CONFORMALLY FLAT 3-MANIFOLDS WITH

CONSTANT SCALAR CURVATURE II

Qing-Ming Cheng*, Susumu Ishikawa** & Katsuhiro Shiohama**

Abstract. This paper is the continuation of [1], in which we studied complete locally conformally flat Riemannian manifolds with constant scalar curvature. We shall prove that there exist no complete locally conformally flat 3-dimensional Riemannian manifolds with constant negative scalar curvature possessing Ricci curvature tensor of constant squared norm S satisfying \frac{r^{2}}{3} + ( \frac{7}{113} )^{2} \frac{2r^{2}}{3} < S < \frac{r^{2}}{3} + ( \frac{35}{137} )^{2} \frac{r^{2}}{6} .

1. Introduction

A well known result in Riemannian geometry states that every compact ndimensional Riemannian manifold can be deformed to a Riemannian manifold with constant scalar curvature by a conformal transformation (see [3], [5], [6], etc.). This result was obtained through the application of the so-called Yamabe equation. The Yamabe equation has a deep connection with locally conformally flat Riemannian manifolds (see [4]). Let us begin by recalling that M^{n} is by definition locally conformally flat if and only if every point on M^{n} admits a coordinate neighborhood with coordinates u^{1}, ⋯, u^{n} in which the Riemannian metric can be expressed as

ds^{2} = ∑_{i=1}^{n} λ^{2}(u)(du^{i})^{2},

where λ(u) is a positive function defined in the coordinate neighborhood. There are many examples of locally conformally flat Riemannian manifolds (see [4], etc.).

The study of global properties on locally conformally flat Riemannian manifolds was initiated by Kuiper, who studied their fundamental groups. Recently, Schoen and Yau also investigated the global properties on the locally conformally flat Riemannian manifolds by using the Yamabe equation. In the previous paper [1], we have classified in dimension 3 the class of such Riemannian manifolds with a certain additional properties. Let r and S be respectively the scalar curvature and the squared norm of Ricci curvature tensor of a complete locally conformally flat n-dimensional Riemannian manifold M^{n}. Let M^{n}(c) be n-dimensional space form of constant sectional curvature c. The following results have been established in [1]:

Key words and phrases. conformally flat manifold, constant scalar curvature and Ricci curvature tensor.

*Research partially Supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture and by Grant-in-Aid for Scientific Research from Josai University

**Research partially Supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture

1991 Mathematics Subject Classification 53C20

Let M^{3} be a complete locally conformally flat 3-dimensional Riemannian manifold of both r and S being constant. If r ≥ 0, then

(1) M^{3} is isometric to a space form of constant sectional curvature

or

(2) M^{3} is isometric to a Riemannian product M^{2}(c) × N^{1} where N^{1} = R^{1} or S^{1} for some c ≥ 0.

If r < 0, then

(3) M is isometric to a space form of constant negative curvature

or

(4) \frac{r^{2}}{3} < S ≤ \frac{r^{2}}{2} .

We have also proved in [1] that there exists no compact locally conformally flat Riemannian 3-manifold M^{3} with the following two conditions:

(a) r is a negative constant

(b) S is constant and all eigenvalues of the Ricci curvature tensor at each point

of M^{3} are distinct. Typical examples are stated as follows.

Example 1. H^{3}(r/6) (r < 0) is a complete locally conformally flat 3-dimensional

Riemannian manifold of S = \frac{r^{2}}{3} .

Example 2. H^{2}(c)×R (c < 0) is a complete locally conformally flat 3-dimensional

Riemannian manifold of S = \frac{r^{2}}{2} .

However, it has not yet been known if there exists the complete locally conformally flat 3-dimensional Riemannian manifold on which r is a negative constant

and S is constant such that \frac{r^{2}}{3} < S < \frac{r^{2}}{2} . The following conjecture has been proposed in [1]

Conjecture. Let M^{3} be a complete locally conformally flat 3-dimensional Riemannian manifold with the constant negative scalar curvature and the Ricci curvature tensor of constant squared norm. Then, M^{3} is isometric to either a space form of constant negative curvature or the Riemannian product H^{2}(c) × N^{1} (c < 0).

