
+ 4 ∂_{1}f {cos ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f) − sin ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f)}
+ 6 f^{5} (∂_{4}u + ∂_{3}v){cos ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f) + sin ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f)}],

(3.8)

ρ_{23} = \frac{1}{4 f^{2}} [f^{6}\{cos ϕ ∂_{3}(∂_{3}u − ∂_{4}v) − sin ϕ ∂_{4}(∂_{3}u − ∂_{4}v)\}
+ 4 ∂_{2}f {cos ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f) + sin ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f)}
+ 6 f^{5} (∂_{3}u − ∂_{4}v){cos ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f) − sin ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f)}],

(3.9)

ρ_{24} = \frac{1}{2}f ∂_{2}f(∂_{3}u − ∂_{4}v) − \frac{1}{2}f ∂_{1}f(∂_{4}u + ∂_{3}v)
+ sin 2ϕ {(∂_{4}f + v ∂_{1}f − u ∂_{2}f)^{2} − (∂_{3}f − u ∂_{1}f − v ∂_{2}f)^{2}}
− 2 cos 2ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f)(∂_{4}f + v ∂_{1}f − u ∂_{2}f)
+ f cos ϕ sin ϕ ∂_{4}(∂_{4}f + v ∂_{1}f − u ∂_{2}f) − f cos^{2} ϕ ∂_{4}(∂_{3}f − u ∂_{1}f − v ∂_{2}f)
+ f sin^{2} ϕ ∂_{3}(∂_{4}f + v ∂_{1}f − u ∂_{2}f) − f cos ϕ sin ϕ ∂_{3}(∂_{3}f − u ∂_{1}f − v ∂_{2}f)
+ f(u cos ϕ + v sin ϕ) { cos ϕ ∂_{2}(∂_{3}f − u ∂_{1}f − v ∂_{2}f)
− sin ϕ ∂_{2}(∂_{4}f + v ∂_{1}f − u ∂_{2}f) }
+ f(u sin ϕ − v cos ϕ) { cos ϕ ∂_{1}(∂_{3}f − u ∂_{1}f − v ∂_{2}f)
− sin ϕ ∂_{1}(∂_{4}f + v ∂_{1}f − u ∂_{2}f) } ,

(3.10)

ρ_{34} = −\frac{1}{4 f^{2}} [f^{6}\{sin ϕ ∂_{3}(∂_{3}u − ∂_{4}v) + cos ϕ ∂_{4}(∂_{3}u − ∂_{4}v)\}
+ 4 ∂_{2}f {sin ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f) − cos ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f)}
+ 6 f^{5} (∂_{3}u − ∂_{4}v){sin ϕ (∂_{3}f − u ∂_{1}f − v ∂_{2}f) + cos ϕ (∂_{4}f + v ∂_{1}f − u ∂_{2}f)}].

In order that our almost Kähler manifold M(f, u, v, ϕ) is Einstein, it is necessary that ρ_{ij} = 0 for i ≠ j. From (3.6), if

∂_{3}u − ∂_{4}v = 0 or ∂_{4}u + ∂_{3}v = 0

and

∂_{1}f = 0 or ∂_{2}f = 0,

then ρ_{13} = 0. Taking account of the Nurowski-Przanowski example, we assume that

(3.11)  ∂_{4}u + ∂_{3}v = 0 and ∂_{2}f = 0.

By (3.5) and (3.7), if

(3.12)  ∂_{3}f = u∂_{1}f and ∂_{4}f = −v∂_{1}f,

then ρ_{12} = ρ_{14} = ρ_{24} = 0. Further, from (3.8), if

(3.13)  ∂_{3}u − ∂_{4}v = λ(= costant),

then ρ_{23} = ρ_{34} = 0.

In this case, (3.1) ∼ (3.4) reduce to

(3.14)  ρ_{11} = −\frac{1}{4 f^{4}} [10 (∂_{1}f)^{2} − 2 f ∂_{1}^{2}f],

(3.15)  ρ_{22} = −\frac{1}{4 f^{4}} [6 (∂_{1}f)^{2} − 2 f ∂_{1}^{2}f + λ^{2}f^{10}],

(3.16)  ρ_{33} = \frac{1}{4 f^{4}} [6 (∂_{1}f)^{2} − 2 f ∂_{1}^{2}f + λ^{2}f^{10}],

(3.17)  ρ_{44} = −\frac{1}{4 f^{4}} [6 (∂_{1}f)^{2} − 2 f ∂_{1}^{2}f + λ^{2}f^{10}].

Since ρ_{22} = −ρ_{33}, it must be ρ = 0 for (M, J, g) is Einstein. Comparing (3.14) with (3.15), we find

4(∂_{1}f)^{2} = λ^{2}f^{10} and 2∂_{1}f = ±λf^{5}.

If 2∂_{1}f = −λf^{5}, then N = 0 by (2.9), and J is integrable. So, we suppose

(3.18)  2∂_{1}f = λf^{5}, λ ≠ 0.

From (3.18), we obtain that f is of in the following form:

(3.19)  f = \{−2λx_{1} − φ(x_{3}, x_{4})\}^{− \frac{1}{4}} ,

where φ is an arbitrary function of x_{3} and x_{4}.

By (3.12), we have

(3.20)  ∂_{3}φ = 2λu and ∂_{4}φ = −2λv.

