It is well known that every K¨ahler metric ds

On a certain change of K¨ ahler metrics Tsuyoshi Yamazaki and Yoshiyuki Watanabe

1 ． Introduction

It is well known that every K¨ahler metric ds

= 2Σg

dz

d¯ z

is locally expressible in the form

g

= ∂

ρ

∂z

∂ z ¯

,

with respect to local complex coordinates (z

), α = 1, · · · , n, where ρ = ρ(z, z) is a real valued function, called (K¨ahler) potential. ¯

The purpose of this paper is to initiate a study on the K¨ahler metric ˜ g

, changing a K¨ahler metric g

as follows:

˜

g

= f(ρ)g

+ f

(ρ)ρ

ρ

, (1.1)

2000Mathematics Subject Classification. 53B35, 53C55.

Key words and phrases. K¨ahler metrics, a change of K¨ahler metrics, Ricci tensors, constant scalar curvature.

where ρ

= ∂ρ

∂z

, f = f (ρ) and f

(ρ) = df

dρ satisfy f > 0, f + lf

> 0, (1.2)

denoting l = g

ρ

ρ

= ρ

ρ

and ρ

= g

ρ

. In fact, it can be easily checked that ˜ g

satisfy the K¨ahler condition

∂ g ˜

∂¯ z

= ∂˜ g

∂¯ z

. (1.3)

In particular, if the g

is a (locally) flat K¨ahler metric, that is, g

=

∂

t

∂z

∂ z ¯

= δ

, where t = Σz

z ¯

, then we have

˜

g

= ∂

f(t)

∂z

∂¯ z

= ˙ f (t)δ

+ ¨ f (t)¯ z

z

, (1.4)

denoting ˙ f(t) = df

dt and ¨ f(t) = d

f

dt

. ( Tachibana and Liu [2], Watanabe [3], [4] and Watanabe and Mori [5]). Now, let g

be a Fubini metric (cf.

Tachibana [1]) and we study the K¨ahler metric ˜ g

, changed by (1.1). In particular, we shall generalize some results in Watanabe [3]. The main theorem is the following

Theorem 1. Let D

be C

or the ball B

= {(z

) ∈ C

| Σz

z ¯

< r

} around the origin of C

, whose radius is given by r = 2

p

|k| (k 6= 0). Let (D

, g

) be a Fubini space. If the scalar curvature R ˜ of the K¨ahler metric

˜

g

, changed by (1.1), satisfies the condition R ˜ = constant, (1.5)

then (D

, g ˜

) is of constant holomorphic curvature or flat.

Preliminary facts will be given in Section 2. In Section 3, we shall com-

pute the curvature tensors of the K¨ahler metric, changed by (1.1). In

Sections 4, 5, and 6, we shall study the K¨ahler metric ˜ g

, changing a Fu-

bini metric in the form (1.1). In Section 5, we shall consider such a K¨ahler

manifold, satisfying the condition:

∇ ˜

∇ ˜

R ˜

− ∇ ˜

∇ ˜

R ˜

= 0, (1.6)

where ˜ R

is the Ricci tensor of ˜ g

. The last section is devoted to prove the main theorem.

2．Preliminaries

We agree to adopt the summation convention and the following range of indices throughout the paper:

1 ≤ i, j, k, · · · ≤ 2n, 1 ≤ α, β, γ, · · · ≤ n.

Consider a complex n dimensional K¨ahler manifold with metric ds

= Σg

dz

dz

,

(2.1)

where (z

) are local complex coordinates and ¯ z

= z

(=conjugate of z

).

g

satisfy the conditions:

g

= g

= 0, (2.2)

and (1.3). Then (2.1) becomes

ds

= 2Σg

dz

d¯ z

. (2.3)

g

satisfy the corresponding equations to (2.2). The Christoffel symbols Γ

vanish except

Γ

= g

∂g

∂z

, (2.4)

and their conjugates. As to the curvature tensor R

, only the components of the form R

and their conjugates are different from zero, and it holds that

R

= ∂Γ

∂¯ z

,

(2.5)

from which the Ricci tensor R

is given by R

= R

= − ∂Γ

∂ z ¯

, (2.6)

R

= R

= 0.

(2.7)

The scalar curvature R = g

R

is 2g

R

.

A K¨ahler manifold is called a space of constant holomorphic curvature k if its curvature tensor satisfies

R

= k(g

g

+ g

g

), denoting k = R

2n(n + 1) .