The purpose of present article is to study complete locally conformally flat 3dimensional Riemannian manifolds with constant negative scalar curvature r possessing Ricci curvature tensor of constant squared norm S. Our Main Theorem strongly supports the validity of the above conjecture.

Main Theorem. There exist no complete locally conformally flat 3-dimensional Riemannian manifolds with negative constant scalar curvature r and the Ricci cur-

vature tensor with constant squared norm S satisfying \frac{r^{2}}{3} + (\frac{7}{113})^{2} \frac{2r^{2}}{3} < S <

\frac{r^{2}}{3} + (\frac{35}{137})^{2} \frac{r^{2}}{6} .

Acknowledgement. Authors would like to express their gratitude to the referee for his valuable suggestion.

2. Preliminaries

Let M^{n} be an n-dimensional Riemannian manifold. We choose a local frame of orthonormal vector fields {e_{1}, ⋯, e_{n}} adapted to the Riemannian metric of M^{n} and the dual coframe {ω_{1}, ⋯, ω_{n}}. Then the connection forms {ω_{ij}} of M^{n} are characterized by the structure equations:

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

dω_{ij} = − ∑_{k} ω_{ik} ∧ ω_{kj} + \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l},

where R_{ijkl} are the components of curvature tensor of M^{n}.

Let C_{ijkl} denote the components of Weyl tensor of M^{n}, that is,

C_{ijkl} = R_{ijkl} − \frac{1}{n − 2}(R_{ik}δ_{jl} − R_{il}δ_{jk} + R_{jl}δ_{ik} − R_{jk}δ_{il})

+ \frac{r}{(n − 1)(n − 2)}(δ_{ik}δ_{jl} − δ_{il}δ_{jk}),

where R_{ij} and r are the components of Ricci curvature tensor and the scalar curvature of M^{n}, respectively. Let ∇ denote the covariant differentiation on M^{n}. By writing R_{ij,k} = ∇_{k}R_{ij} and R_{ij,kl} = ∇_{l}∇_{k}R_{ij}, etc., we have the Ricci formulas

R_{ij,kl} − R_{ij,lk} = ∑_{t} R_{tj}R_{tikl} + ∑_{t} R_{it}R_{tjkl},

R_{ij,klm} − R_{ij,kml} = ∑_{t} R_{tj,k}R_{tilm} + ∑_{t} R_{it,k}R_{tjlm} + ∑_{t} R_{ij,t}R_{tklm}.

The Bek tensor with the components C_{ijk} is defined by

C_{ijk} = \frac{1}{n − 2}(R_{ij,k} − R_{ik,j}) − \frac{1}{2(n − 1)(n − 2)}(δ_{ij}r_{k} − δ_{ik}r_{j}),

where r_{k} = ∇_{k}r.

For n ≥ 4, we know that M^{n} is a locally conformally flat Riemannian manifold if and only if C_{ijkl} = 0, and that C_{ijk} = 0 on M^{n} in this case.

When n = 3, we always have C_{ijkl} = 0 on M^{3} and that M^{3} is a locally conformally flat Riemannian manifold if and only if C_{ijk} = 0. Hence, if M^{n} is a locally conformally flat n-dimensional Riemannian manifold, then

(2.1)  R_{ijkl} = \frac{1}{n − 2}(R_{ik}δ_{jl} − R_{il}δ_{jk} + R_{jl}δ_{ik} − R_{jk}δ_{il})

− \frac{r}{(n − 1)(n − 2)}(δ_{ik}δ_{jl} − δ_{il}δ_{jk}).

Moreover if the scalar curvature r is constant, then we obtain

(2.2)  R_{ij,k} = R_{ik,j}.