Taking account of (3.13) and (3.20), we deduce

(3.21)  ∂_{3}^{2}φ + ∂_{4}^{2}φ = 2λ^{2}.

Here, we suppose that φ is a polynomial of degree 2, for simplicity:

φ = ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k.

Then by (3.20) and (3.21), we have

u = \frac{1}{λ}(ax_{3} + bx_{4} + d),

v = −\frac{1}{λ}(bx_{3} + cx_{4} + e),

a + c = λ^{2}(> 0).

Summing up the above arguements, we obtain the following Theorem 3.1. Let

u = ±\frac{1}{\sqrt{a + c}}(ax_{3} + bx_{4} + d),

v = ∓\frac{1}{\sqrt{a + c}}(bx_{3} + cx_{4} + e),

f = \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{1}{4}} ,

where a, b, c, d, e, k are arbitrary constants such that a + c > 0. Then Riemannian metric g given by (2.4) is a Ricci flat, i.e., M(f, u, v, ϕ) is a Ricci flat strictly almost Kähler manifold.

Note that the Nurowski-Przanowski’s Ricci flat example is obtained by putting a = c = 8, λ = −4, b = d = e = k = 0.

Next, we assume that

(3.22)  ∂_{3}u − ∂_{4}v = 0 and ∂_{1}f = 0.

By (3.8) and (3.10), if

(3.23)  ∂_{3}f = v∂_{2}f and ∂_{4}f = u∂_{2}f,

then ρ_{23} = ρ_{24} = ρ_{34} = 0. Further, from (3.5), if

(3.24)  ∂_{3}u + ∂_{4}v = μ(= costant),

then ρ_{12} = ρ_{14} = 0.

In this case, (3.1) ∼ (3.4) reduce to

(3.25)  ρ_{11} = \frac{1}{4 f^{4}} [6 (∂_{2}f)^{2} − 2 f ∂_{2}^{2}f + μ^{2}f^{10}],

(3.26)  ρ_{22} = −\frac{1}{4 f^{4}} [6 (∂_{2}f)^{2} − 2 f ∂_{2}^{2}f + μ^{2}f^{10}],

(3.27)  ρ_{33} = −\frac{1}{4 f^{4}} [10 (∂_{2}f)^{2} − 2 f ∂_{2}^{2}f],

(3.28)  ρ_{44} = −\frac{1}{4 f^{4}} [6 (∂_{2}f)^{2} − 2 f ∂_{2}^{2}f + μ^{2}f^{10}].

By the same arguement as above, we have

(3.29)  f = \{−2μx_{2} − φ(x_{3}, x_{4})\}^{− \frac{1}{4}} ,

and suppose that

φ = ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k.

Then, we find

u = \frac{1}{μ}(bx_{3} + cx_{4} + e),

v = \frac{1}{μ}(ax_{3} + bx_{4} + d),

a + c = μ^{2}(> 0).

Consequently, we obtain the following Theorem 3.2. Let

u = ±\frac{1}{\sqrt{a + c}}(bx_{3} + cx_{4} + e),

v = ±\frac{1}{\sqrt{a + c}}(ax_{3} + bx_{4} + d),

f = \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{1}{4}} ,

where a, b, c, d, e, k are arbitrary constants such that a + c > 0. Then Riemannian metric g given by (2.4) is a Ricci flat, i.e., M(f, u, v, ϕ) is a Ricci flat strictly almost Kähler manifold.

In the next section, we shall show that the almost Kähler manifolds M(f, u, v, ϕ) in Theorems 3.1 and 3.2 are also of pointwise constant holomorphic sectional curvature and weakly ∗-Einstein. Remark 3.3. By considering the integrable case: 2∂_{1}f = −λf^{5} or 2∂_{2}f = −μf^{5}, we get the Ricci flat Kähler manifolds,

Let

u = ±\frac{1}{\sqrt{a + c}}(ax_{3} + bx_{4} + d),

v = ∓\frac{1}{\sqrt{a + c}}(bx_{3} + cx_{4} + e),

f = \{±2\sqrt{a + c} x_{1} + (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{1}{4}} ,

or

u = ±\frac{1}{\sqrt{a + c}}(bx_{3} + cx_{4} + e),

v = ±\frac{1}{\sqrt{a + c}}(ax_{3} + bx_{4} + d),

f = \{±2\sqrt{a + c} x_{2} + (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{1}{4}} .

Then M(f, u, v, ϕ) are Ricci flat Kähler manifolds.

4 Examples of almost Kähler 4-manifolds with pointwise constant holomorphic sectional curvature

In this section, we construct 4-dimensional almost Kähler manifolds of pointwise constant holomorphic sectional curvature.

First, we show that the almost Kähler manifolds M(f, u, v, ϕ) in Theorem 3.1 have pointwise constant holomorphic sectional curvature. Indeed, by direct computations, we obtain

(4.1)

A_{13} = −A_{24} =

−\frac{1}{2}(a + c) \{∓2\sqrt{a + c} x_{1} = −(ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

A_{31} = −A_{42}

= \frac{3}{2}(a + c) \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

A_{ij} = 0 (otherwise),

(4.2)

R_{1212} = R_{1234} = R_{1414} = R_{1423} = R_{2323} = R_{3434}

= −\frac{1}{4}(a + c) \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

R_{1313} = −R_{1324} = R_{2424}

= \frac{1}{2}(a + c) \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

R_{ijkl} = 0 (otherwise i < j, k < l),

(4.3) \begin{matrix}ρ_{11}^{∗} = ρ_{22}^{∗} = ρ_{33}^{∗} = ρ_{44}^{∗} \\ = \frac{1}{2}(a + c) \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} , \\ ρ_{ij}^{∗} = 0  (i ≠ j), \end{matrix}

(4.4) τ^{∗} = 2(a + c) \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} .