3．Curvature tensors

For a K¨ahler metric g

= ∂

ρ

∂z

∂ z ¯

, we denote the curvature tensor, the Ricci tensor and the scalar curvature by R

, R

and R, respectively.

We are now going to compute the curvature tensor ˜ R

, the Ricci tensor R ˜

and the scalar curvature ˜ R of the K¨ahler metric ˜ g

, given by

˜

g

= f (ρ)g

+ f

ρ

ρ

, (3.1)

where the dash means differentiation with respect to ρ. As the metric is positive definite, the function should satisfy (1.2). Then ˜ g

are given by

˜ g

= 1

f

³

g

− f

f + lf

ρ

ρ

. (3.2)

From (2.4), we have

Γ ˜

= ˜ g

∂˜ g

∂z

= Γ

+ f

f (ρ

δ

+ ρ

δ

) + f f

− 2f

f (f + lf

) ρ

ρ

ρ

+ f

f + lf

ρ

∇

ρ

,

denoting ∇

ρ

= ∂ρ

∂z

− Γ

ρ

. After some computations, we have

(3.3) ˜ R

= ∂ Γ ˜

∂ z ¯

= R

+ f f

− f

f

ρ

(ρ

δ

+ ρ

δ

) + f

f (g

δ

+ g

δ

) + f (f + lf

)(f f

− 2f

)

− (f f

− 2f

)(3f f

+ lf f

+ lf

)

f

(f + lf

)

ρ

ρ

ρ

ρ

+ f f

− 2f

f (f + lf

) ρ

(ρ

g

+ ρ

g

) − f

(f f

− 2f

)

f (f + lf

)

ρ

ρ

ρ

ρ

∇

ρ

+ f f

− 2f

f (f + lf

) ρ

ρ

g

∇

ρ

+ f f

− 2f

(f + lf

)

ρ

ρ

∇

ρ

− f

(f + lf

)

(ρ

∇

ρ

)ρ

∇

ρ

+ f

f + lf

(∇

ρ

)g

∇

ρ

− f

f + lf

ρ

ρ

R

. Next, from (2.6) we have

(3.4) ˜ R

= − R ˜

= R

− (n + 1)f

f g

− f f

− 2f

f (f + lf

) lg

− (n + 1)(f f

− f

) f

ρ

ρ

− f (f + lf

)(f f

− 2f

)

− (f f

− 2f

)(3f f

+ lf f

+ lf

) f

(f + lf

)

lρ

ρ

− f f

− 2f

f (f + lf

) ρ

ρ

− f f

− 2f

(f + lf

)

ρ

ρ

∇

ρ

− f f

− 2f

(f + lf

)

ρ

ρ

∇

ρ

+ f

(f + lf

)

(ρ

∇

ρ

)ρ

∇

ρ

− f

f + lf

(∇

ρ

)g

∇

ρ

+ f

f + lf

ρ

ρ

R

. Finally, the scalar curvature ˜ R is given by

(3.5) ˜ R = 2˜ g

R ˜

= 1

f R − 2(n + 1)(f f

− f

)

f

l

− 2f (f + lf

)(f f

− 2f

)

− 2(f f

− 2f

)(3f f

+ lf f

+ lf

)

f

(f + lf

)

l

− 2(f f

− 2f

)

f

(f + lf

) l − 2n(n + 1)f

f

− 2n(f f

− 2f

)

f

(f + lf

) l − 2(f f

− 2f

)

f(f + lf

)

(ρ

ρ

∇

ρ

+ ρ

ρ

∇

ρ

)

− 2f

f (f + lf

) g

g

(∇

ρ

)∇

ρ

+ 2f

f (f + lf

)

g

(ρ

∇

ρ

)ρ

∇

ρ

− f

f(f + lf

) lR + 2(n + 1)f

(f f

− f

)

f

(f + lf

) l

+ 2(n + 1)f

f

(f + lf

) l

+ 2f f

(f + lf

)(f f

− 2f

)

− 2f

(f f

− 2f

)(3f f

+ lf f

+ lf

)

f

(f + lf

)

l

+ 2f

(f f

− 2f

)

f (f + lf

)

lρ

ρ

∇

ρ

+ 4f

(f f

− 2f

) f

(f + lf

)

l

+ 2f

(f f

− 2f

)

f (f + lf

)

lρ

ρ

∇

ρ

− 2f

f(f + lf

)

(ρ

ρ

∇

ρ

)ρ

ρ

∇

ρ

+ 2f

f (f + lf

)

g

(ρ

∇

ρ

)ρ

∇

ρ

− 2f

f (f + lf

)

ρ

ρ

ρ

ρ

R

.