From now on, we assume n = 3. Let λ_{i} for i = 1, 2, 3 be the eigenvalues of Ricci

curvature tensor (R_{ij}) of M^{3} and define μ_{i} := λ_{i} − \frac{r}{3}. We put B = S − \frac{r^{2}}{3} . The following has been proved in [1]:

Proposition. Let M^{3} be a locally conformally flat Riemannian manifold of dimension 3. If the scalar curvature and the squared norm of Ricci curvature tensor of M^{3} are constant, then we have

(2.3)  ∑_{i,j,k} R_{ij,k}^{2} = −3 ∑_{i} μ_{i}^{3} − \frac{r}{2}B,

(2.4)  ∑_{i,j,k,l} R_{ij,kl}^{2} = −24 ∑_{i,j,k} μ_{i}R_{ij,k}^{2} − \frac{3}{4}r ∑_{i,j,k} R_{ij,k}^{2} − \frac{9}{4}B(B − \frac{r^{2}}{6} ).

In order to prove the Main Theorem, the Theorem 3 in [1] will play an important role. We restate it in the following:

Theorem 3 (see [1]). Let M^{3} be a complete locally conformally flat 3-dimensional Riemannian manifold with constant scalar curvature r and constant squared norm S of the Ricci curvature tensor. If r < 0, then we have

sup ∑_{i=1}^{3} μ_{i}^{3} = \frac{B^{\frac{3}{2}}}{\sqrt{6}} .

3. Proof of Main Theorem

In order to prove Main Theorem, first, we discuss the following special case:

Theorem 3.0. There exist no locally conformally flat 3-dimensional Riemannian manifolds with the negative constant scalar curvature r and Ricci curvature tensor

of constant squared norm S satisfying \frac{r^{2}}{3} + (\frac{7}{113})^{2} \frac{2r^{2}}{3} < S < \frac{r^{2}}{3} + (\frac{35}{137})^{2} \frac{r^{2}}{6} such

that ∑_{i}(λ_{i} − \frac{r}{3})^{3} = \frac{(S − \frac{r^{2}}{3} )^{\frac{3}{2}}}{\sqrt{6}} holds at some point p.

From the Lemma 1 in [1], we know that

−\frac{B^{\frac{3}{2}}}{\sqrt{6}} ≤ ∑_{i} μ_{i}^{3} ≤ \frac{B^{\frac{3}{2}}}{\sqrt{6}} ,

where the equality holds if and only if at least two of μ_{i} are equal. From the assumption of Theorem 3.0, we have, at a point p, ∑_{i} μ_{i}^{3} = \frac{B^{\frac{3}{2}}}{\sqrt{6}} . Hence, for the purpose of proving this Theorem 3.0, we can assume that at the point p

(3.1)  λ_{1} = λ_{2} = λ and λ_{3} = r − 2λ,

that is,

(3.2)  μ_{1} = μ_{2} = μ and μ_{3} = −2μ.

If μ = 0 at the point p, then μ_{1} = μ_{2} = μ_{3} = 0 at p. Hence, S − \frac{r^{2}}{3} = B =

∑_{i} μ_{i}^{2} = 0 at p. Since S > \frac{r^{2}}{3} +(\frac{7}{113})^{2} \frac{2r^{2}}{3} , that is, B > (\frac{7}{113})^{2} \frac{2r^{2}}{3} , we can assume μ ≠ 0 at the point p. Then we obtain from (3.2) at p

(3.3)  B = 6μ^{2}.

Let {e_{1}, e_{2}, e_{3}} be an orthonormal basis of tangent space T_{p}M at p. Thus we have at p

(R_{ij}) = ( \begin{matrix}λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & r − 2λ \end{matrix} )

and

(τ_{ij}) = ( \begin{matrix}μ & 0 & 0 \\ 0 & μ & 0 \\ 0 & 0 & −2μ \end{matrix} ) ,

where τ_{ij} = R_{ij} − \frac{r}{3}δ_{ij}. Because \{e_{1}, e_{2}\} is a basis of the eigenspace corresponding to the λ-eigenvalue of (R_{ij}) at p, we can assume R_{12,3} = 0 at p, rotating it if necessary.