By virtue of Proposition 2.1, (4.1), (4.2), (4.3), and (4.4), we have the following Theorem 4.1. Let M(f, u, v, ϕ) be the Ricci flat strictly almost Kähler manifold in Theorem 3.1, i.e., functions u, v and f are given by

u = ±\frac{1}{\sqrt{a + c}}(ax_{3} + bx_{4} + d),

v = ∓\frac{1}{\sqrt{a + c}}(bx_{3} + cx_{4} + e),

f = \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{1}{4}} .

Then M(f, u, v, ϕ) is of pointwise constant holomorphic sectional curvature

c = \frac{1}{4}(a + c) \{∓2\sqrt{a + c} x_{1} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

and weakly ∗-Einstein.

Next, let M(f, u, v, ϕ) be the almost Kähler manifold in Theorem 3.2 Then, similarly we obtain

(4.5)

A_{13} = −A_{24}

= −\frac{3}{2}(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

A_{31} = −A_{42}

= \frac{1}{2}(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

A_{ij} = 0 (otherwise),

(4.6)

R_{1212} = R_{1234} = R_{1414} = R_{1423} = R_{2323} = R_{3434}

= −\frac{1}{4}(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

R_{1313} = −R_{1324} = R_{2424}

= \frac{1}{2}(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

R_{ijkl} = 0 (otherwisei < j, k < l),

(4.7) \begin{matrix}ρ_{11}^{∗} = ρ_{22}^{∗} = ρ_{33}^{∗} = ρ_{44}^{∗} \\ = \frac{1}{2}(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} , \\ ρ_{ij}^{∗} = 0  (i ≠ j), \end{matrix}

(4.8) τ^{∗} = 2(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} .

Therefore we have the following Theorem 4.2. Let M(f, u, v, ϕ) be the Ricci flat strictly almost Kähler manifold in Theorem 3.2, i.e., u, v and f are given by

u = ±\frac{1}{\sqrt{a + c}}(bx_{3} + cx_{4} + e),

v = ±\frac{1}{\sqrt{a + c}}(ax_{3} + bx_{4} + d),

f = \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{1}{4}} .

Then M(f, u, v, ϕ) is of pointwise constant holomorphic sectional curvature

c = \frac{1}{4}(a + c) \{∓2\sqrt{a + c} x_{2} − (ax_{3}^{2} + 2bx_{3}x_{4} + cx_{4}^{2} + 2dx_{3} + 2ex_{4} + k)\}^{− \frac{3}{2}} ,

and weakly ∗-Einstein.

Now, we shall construct other examples. In the previous paper [5], we assumed that u = v = 0 and ϕ = 0 for the sake of simplicity. Here, we shall consider the case where u = α, v = β (α and β are constants) and arbitrary ϕ. Then, with respect to the unitary frame {e_{i}} in (2.6) and the coordinates system (ξ_{i}) in (2.10), we have the following expressions:

(4.9)

A_{11} = \frac{3˜∂_{1}f ˜∂_{2}f}{f^{4}} − \frac{˜∂_{1} ˜∂_{2}f}{f^{3}} − cos 2ϕ ˜∂_{3}f ˜∂_{4}f − \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\},
A_{12} = \frac{cos ϕ}{f^{2}}\{2˜∂_{1}f ˜∂_{3}f − 2˜∂_{2}f ˜∂_{4}f + f ˜∂_{2} ˜∂_{4}f\}
− \frac{sin ϕ}{f^{2}}\{2˜∂_{2}f ˜∂_{3}f + 2˜∂_{1}f ˜∂_{4}f − f ˜∂_{2} ˜∂_{3}f\},
A_{13} = −\frac{1}{f^{4}} \{(˜∂_{1}f)^{2} − 2(˜∂_{2}f)^{2} + f ˜∂_{2}^{2}f\} − \{cos ϕ ˜∂_{3}f − sin ϕ ˜∂_{4}f\}^{2},
A_{14} = −\frac{cos ϕ}{f^{2}}\{˜∂_{2}f ˜∂_{3}f + ˜∂_{1}f ˜∂_{4}f − f ˜∂_{2} ˜∂_{3}f\}
− \frac{sin ϕ}{f^{2}}\{˜∂_{1}f ˜∂_{3}f − ˜∂_{2}f ˜∂_{4}f + f ˜∂_{2} ˜∂_{4}f\},
A_{21} = −\frac{cos ϕ}{f^{2}}\{2˜∂_{2}f ˜∂_{4}f − f ˜∂_{1} ˜∂_{3}f\} − \frac{sin ϕ}{f^{2}}\{2˜∂_{2}f ˜∂_{3}f + f ˜∂_{1} ˜∂_{4}f\},