4 ． A K¨ ahler manifold of constant holomorphic curvature A K¨ahler metric with non zero constant holomorphic curvature k defined in the complex number n-space C

is called a Fubini metric. The potential function is given by

ρ = 2

k log

1 + k 4 t

, denoting t = Σz

z ¯

. Then putting Θ = 1 + k

4 t, we have g

= 1

2Θ

³

Θδ

− k 4 z ¯

z

. (4.1)

First from (4.1), we have

g

= 1

2e

δ

− k 2 ρ

ρ

, and

ρ

= ∂ρ

∂z

= z ¯

2Θ , (4.2)

from which we obtain

g

= 2Θ

δ

+ k 4 z

z ¯

, (4.3)

ρ

= g

ρ

= z

Θ, (4.4)

e

= l

= 1 + k 2 l > 0, (4.5)

and

l = ρ

ρ

= t 2 . Taking account of (2.4), we have

Γ

= − k

2 (δ

ρ

+ δ

ρ

).

(4.6)

From (4.1), (4.2) and (4.6), we have

∇

ρ

= k

2 ρ

ρ

.

(4.7)

Taking account of (2.5), we have R

= − k

2 (δ

g

+ δ

g

).

(4.8)

Moreover taking account of (2.6), we have R

= (n + 1)k

2 g

. (4.9)

Finally, taking account of (4.9), we have

R = n(n + 1)k.

In the sequel, we consider the K¨ahler metric ˜ g

, deforming g

in (1.1).

Then we compute the following

˜

g

= f

2e

δ

+

f

− k

2 f

ρ

ρ

, (4.10)

Γ ˜

=

f

f − k

2

´

(ρ

δ

+ ρ

δ

) + 2(f f

− 2f

+

f f

)

f(f + lf

) e

ρ

ρ

ρ

, (4.11)

(4.12) ˜ R

=

f

f − k

2

´

(g

δ

+ g

δ

) + f f

− f

f

(δ

ρ

ρ

+ δ

ρ

ρ

)

+ F

f (f + lf

) (ρ

ρ

g

+ ρ

ρ

g

) +

·

1

f

(f + lf

)

F

f (f +lf

)−F (3f f

+lf

+lf f

)

+

k2

F (f + lf

)

¸

ρ

ρ

ρ

ρ

, where we denote F by

F = f f

− 2f

+ k 2 f f

. (4.13)

Moreover, we obtain

R ˜

=

½

(n + 1)

k 2 − f

f

´

− F l f(f + lf

)

¾

g

−

·

(n + 1)(f f

− f

)

f

+ F

f(f + lf

) + l f

(f + lf

)

n

F

f (f + lf

)

−F (3f f

+ lf

+ lf f

)

+

k2

lF (f + lf

)

¸

ρ

ρ

, and

R ˜

= µg

+ µ

ρ

ρ

, (4.14)

where µ is a function given by µ = (n + 1)

k

2 − f

f

´

− F l f (f + lf

) . (4.15)

Finally, the scalar curvature ˜ R is given by R ˜ = 2nµ

f + 2l(f µ

− f

µ) f (f + lf

) . (4.16)

5．A K¨ ahler metric satisfying a certain condition

Let D

be C

or the ball B

= {(z

) ∈ C

| Σz

z ¯

< r

} around the origin O of C

, whose radius is given by r = 2

p

|k| (k 6= 0). Let (D

, g

) be a Fubini space. In this section, let us consider the K¨ahler metric ˜ g, satifying the condition (1.6). Then by the Ricci’s formula, we have

R ˜

R ˜

+ ˜ R

R ˜

= 0, (5.1)

from which

R ˜

R ˜

+ ˜ R

R ˜

+ ˜ R

R ˜

+ ˜ R

R ˜

= 0.

Thus (5.1) is equivalent to

R ˜

R ˜

+ ˜ R

R ˜

= 0 (conj.), (5.2)

by virtue of (4.12) and (4.14).