Lemma 3.1. At the point p, we have

(3.4)  R_{22,k} = −R_{11,k}, R_{33,k} = 0 for k = 1, 2, 3,
(3.5)  ∑_{i,j,k} R_{ij,k}^{2} = 6R_{11,3}^{2} + 4(R_{11,1}^{2} + R_{22,2}^{2}) = 3B(μ − \frac{r}{6}).

Proof. By differentiating ∑_{i} R_{ii} = r and ∑_{i,j} R_{ij}^{2} = S at p, we have respectively

∑_{i} R_{ii,k} = 0 and ∑_{i} R_{ii}R_{ii,k} = 0.

Hence, at p, we have R_{22,k} = −R_{11,k} and R_{33,k} = 0 for k = 1, 2, 3. Therefore, by using (2.3), we obtain also at p

∑_{i,j,k} R_{ij,k}^{2} = 6R_{11,3}^{2} + 4(R_{11,1}^{2} + R_{22,2}^{2}) = 3B(μ − \frac{r}{6}).

This completes the proof of Lemma 3.1. □

Let us denote A_{1} and A_{2} by

(3.6)  A_{1} = R_{11,3}^{2} and A_{2} = R_{11,1}^{2} + R_{22,2}^{2}.

Then (3.5) in Lemma 3.1 means

(3.7)  6A_{1} + 4A_{2} = 3B(μ − \frac{r}{6}).

Lemma 3.2. If {i, j, k, l} is not a permutation of {1, 1, 3, 3} or {2, 2, 3, 3}, then, at p,

R_{ij,kl} = R_{ij,lk}.

Proof. From the Ricci formulas and (2.1), we obtain at p

R_{ij,kl} − R_{ij,lk} = (μ_{i} − μ_{j})(μ_{i} + μ_{j} + \frac{r}{6})(δ_{ik}δ_{jl} − δ_{il}δ_{jk}).

If {i, j, k, l} is not a permutation of {1, 1, 3, 3} or {2, 2, 3, 3}, we have at p

R_{ij,kl} = R_{ij,lk}.

Thus, we have finished the proof of Lemma 3.2. □

Lemma 3.3. At the point p,

R_{11,11} = R_{22,22}, R_{33,33} = \frac{2A_{1}}{3μ} , R_{33,12} = 0, R_{33,13} = \frac{2R_{11,1}R_{11,3}}{3μ} ,

R_{33,23} = −\frac{2R_{22,2}R_{11,3}}{3μ} , R_{33,11} = R_{33,22} = \frac{2(A_{1} + A_{2})}{3μ}, R_{11,33} = R_{22,33} = −\frac{A_{1}}{3μ} .

Proof. Since ∑_{i} R_{ii} = r and ∑_{i,j} R_{ij}^{2} = S are constant, we have, by differentiating

at p them,

(3.8)  ∑_{i} R_{ii,kl} = 0,

and

∑_{i} μ_{i}R_{ii,kl} + ∑_{i,j} R_{ij,k}R_{ij,l} = 0.

Hence, by direct calculation it follows that

R_{33,kl} = \frac{1}{3μ} ∑_{i,j} R_{ij,k}R_{ij,l}.