A_{22} = \frac{˜∂_{1}f ˜∂_{2}f}{f^{4}} − cos 2ϕ ˜∂_{3}f ˜∂_{4}f − \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
− f cos 2ϕ ˜∂_{3} ˜∂_{4}f − \frac{f sin 2ϕ}{2}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\},
A_{23} = −\frac{cos ϕ}{f^{2}}\{˜∂_{2}f ˜∂_{3}f − ˜∂_{1}f ˜∂_{4}f − f ˜∂_{2} ˜∂_{3}f\}
+ \frac{sin ϕ}{f^{2}}\{˜∂_{1}f ˜∂_{3}f + ˜∂_{2}f ˜∂_{4}f − f ˜∂_{2} ˜∂_{4}f\},

A_{24} = \frac{(˜∂_{2}f)^{2}}{f^{4}} + \{cos ϕ ˜∂_{4}f + sin ϕ ˜∂_{3}f\}^{2} − \frac{f}{2} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\}
− \frac{f cos 2ϕ}{2}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\} + f sin 2ϕ ˜∂_{3} ˜∂_{4}f,
A_{31} = −\frac{1}{f^{4}} \{2(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2} − f ˜∂_{1}^{2}f\} + \{cos ϕ ˜∂_{4}f + sin ϕ ˜∂_{3}f\}^{2},

130 T. Sato

A_{32} = \frac{cos ϕ}{f^{2}}\{˜∂_{2}f ˜∂_{3}f + ˜∂_{1}f ˜∂_{4}f − f ˜∂_{1} ˜∂_{4}f\}
+ \frac{sin ϕ}{f^{2}}\{˜∂_{1}f ˜∂_{3}f − ˜∂_{2}f ˜∂_{4}f − f ˜∂_{1} ˜∂_{3}f\},
A_{33} = −\frac{3˜∂_{1}f ˜∂_{2}f}{f^{4}} + \frac{˜∂_{1} ˜∂_{2}f}{f^{3}} + cos 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\},
A_{34} = \frac{cos ϕ}{f^{2}}\{2˜∂_{1}f ˜∂_{3}f − 2˜∂_{2}f ˜∂_{4}f − f ˜∂_{1} ˜∂_{3}f\}
− \frac{sin ϕ}{f^{2}}\{2˜∂_{2}f ˜∂_{3}f + 2˜∂_{1}f ˜∂_{4}f − f ˜∂_{1} ˜∂_{4}f\},
A_{41} = −\frac{cos ϕ}{f^{2}}\{˜∂_{2}f ˜∂_{3}f − ˜∂_{1}f ˜∂_{4}f + f ˜∂_{1} ˜∂_{4}f\}
+ \frac{sin ϕ}{f^{2}}\{˜∂_{1}f ˜∂_{3}f + ˜∂_{2}f ˜∂_{4}f − f ˜∂_{1} ˜∂_{3}f\},
A_{42} = −\frac{(˜∂_{1}f)^{2}}{f^{4}} − \{cos ϕ˜∂_{3}f − sin ϕ˜∂_{4}f\}^{2} + \frac{f}{2} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\}
− \frac{f cos 2ϕ}{2}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\} + f sin 2ϕ ˜∂_{3} ˜∂_{4}f,
A_{43} = \frac{cos ϕ}{f^{2}}\{2˜∂_{1}f ˜∂_{3}f − f ˜∂_{2} ˜∂_{4}f\} − \frac{sin ϕ}{f^{2}}\{2˜∂_{1}f ˜∂_{4}f + f ˜∂_{2} ˜∂_{3}f\},
A_{44} = −\frac{˜∂_{1}f ˜∂_{2}f}{f^{4}} + cos 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
+ f cos 2ϕ ˜∂_{3} ˜∂_{4}f + \frac{f sin 2ϕ}{2}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\}.

Furthermore, we find from (2.11) (4.10)

R_{1212} = \frac{1}{2f^{4}} \{3(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2} − f ˜∂_{1}^{2}f\} − \frac{cos 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
+ sin 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{f}{4} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} − \frac{f cos 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\}
+ \frac{f sin 2ϕ}{2} ˜∂_{3} ˜∂_{4}f,
R_{3434} = \frac{1}{2f^{4}} \{−(˜∂_{1}f)^{2} + 3(˜∂_{2}f)^{2} − f ˜∂_{2}^{2}f\} + \frac{cos 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
− sin 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{f}{4} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} + \frac{f cos 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\}
− \frac{f sin 2ϕ}{2} ˜∂_{3} ˜∂_{4}f,