Now substituting (4.8) and (4.10) into the left hand side of (5.2), we can

see that it reduces to the following.

(5.2)

½

µ

f

f − k 2

´

+ lµ

F

f (f + lf

) + µF

f (f + lf

) − µ(f f

− f

) f

¾

(ρ

ρ

g

−ρ

ρ

g

) = 0.

Now we assume that n ≥ 2. Then we have

½

µ

f

f − k 2

´

+ lµ

F

f (f + lf

) + µF

f (f + lf

) − µ(f f

− f

) f

¾

= 0, (5.3)

taking account of f (ρ) ∈ C

(R). (5.3) gives

µ

f

f

− k

2 f

+ lµ

f F

f + lf

− lµf

F

f + lf

− µf

f

− k

2 f

= 0.

Thus we obtain

½

f

− k

2 f + lF f + lf

¾

(µ

f − µf

) = 0.

(5.4)

In the followings, we consider two cases, that is, Case where f

− k 2 f + lF

f + lf

= 0 and Case where µ

f − µf

= 0.

Case : Assume that f

− k

2 f + lF

f + lf

= 0 in an open subdomain ∆

of D

. Taking account of (4.13),

we have

lf

f + f

f − lf

f

− k

2 = 0.

(5.5)

Putting g = f

f (= (logf )

) in (5.5), we have lg

+ g − k

2 = 0.

(5.6)

Putting h = lg in (5.6), we have h

− k

2 h − k 2 = 0.

(5.7)

The general solution of (5.7) is given by

h = ce

− 1,

where c is an integral constant. Therefore the general solution of (5.6) is given by

lg = ce

− 1,

where c is an integral constant. Therefore the general solution of (5.5) is given by

f = ae

µ

1 − e

k

¶_b−1

, (5.8)

where a, b are integral constants.

If b 6= 1, then f and f + lf

satisfy

f (0) = 0, (f + lf

)(0) = 0,

because of ρ(0) = 0. Thus in this case the solution does not satisfy the condition (1.2) where ∆

3 O.

Next, suppose that b = 1. Then from (1.2), we see that a > 0 where

∆

3 O, that is,

(5.8)

f = ae

(a > 0), where ∆

3 O.

Case : Suppose that

µ

f − µf

= 0 (5.9)

holds in a subdomain ∆

of D

. First by (4.10) and (4.14) we see that the potential function satisfying (5.9) gives an Einstein metric. By (4.15), we have

(5.10) F

f (f + lf

)

½³

n + 2 + k

2 l

f

+ nlf f

¾

+ l

(f + lf

)

½

F

f (f + lf

) − F

3f f

+ lf

+ lf f

+ k

2 lf f

= 0.

Setting

σ = F

f (f + lf

) , (5.11)

we have

lf σ

+

n + 2 + k

2 l

f + nlf

¾

σ = 0.

(5.12)

If ∆

contains the origin O, putting l(0) = 0 in (5.12) we have σ(0) = 0,

(5.13)

because of f(0) > 0.

If l > 0, then multiplying (5.12) by l

f

, we have

l

f

σ

+

n + 2 + k

2 l

l

f

σ + nl

f

f

σ = 0, which implies

(l

f

σ)

− (n + 1)k

2 (l

f

σ) = 0.

Thus the general solution is given by

l

f

σ = ce

, (5.14)

where c is an integral constant. Since the function f satisfies (5.12), by the argument of continuity of the left hand side of (5.14), we can conclude that c = 0 if ∆

contains the origin O. Thus we have

σ(ρ) = 0, (5.15)

together with (5.14) if ∆

3 O. Since

µ

f

−

f f

¶₀

= F

f

, we have f

− k

2 f − Cf

= 0 (C = constant) (5.16)

together with (5.11). (5.16) has the general solution f

= 1

ce

+ d ,

(5.17)

where constants c(> 0) and d satisfy the condition c + d > 0, that is, when O ∈ ∆

,

f

= 1

ce

+ d (c > 0, c + d > 0).

(5.18)

Note that in the case of (5.18) the corresponding K¨ahler manifold (∆

, g) ˜ is of constant holomorphic curvature dk

2 .

Finally, the function f given by (5.8)

does not satisfy (5.17) with d 6=

0 and can be smoothly connected only the solution of (5.18) for d = 0.