That is, by using (3.7), it follows that

R_{33,12} = 0, R_{33,33} = \frac{1}{3μ} ∑_{i,j} R_{ij,3}^{2} = \frac{2A_{1}}{3μ} ,

R_{33,13} = \frac{1}{3μ} ∑_{i,j} R_{ij,1}R_{ij,3} = \frac{2R_{11,1}R_{11,3}}{3μ} ,

R_{33,23} = \frac{1}{3μ} ∑_{i,j} R_{ij,2}R_{ij,3} = −\frac{2R_{22,2}R_{11,3}}{3μ} ,

R_{33,11} = R_{33,22} = \frac{1}{3μ} ∑_{i,j} R_{ij,1}^{2} = \frac{2(A_{1} + A_{2})}{3μ},

R_{11,33} = R_{22,33} = R_{33,11} + (μ_{1} − μ_{3})(μ_{1} + μ_{3} + \frac{r}{6}) = −\frac{A_{1}}{3μ}

at the point p. According to (3.8), we have also at p

R_{11,11} = R_{22,22}.

This completes the proof of Lemma 3.3. □

Lemma 3.4. At the point p, we have

μ^{2}{2(R_{11,11}^{2} + R_{11,12}^{2}) + 3(R_{11,23}^{2} + R_{22,13}^{2}) + R_{11,13}^{2} + R_{22,23}^{2}} + 2(A_{1} + A_{2})μR_{11,11}

= −\frac{58A_{1}^{2} + 112A_{1}A_{2} + 48A_{2}^{2}}{36} − 4μBA_{2} + \frac{3μ^{2}}{16} (r^{2} − 3rμ − 3B)B.

Proof. By calculating ∑_{i,j,k} R_{ij,k1}^{2}, ∑_{i,j,k} R_{ij,k2}^{2} and ∑_{i,j,k} R_{ij,k3}^{2} separately, we have at p

(3.9) ∑_{i,j,k} R_{ij,k1}^{2} = 6R_{12,31}^{2} + ∑_{i} R_{ii,i1}^{2} + 3 ∑_{i≠k} R_{ii,k1}^{2}

= 3(R_{11,21}^{2} + R_{11,31}^{2} + R_{22,11}^{2} + R_{22,31}^{2} + R_{33,11}^{2} + R_{33,21}^{2})

+ 6R_{11,23}^{2} + R_{11,11}^{2} + R_{22,21}^{2} + R_{33,31}^{2}

= 3(2R_{11,23}^{2} + R_{11,13}^{2} + R_{22,13}^{2}) + R_{33,31}^{2}

+ 6(R_{11,11}R_{33,11} + R_{33,11}^{2}) + 4(R_{11,12}^{2} + R_{11,11}^{2}),

(3.10) ∑_{i,j,k} R_{ij,k2}^{2} = 6R_{12,32}^{2} + ∑_{i} R_{ii,i2}^{2} + 3 ∑_{i≠k} R_{ii,k2}^{2}

= 3(R_{11,22}^{2} + R_{11,32}^{2} + R_{22,12}^{2} + R_{22,32}^{2} + R_{33,12}^{2} + R_{33,22}^{2})

+ 6R_{22,13}^{2} + R_{11,12}^{2} + R_{22,22}^{2} + R_{33,32}^{2}

= 3(2R_{11,11}R_{33,11} + R_{33,11}^{2} + R_{33,22}^{2}) + 4(R_{11,12}^{2} + R_{11,11}^{2})

+ 3(2R_{22,13}^{2} + R_{11,23}^{2} + R_{22,23}^{2}) + R_{33,23}^{2},

(3.11) ∑_{i,j,k} R_{ij,k3}^{2} = 6R_{12,33}^{2} + ∑_{i} R_{ii,i3}^{2} + 3 ∑_{i≠k} R_{ii,k3}^{2}

= 3(R_{11,23}^{2} + R_{11,33}^{2} + R_{22,13}^{2} + R_{22,33}^{2} + R_{33,13}^{2} + R_{33,23}^{2})

+ R_{11,13}^{2} + R_{22,23}^{2} + R_{33,33}^{2}

= 3(R_{11,23}^{2} + R_{22,13}^{2}) + (R_{11,13}^{2} + R_{22,23}^{2}) + (\frac{2A_{1}}{3μ})^{2}

+ 3\{2(\frac{−A_{1}}{3μ})^{2} + (\frac{2R_{11,1}R_{11,3}}{3μ})^{2} + (\frac{−2R_{22,2}R_{11,3}}{3μ})^{2}\}.