R_{1234} = 0,
R_{1324} = 0,
R_{1423} = 0,
R_{1313} = \frac{1}{2f^{4}} \{−(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2} + f ˜∂_{1}^{2}f + f ˜∂_{2}^{2}f\} + \frac{1}{2}\{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\},
R_{1414} = \frac{1}{2f^{4}} \{3(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2} − f ˜∂_{1}^{2}f\} + \frac{cos 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
− sin 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{f}{4} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} + \frac{f cos 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\}
− \frac{f sin 2ϕ}{2} ˜∂_{3} ˜∂_{4}f,
R_{2323} = \frac{1}{2f^{4}} \{−(˜∂_{1}f)^{2} + 3(˜∂_{2}f)^{2} − f ˜∂_{2}^{2}f\} − \frac{cos 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
+ sin 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{f}{4} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} − \frac{f cos 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\}
+ \frac{f sin 2ϕ}{2} ˜∂_{3} ˜∂_{4}f,
R_{2424} = \frac{1}{2f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} + \frac{1}{2}\{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\} − \frac{f}{2} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\},
R_{1334} = \frac{cos ϕ}{2f} ˜∂_{1} ˜∂_{3}f − \frac{sin ϕ}{2f} ˜∂_{1} ˜∂_{4}f,
R_{2434} = −\frac{cos ϕ}{f^{2}} ˜∂_{2}f ˜∂_{4}f − \frac{sin ϕ}{f^{2}} ˜∂_{2}f ˜∂_{3}f + \frac{cos ϕ}{2f} ˜∂_{2} ˜∂_{4}f + \frac{sin ϕ}{2f} ˜∂_{2} ˜∂_{3}f,
R_{1213} = −\frac{cos ϕ}{2f} ˜∂_{2} ˜∂_{4}f − \frac{sin ϕ}{2f} ˜∂_{2} ˜∂_{3}f,
R_{1224} = \frac{cos ϕ}{f^{2}} ˜∂_{1}f ˜∂_{3}f − \frac{sin ϕ}{f^{2}} ˜∂_{1}f ˜∂_{4}f − \frac{cos ϕ}{2f} ˜∂_{1} ˜∂_{3}f + \frac{sin ϕ}{2f} ˜∂_{1} ˜∂_{4}f,
R_{1434} = \frac{1}{2f^{4}} \{4˜∂_{1}f ˜∂_{2}f − f ˜∂_{1} ˜∂_{2}f\},
R_{2334} = − cos 2ϕ ˜∂_{3}f ˜∂_{4}f − \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
− \frac{f cos 2ϕ}{2} ˜∂_{3} ˜∂_{4}f − \frac{f sin 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\},
R_{1214} = cos 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}
+ \frac{f cos 2ϕ}{2} ˜∂_{3} ˜∂_{4}f + \frac{f sin 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\},

R_{1223} = −\frac{1}{2f^{4}} \{4˜∂_{1}f ˜∂_{2}f − f ˜∂_{1} ˜∂_{2}f\},
R_{1323} = −\frac{cos ϕ}{2f} ˜∂_{1} ˜∂_{4}f − \frac{sin ϕ}{2f} ˜∂_{1} ˜∂_{3}f,
R_{2324} = −\frac{cos ϕ}{f^{2}} ˜∂_{2}f ˜∂_{3}f + \frac{sin ϕ}{f^{2}} ˜∂_{2}f ˜∂_{4}f + \frac{cos ϕ}{2f} ˜∂_{2} ˜∂_{3}f − \frac{sin ϕ}{2f} ˜∂_{2} ˜∂_{4}f,
R_{1314} = −\frac{cos ϕ}{2f} ˜∂_{2} ˜∂_{3}f + \frac{sin ϕ}{2f} ˜∂_{2} ˜∂_{4}f,
R_{1424} = −\frac{cos ϕ}{f^{2}} ˜∂_{1}f ˜∂_{4}f − \frac{sin ϕ}{f^{2}} ˜∂_{1}f ˜∂_{3}f + \frac{cos ϕ}{2f} ˜∂_{1} ˜∂_{4}f + \frac{sin ϕ}{2f} ˜∂_{1} ˜∂_{3}f.

Then, by (4.1), we obtain easily

(4.11)  \begin{aligned}τ & = & \frac{1}{f^{3}} \{˜∂_{1}^{2}f + ˜∂_{2}^{2}f\} − \frac{4}{f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} \\ & − & 2\{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\} − f\{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\}, \end{aligned}

(4.12)  τ^{∗} = \frac{1}{f^{3}} \{˜∂_{1}^{2}f + ˜∂_{2}^{2}f\} − \frac{2}{f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} − f\{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\},

(4.13)  τ^{∗} − τ = \frac{2}{f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} + 2\{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\}.

By (4.13), we see that M(f, α, β, ϕ) is non-Kählerian if and only if f is not constant. By Proposition 2.1, (4.9), (4.1) and (4.13), we find the following Lemma 4.3. The almost Kähler manifold M(f, α, β, ϕ) is of pointwise constant holomorphic sectional curvature c = c(p) if and only if

(4.14)

\frac{1}{2f^{3}} \{˜∂_{1}^{2}f − ˜∂_{2}^{2}f\} + \frac{f cos 2ϕ}{2}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\} − f sin 2ϕ ˜∂_{3} ˜∂_{4}f

= \frac{2}{f^{4}} \{(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2}\} − cos 2ϕ\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\} + 2 sin 2ϕ ˜∂_{3}f ˜∂_{4}f,

(4.15)  c = −\frac{1}{8}(τ^{∗} − τ) = −\frac{1}{4f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} − \frac{1}{4}\{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\},

(4.16)  \begin{matrix}\frac{1}{f^{3}} \{˜∂_{1}^{2}f + ˜∂_{2}^{2}f\} − f\{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} \\ = \frac{1}{f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} − \{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\}, \end{matrix}