Conversely the solution of (5.18) for d 6= 0 or a 6= 1

c does not satisfy (5.8)

. Thus we obtain the following

Theorem 2. Let D

be C

or the ball B

= {(z

) ∈ C

| Σz

z ¯

< r

} around the origin of C

, whose radius is given by r = 2

p

|k| (k 6= 0). Let (D

, g

) be a Fubini space. If the K¨ahler metric ˜ g

, changed by (1.1) satisfies the condition (1.6), then it is of constant holomorphic curvature or flat.

6．Proof of the main theorem

By assumption ˜ R is constant in (4.16). Multiplying (4.16) by f (f +lf

), we have

R ˜

2 f (f + lf

) = lµ

f + nf µ + (n − 1)lf

µ.

(6.1)

Putting l = 0 in (6.1), we have R ˜

2 f (0) − nµ(0) = 0, (6.2)

because of f (0) > 0. For l > 0, we multiply (6.1) by l

f

. Then we have

R ˜

2 (f

l

+ l

f

f

) = l

f

µ

+ nl

f

µ + (n − 1)l

f

f

µ.

By

³

R ˜

2n l

f

= R ˜

2 (f

l

+ l

f

f

) + nk 2

³

R ˜ 2n l

f

and

(l

f

µ)

= l

f

µ

+ nl

f

µ + (n − 1)l

f

f

µ + nk

2 (l

f

µ), we obtain

³

R ˜

2n l

f

− l

f

µ

= nk 2

³

R ˜

2n l

f

− l

f

µ

, from which

R ˜

2n l

f

= l

f

µ + Ce

, where C is an integral constant. Then we have

l

f

R ˜

2n f − µ

= Ce

. (6.3)

But we can see that C = 0, taking limit of the left hand side of (6.3) of l as l tends to 0. Therefore we have

R ˜

2n f

(f + lf

) + (n + 1)

f

f − k

2

´

f (f + lf

) (6.4)

+l

f f

− 2f

+ k

2 f f

= 0.

Putting l = 0 in (6.4), we have f

(0) − k

2 f (0) + R ˜

2n(n + 1) f (0)

= 0.

(6.5)

Multiplying (6.4) by f

l

, we have R ˜

2n (f

l

+ f

l

f

) + (n + 1)f

l

f

+ l

f

f

+ (n − 1)f

l

f

− (n + 1)k

2 f

l

− nk

2 f

l

f

= 0,

from which

½

l

f

f

− k

2 f + R ˜

2n(n + 1) f

0

= (n + 1)k

2 l

f

f

− k

2 f + R ˜

2n(n + 1) f

, that is,

l

f

f

− k

2 f + R ˜

2n(n + 1) f

= ˜ Ce

, (6.6)

where ˜ C is an integral constant. But we have C ˜ = 0

taking limit of the left hand side of (6.6) of l as l tends to 0. Thus we have f

− k

2 f + kf

= f

− k

2 f + R ˜

2n(n + 1) f

= 0, (6.7)

together with (6.5).

Thus the general solution of (6.7) is given by

f = 1

ae

+ b ,

where constants a(> 0) and b satisfy the condition a + b > 0. Now, if b = 0, then the corresponding K¨ahler metric is flat. Next, if b 6= 0, the corresponding K¨ahler metric is of constant holomorphic curvature and the holomorphic curvature is R ˜

2n(n + 1) (cf.(4.12)) because of an elementary calculation. Thus as Theorem 2, this proves the main theorem.

References

[1] S. Tachibana, On the characteristic function of spaces of constant holomorphic curvature, Colloq. Math., 26(1972), 149-155.

[2] S. Tachibana and R.C. Liu, Notes on K¨ahlerian metrics with vanishing

Bochner curvature tensor, K¯odai Math. Sem. Rep., 22(1970), 313-321.

[3] Y. Watanabe, K¨ahlerian metrics given by certain smooth potential functions, Kodai Math J., 5(1982), 329-338.

[4] Y. Watanabe, Unitary-symmetric K¨ahlerian manifolds and pointed Blaschke manifolds, Tsukuba J. Math., 12(1988), 129-148.

[5] Y. Watanabe and H. Mori, Geometry on unitary-symmetric K¨ahler manifold, Math. J. Toyama Univ., 16(1993), 135-193.

Department of Mathematics Faculty of Science

Toyama University Toyama, 930-8555, Japan

(Received May 7, 2003)