Here, we used the relations R_{11,11}+R_{22,11}+R_{33,11} = 0, R_{33,21} = 0, R_{22,21} = −R_{11,21} and so on in (3.8) and Lemma 3.3. Therefore, by summing up these equations (3.9),(3.10) and (3.11), we obtain at p

(3.12)  ∑_{i,j,k,l} R_{ij,kl}^{2} = 12(R_{11,23}^{2} + R_{22,13}^{2}) + 4(R_{11,13}^{2} + R_{22,23}^{2})

+ 8(R_{11,12}^{2} + R_{11,11}^{2}) + \frac{10A_{1}^{2} + 12A_{1}A_{2}}{9μ^{2}}

+ \frac{16(A_{1} + A_{2})^{2}}{3μ^{2}} + \frac{4A_{1}A_{2}}{9μ^{2}} + \frac{8(A_{1} + A_{2})}{μ}R_{11,11}.

The other hand, by using (2.4), we have at p

(3.13)  ∑_{i,j,k,l} R_{ij,kl}^{2} = −24(4μA_{2}) − \frac{3r}{4} 3B(μ − \frac{r}{6}) − \frac{9}{4}B(B − \frac{r^{2}}{6} )

= −96μA_{2} + \frac{3B}{4} (r^{2} − 3rμ − 3B).

Hence, from (3.12) and (3.13), we obtain at p μ^{2}{2(R_{11,11}^{2} + R_{11,12}^{2}) + 3(R_{11,23}^{2} + R_{22,13}^{2}) + R_{11,13}^{2} + R_{22,23}^{2}} + 2(A_{1} + A_{2})μR_{11,11}

= −\frac{58A_{1}^{2} + 112A_{1}A_{2} + 48A_{2}^{2}}{36} − 4μBA_{2} + \frac{3μ^{2}}{16} (r^{2} − 3rμ − 3B)B.

This proves Lemma 3.4. □ Lemma 3.5. At the point p, it follows that

∑_{i,j,k} R_{ij,k}R_{ij,k33} = −3(R_{11,23}^{2} + R_{22,13}^{2}) − (R_{11,13}^{2} + R_{22,23}^{2})

− \frac{9}{2}(2 ∑_{i,j} μ_{i}R_{ij,3}^{2} + ∑_{i} μ_{i}^{2}R_{ii,33}) − \frac{10A_{1}^{2} + 12A_{1}A_{2}}{9μ^{2}} .

Proof. By differentiating at p

∑_{i,j,k} R_{ij,k}^{2} = −3 ∑_{i} μ_{i}^{3} − \frac{r}{2}B,

obtained in Proposition, we have

(3.14)  2 ∑_{i,j,k} R_{ij,k}R_{ij,k33} + 2 ∑_{i,j,k} R_{ij,k3}^{2} = −3∇_{3}∇_{3} ∑_{i} μ_{i}^{3}

and we obtain at p, by directly calculating,

(3.15)  ∇_{3}∇_{3} ∑_{i} μ_{i}^{3} = 6 ∑_{i,j} μ_{i}R_{ij,3}^{2} + 3 ∑_{i} μ_{i}^{2}R_{ii,33}.

Thus, from (3.11), (3.14) and (3.15), we conclude

∑_{i,j,k} R_{ij,k}R_{ij,k33} = −3(R_{11,23}^{2} + R_{22,13}^{2}) − (R_{11,13}^{2} + R_{22,23}^{2})

− \frac{9}{2}(2 ∑_{i,j} μ_{i}R_{ij,3}^{2} + ∑_{i,j} μ_{i}^{2}R_{ii,33}) − \frac{10A_{1}^{2} + 12A_{1}A_{2}}{9μ^{2}} ,

that is, the proposed relation in Lemma 3.5 holds. □

Lemma 3.6. At the point p, we have

∑_{i,j,k} R_{ij,k}R_{33,ijk} = \frac{4}{3μ^{2}} (2A_{1}^{2} − A_{2}^{2}) + \frac{4A_{1}}{μ} R_{11,11}.