(4.17)

cos ϕ{˜∂_{1} ˜∂_{3}f + ˜∂_{2} ˜∂_{4}f} − sin ϕ{˜∂_{1} ˜∂_{4}f − ˜∂_{2} ˜∂_{3}f}

= \frac{2 cos ϕ}{f} ˜∂_{2}f ˜∂_{4}f + \frac{2 sin ϕ}{f} ˜∂_{2}f ˜∂_{3}f,

(4.18)

cos ϕ{˜∂_{1} ˜∂_{3}f + ˜∂_{2} ˜∂_{4}f} − sin ϕ{˜∂_{1} ˜∂_{4}f − ˜∂_{2} ˜∂_{3}f}

= \frac{2 cos ϕ}{f} ˜∂_{1}f ˜∂_{3}f − \frac{2 sin ϕ}{f} ˜∂_{1}f ˜∂_{4}f,

(4.19)

\frac{˜∂_{1} ˜∂_{2}f}{f^{3}} − f cos 2ϕ˜∂_{3} ˜∂_{4}f − \frac{f sin 2ϕ}{2}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\}

= \frac{4˜∂_{1}f ˜∂_{2}f}{f^{4}} + 2 cos 2ϕ˜∂_{3}f ˜∂_{4}f + sin 2ϕ\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\},

(4.20)

cos ϕ{˜∂_{1} ˜∂_{4}f − ˜∂_{2} ˜∂_{3}f} + sin ϕ{˜∂_{1} ˜∂_{3}f + ˜∂_{2} ˜∂_{4}f}

= \frac{cos ϕ}{f}\{˜∂_{1}f ˜∂_{4}f − ˜∂_{2}f ˜∂_{3}f\} + \frac{sin ϕ}{f}\{˜∂_{1}f ˜∂_{3}f + ˜∂_{2}f ˜∂_{4}f\}.

To obtain solutions of the above equations, we first assume that f = f(ξ_{1}, ξ_{2}) is a function of two variables ξ_{1}, ξ_{2}. Then (4.14) ∼ (4.20) reduce to

(4.21)  ˜∂_{1}^{2}f − ˜∂_{2}^{2}f = \frac{4}{f} \{(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2}\},

(4.22)  c = −\frac{1}{8}(τ^{∗} − τ) = −\frac{1}{4f^{4}} \{(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\},

(4.23)  f(˜∂_{1}^{2}f + ˜∂_{2}^{2}f) = (˜∂_{1}f)^{2} + (˜∂_{2}f)^{2},

(4.24)  f ˜∂_{1} ˜∂_{2}f = 4˜∂_{1}f ˜∂_{2}f.

By the same way as [5], we see that f = K(ξ_{1}^{2} + ξ_{2}^{2})^{− \frac{1}{3}} satisfies (4.21), (4.23) and (4.24), where K is a constant.

Next, we assume that f = f(ξ_{3}, ξ_{4}) is a function of ξ_{3}, ξ_{4}. Then (4.14) ∼ (4.15) reduce to

(4.25)  f(˜∂_{3}^{2}f − ˜∂_{4}^{2}f) = −2{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}},

(4.26)  c = −\frac{1}{8}(τ^{∗} − τ) = −\frac{1}{4}\{(˜∂_{3}f)^{2} + (˜∂_{4}f)^{2}\},

(4.27)  f(˜∂_{3}^{2}f + ˜∂_{4}^{2}f) = (˜∂_{3}f)^{2} + (˜∂_{4}f)^{2},

(4.28)  f ˜∂_{3} ˜∂_{4}f = −2˜∂_{3}f ˜∂_{4}f.

Similarly, we can easily see that f = L(ξ_{3}^{2} + ξ_{4}^{2})^{\frac{1}{3}} satisfies (4.25),(4.27) and (4.28), where L is a constant.

Consequently, we obtain the following Theorem 4.4. Let u = α, v = β (α, β are constants) and

f = K\{(x_{1} + αx_{3} − βx_{4})^{2} + (x_{2} + βx_{3} + αx_{4})^{2}\}^{− \frac{1}{3}} ,

or

f = L(x_{3}^{2} + x_{4}^{2})^{\frac{1}{3}} ,

where K and L are non-zero constants. Then the almost Kähler manifold M(f, u, v, ϕ) is of pointwise constant holomorphic sectional curvature with

c = −\frac{1}{9K^{2}} \{(x_{1} + αx_{3} − βx_{4})^{2} + (x_{2} + βx_{3} + αx_{4})^{2}\}^{− \frac{1}{3}}

or

c = −\frac{1}{9}L^{2}(x_{3}^{2} + x_{4}^{2})^{− \frac{1}{3}} ,

respectively.

We note that the pointwise constant c in the above theorem is negative contrary to the ones in Theorems 4.1 and 4.2.

5 Weakly ∗-Einstein examples

In this section, we provide weakly ∗-Einstein almost Kähler 4-manifolds. In §4, we have already shown that the Ricci flat almost Kähler manifolds M(f, u, v, ϕ) in Theorems 3.1 and 3.2 are weakly ∗-Einstein.

Here, we consider the almost Kähler manifolds M(f, α, β, ϕ) for any constants α, β. In this case, by using the coordinates (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) of (2.10), the components of the Ricci ∗-tensor with respect to the unitary frame {e_{i}} in (2.6) is given by

ρ_{11}^{∗} = ρ_{22}^{∗} = \frac{˜∂_{1}^{2}f}{2f^{3}} − \frac{f}{4} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} + \frac{f cos 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\} − \frac{f sin 2ϕ}{2} ˜∂_{3} ˜∂_{4}f

+\frac{1}{2f^{4}} \{−3(˜∂_{1}f)^{2} + (˜∂_{2}f)^{2}\} + \frac{cos 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\} − sin 2ϕ ˜∂_{3}f ˜∂_{4}f,

ρ_{33}^{∗} = ρ_{44}^{∗} = \frac{˜∂_{2}^{2}f}{2f^{3}} − \frac{f}{4} \{˜∂_{3}^{2}f + ˜∂_{4}^{2}f\} − \frac{f cos 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\} + \frac{f sin 2ϕ}{2} ˜∂_{3} ˜∂_{4}f

+\frac{1}{2f^{4}} \{(˜∂_{1}f)^{2} − 3(˜∂_{2}f)^{2}\} − \frac{cos 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\} + sin 2ϕ ˜∂_{3}f ˜∂_{4}f,

ρ_{12}^{∗} = ρ_{21}^{∗} = ρ_{34}^{∗} = ρ_{43}^{∗} = 0,

ρ_{13}^{∗} = ρ_{42}^{∗} = ρ_{24}^{∗} = ρ_{31}^{∗} = \frac{˜∂_{1} ˜∂_{2}f}{2f^{3}} − 2\frac{˜∂_{1}f ˜∂_{2}f}{f^{4}} − \frac{f cos 2ϕ}{2} ˜∂_{3} ˜∂_{4}f − cos 2ϕ ˜∂_{3}f ˜∂_{4}f

−\frac{f sin 2ϕ}{4}\{˜∂_{3}^{2}f − ˜∂_{4}^{2}f\} − \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\},

ρ_{14}^{∗} = −ρ_{32}^{∗} = \frac{cos ϕ}{2f}(˜∂_{1} ˜∂_{3}f − ˜∂_{2} ˜∂_{4}f) − \frac{sin ϕ}{2f}(˜∂_{2} ˜∂_{3}f + ˜∂_{1} ˜∂_{4}f),

ρ_{23}^{∗} = −ρ_{41}^{∗} = \frac{cos ϕ}{2f}(˜∂_{1} ˜∂_{3}f − ˜∂_{2} ˜∂_{4}f) − \frac{cos ϕ}{f^{2}}(˜∂_{1}f ˜∂_{3}f − ˜∂_{2}f ˜∂_{4}f)

−\frac{sin ϕ}{2f}(˜∂_{2} ˜∂_{3}f + ˜∂_{1} ˜∂_{4}f) + \frac{sin ϕ}{f^{2}}(˜∂_{2}f ˜∂_{3}f + ˜∂_{1}f ˜∂_{4}f),

By (5.1), we have the following Lemma 5.1. The almost Kähler 4-manifold M(f, α, β, ϕ) is weakly ∗-Einstein if and only if

(5.2)

\frac{1}{2f^{3}} (˜∂_{1}^{2}f − ˜∂_{2}^{2}f) + \frac{f cos 2ϕ}{2}(˜∂_{3}^{2}f − ˜∂_{4}^{2}f) − f sin 2ϕ ˜∂_{3} ˜∂_{4}f

= \frac{2}{f^{4}} \{(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2}\} − cos 2ϕ\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\} + 2 sin 2ϕ ˜∂_{3}f ˜∂_{4}f,

(5.3)

\frac{˜∂_{1} ˜∂_{2}f}{2f^{3}} − \frac{f cos 2ϕ}{2} ˜∂_{3} ˜∂_{4}f − \frac{f sin 2ϕ}{4}(˜∂_{3}^{2}f − ˜∂_{4}^{2}f)

= 2\frac{˜∂_{1}f ˜∂_{2}f}{f^{4}} + cos 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\},

(5.4)  cos ϕ(˜∂_{1} ˜∂_{3}f − ˜∂_{2} ˜∂_{4}f) − sin ϕ(˜∂_{2} ˜∂_{3}f + ˜∂_{1} ˜∂_{4}f) = 0,

(5.5) cos ϕ(˜∂_{1} ˜∂_{3}f − ˜∂_{2} ˜∂_{4}f) − sin ϕ(˜∂_{2} ˜∂_{3}f + ˜∂_{1} ˜∂_{4}f)

= \frac{2}{f} \{cos ϕ(˜∂_{1}f ˜∂_{3}f − ˜∂_{2}f ˜∂_{4}f) − sin ϕ(˜∂_{2}f ˜∂_{3}f + ˜∂_{1}f ˜∂_{4}f)\}.

If f is a function of ξ_{1}, ξ_{2}, then (5.2) ∼ (5.5) reduce to

(5.6)  f(˜∂_{1}^{2}f − ˜∂_{2}^{2}f) = 4{(˜∂_{1}f)^{2} − (˜∂_{2}f)^{2}},

(5.7)  f ˜∂_{1} ˜∂_{2}f = 4˜∂_{1}f ˜∂_{2}f.

Since (5.6), (5.7) are same as (4.21), (4.24), we see that

(5.8)  f = K(ξ_{1}^{2} + ξ_{2}^{2})^{− \frac{1}{3}}

is a solution.