Proof. By differentiating ∑_{i} R_{ii} = r and ∑_{i,j} R_{ij}^{2} = S at p, we have,

(3.16)  ∑_{i} R_{ii,klm} = 0

and

∑_{i,j} R_{ij}R_{ij,klm} + ∑_{i,j} R_{ij,m}R_{ij,kl} + ∑_{i,j} R_{ij,k}R_{ij,lm} + ∑_{i,j} R_{ij,l}R_{ij,km} = 0

respectively. Therefore, from (3.16), we obtain at p

(3.17)  ∑_{i,j} R_{ij}R_{ij,klm} = −3μR_{33,klm}.

Hence, it follws at p

(3.18) μ ∑_{k,l,m} R_{kl,m}R_{33,klm} = ∑_{i,j,k,l,m} R_{kl,m}R_{ij,m}R_{ij,kl}

= ∑_{k,l,m} R_{kl,m}R_{11,k}(R_{11,ml} − R_{22,ml}) + 2 ∑_{k,l,m} R_{kl,m}R_{12,k}R_{12,ml}

+ 2 ∑_{k,l,m} R_{kl,m}R_{13,k}R_{13,ml} + 2 ∑_{k,l,m} R_{kl,m}R_{23,k}R_{23,ml}.

Now, by calculating the terms in (3.18) separately, we have at p respectively

(3.19)  ∑_{k,l,m} R_{kl,m}R_{11,k}(R_{11,ml} − R_{22,ml})
= 2(A_{1} + A_{2})(R_{11,11} − R_{22,11}) + 2R_{11,1}R_{11,3}(R_{11,13} − R_{22,13})
+ 2R_{22,2}R_{11,3}(R_{11,23} − R_{22,23}),

2 ∑_{k,l,m} R_{kl,m}R_{12,k}R_{12,ml} (3.20)

= 4A_{2}R_{11,22} − 4R_{22,2}R_{11,3}R_{11,23} + 4R_{11,1}R_{11,3}R_{22,13},

2 ∑_{k,l,m} R_{kl,m}R_{13,k}R_{13,ml} (3.21)

= 2A_{1}(R_{11,33} + R_{33,11}) + 2R_{11,1}R_{11,3}R_{11,13}
− 2R_{11,1}R_{11,3}R_{22,13} − 4R_{22,2}R_{11,3}R_{11,23},

2 ∑_{k,l,m} R_{kl,m}R_{23,k}R_{23,ml} (3.22)

= 2A_{1}(R_{22,33} + R_{33,22}) + 2R_{22,2}R_{11,3}(R_{11,23} − R_{22,23})
+ 4R_{11,1}R_{11,3}R_{22,13}.

From (3.18), (3.19), (3.20), (3.21) and (3.22), we obtain at p, by making use of the relation R_{11,11} + R_{22.11} + R_{33,11} = 0,

μ ∑_{kl,m} R_{kl,m}R_{33,klm} (3.23)

= \frac{4}{3μ}(2A_{1}^{2} − A_{2}^{2}) + 4A_{1}R_{11,11}.

That is, Lemma 3.6 holds. □

Lemma 3.7. At the point p, we obtain

μ^{2}{3(R_{11,23}^{2} + R_{22,13}^{2}) + (R_{11,13}^{2} + R_{22,23}^{2})} (3.24)

= −\frac{10A_{1}^{2} + 12A_{1}A_{2}}{9} − 27A_{1}μ^{3} − \frac{4}{3}(2A_{1}^{2} − A_{2}^{2})

− 3rμ^{2}A_{1} − (4μ + \frac{4}{3}r)μ^{2}A_{2} − 4A_{1}μR_{11,11}.