If f is a function of ξ_{3}, ξ_{4}, then (5.2) ∼ (5.5) reduce to (5.9)

\frac{f cos 2ϕ}{2}(˜∂_{3}^{2}f − ˜∂_{4}^{2}f) − f sin 2ϕ ˜∂_{3} ˜∂_{4}f

= − cos 2ϕ{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}} + 2 sin 2ϕ ˜∂_{3}f ˜∂_{4}f,

(5.10)

−\frac{f cos 2ϕ}{2} ˜∂_{3} ˜∂_{4}f − \frac{f sin 2ϕ}{4}(˜∂_{3}^{2}f − ˜∂_{4}^{2}f)

= cos 2ϕ ˜∂_{3}f ˜∂_{4}f + \frac{sin 2ϕ}{2}\{(˜∂_{3}f)^{2} − (˜∂_{4}f)^{2}\}.

Similarly, we see that

(5.11)  f = L(ξ_{3}^{2} + ξ_{4}^{2})^{\frac{1}{3}}

is a solution of (5.9), (5.10). Thus we have the following Theorem 5.2. Let

f = K\{(x_{1} + αx_{3} − βx_{4})^{2} + (x_{2} + βx_{3} + αx_{4})^{2}\}^{− \frac{1}{3}} ,

or

f = L(x_{3}^{2} + x_{4}^{2})^{\frac{1}{3}} ,

where K and L are non-zero constants. Then the almost Kähler manifold M(f, α, β, ϕ) is weakly ∗-Einstein. The ∗-scalar curvature is given respectively by

τ^{∗} = −\frac{4}{9K^{2}} \{(x_{1} + αx_{3} − βx_{4})^{2} + (x_{2} + βx_{3} + αx_{4})^{2}\}^{− \frac{1}{3}} ,

or

τ^{∗} = −\frac{4}{9}L^{2}(x_{3}^{2} + x_{4}^{2})^{− \frac{1}{3}} .

Now, we give another example. It is easy to see that

(5.12)  f = (a_{1}ξ_{1} + a_{2}ξ_{2} + b_{1})^{− \frac{1}{3}}

is also a solution of (5.6), (5.7), and

(5.13)  f = (a_{3}ξ_{3} + a_{4}ξ_{4} + b_{2})^{\frac{1}{3}}

is a solution of (5.9), (5.10).

From (5.12) and (5.13),

(5.14)  f = (a_{1}ξ_{1} + a_{2}ξ_{2} + b_{1})^{− \frac{1}{3}} (a_{3}ξ_{3} + a_{4}ξ_{4} + b_{2})^{\frac{1}{3}}

satisfies (5.2), (5.3). Moreover, if

(5.15)  (a_{1}a_{3} − a_{2}a_{4}) cos ϕ − (a_{2}a_{3} + a_{1}a_{4}) sin ϕ = 0,

then f in (5.14) also satisfies (5.4) and (5.5). Note that the functions f in (5.12), (5.13) are given as special cases of (5.14) satisfying (5.15).

Consequently, we obtain the following Theorem 5.3. Let

f = \{\frac{a_{3}x_{3} + a_{4}x_{4} + b_{2}}{a_{1}(x_{1} + αx_{3} − βx_{4}) + a_{2}(x_{2} + βx_{3} + αx_{4}) + b_{1}} \}^{\frac{1}{3}} ,

where a_{i} (i = 1, 2, 3, 4) and b_{j} (j = 1, 2) are constants satisfying (5.15). Then the almost Kähler manifold M(f, α, β, ϕ) is weakly ∗-Einstein, and

τ^{∗} = \frac{2(a_{1}^{2} + a_{2}^{2})}{9}\{a_{1}(x_{1} + αx_{3} − βx_{4}) + a_{2}(x_{2} + βx_{3} + αx_{4}) + b_{1}\}^{− \frac{4}{3}}
× (a_{3}x_{3} + a_{4}x_{4} + b_{2})^{− \frac{2}{3}}
+ \frac{2(a_{3}^{2} + a_{4}^{2})}{9}\{a_{1}(x_{1} + αx_{3} − βx_{4}) + a_{2}(x_{2} + βx_{3} + αx_{4}) + b_{1}\}^{− \frac{2}{3}}
× (a_{3}x_{3} + a_{4}x_{4} + b_{2})^{− \frac{4}{3}} .

It should be remarked that the almost Kähler manifold M(f, α, β, ϕ) in Theorem 5.2 is weakly∗-Einstein for all ϕ. On the contrary M(f, α, β, ϕ) in Theorem 5.3 is weakly∗-Einstein for particular values of ϕ and a_{i} (i = 1, 2, 3, 4) satisfying (5.15).

Finally, we note that the Ricci tensors of these examples in Theorem 5.3 are Janti-invariant, i.e., ρ(JX, JY) = −ρ(X, Y), and hence τ = 0.

References

[1] J.T.Cho and K.Sekigawa, Four-dimensional almost Kähler manifolds of pointwise constant holomorphic sectional curvature, Arab. J. Math. Sci. 2(1996), 39-53.

[2] S.I. Goldberg, Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc. 21(1969), 96-100.

[3] P.Nurowski and M.Przanowski, A four-dimensional examples of a Ricci-flat metric admitting almost-Kähler non-Kähler structures, Class, Quantum Grav. 16(1999), L9-L13.

[4] T.Sato, Some remarks on almost Kähler 4-manifolds of pointwise constant holomorphic sectional curvature, Kodai Math. J. 22(1999), 286-303.

[5] T.Sato, An example of an almost Kähler manifold with pointwise constant holomorphic sectional curvature, to appear in Tokyo J. Math.

Faculty of Engineering Kanazawa University Kanazawa, Japan E-mail:tsato@t.kanazawa-u.ac.jp