Proof. We write

R_{ijkl,m} = ∇_{m}R_{ijkl}.

Then, since

R_{33,ijk} − R_{ij,k33}

= ∇_{k}(R_{ij,33} + ∑_{m} R_{mi}R_{m33j} + ∑_{m} R_{3m}R_{mi3j}) − R_{ij,k33}

= R_{ij,33k} + ∑_{m} R_{mi,k}R_{m33j} + ∑_{m} R_{mi}R_{m33j,k}

+ ∑_{m} R_{3m,k}R_{mi3j} + ∑_{m} R_{3m}R_{mi3j,k} − R_{ij,k33}

= R_{ij,3k3} + ∑_{m} R_{mj,3}R_{mi3k} + ∑_{m} R_{im,3}R_{mj3k}

+ ∑_{m} R_{ij,m}R_{m33k} + ∑_{m} R_{mi,k}R_{m33j} + ∑_{m} R_{mi}R_{m33j,k}

+ ∑_{m} R_{3m,k}R_{mi3j} + ∑_{m} R_{3m}R_{mi3j,k} − R_{ij,k33}

= ∇_{3}(R_{ij,k3} + ∑_{m} R_{mj}R_{mi3k} + ∑_{m} R_{im}R_{mj3k}) + ∑_{m} R_{mj,3}R_{mi3k}

+ ∑_{m} R_{im,3}R_{mj3k} + ∑_{m} R_{ij,m}R_{m33k} + ∑_{m} R_{mi,k}R_{m33j}

+ ∑_{m} R_{mi}R_{m33j,k} + ∑_{m} R_{3m,k}R_{mi3j} + ∑_{m} R_{3m}R_{mi3j,k} − R_{ij,k33}

= ∑_{m} R_{mj,3}R_{mi3k} + ∑_{m} R_{im,3}R_{mj3k} + ∑_{m} R_{mj}R_{mi3k,3}

+ ∑_{m} R_{im}R_{mj3k,3} + ∑_{m} R_{mj,3}R_{mi3k} + ∑_{m} R_{im,3}R_{mj3k}

+ ∑_{m} R_{ij,m}R_{m33k} + ∑_{m} R_{mi,k}R_{m33j} + ∑_{m} R_{mi}R_{m33j,k}

+ ∑_{m} R_{3m,k}R_{mi3j} + ∑_{m} R_{3m}R_{mi3j,k},

we obtain at p

∑_{i,j,k} R_{ij,k}(R_{33,ijk} − R_{ij,k33}) (3.25)

= ∑_{i,j,k,m} R_{ij,k}{5R_{mj,3}R_{mi3k} + 2R_{ij,m}R_{m33k}

+ (λ_{i} − λ_{j})R_{ij3k,3} + (λ_{i} − λ_{3})R_{i33j,k}}.

Since M^{3} is a 3-dimensional Riemannian manifold, we have

(3.26)  R_{ijkl} = (R_{ik}δ_{jl} − R_{il}δ_{jk} + δ_{ik}R_{jl} − δ_{il}R_{jk}) − \frac{r}{2}(δ_{ik}δ_{jl} − δ_{il}δ_{jk}).

So, we obtain

(3.27)  R_{ijkl,m} = (R_{ik,m}δ_{jl} − R_{il,m}δ_{jk} + δ_{ik}R_{jl,m} − δ_{il}R_{jk,m}).

Because R_{ij} = λ_{i}δ_{ij} at the point p, by the direct calculations, we also have the following relations

R_{ijkl} = (λ_{i} + λ_{j} − \frac{r}{2})(δ_{ik}δ_{jl} − δ_{il}δ_{jk}), (3.28)

∑_{i,j,k,m} R_{ij,k}R_{mj,3}R_{mi3k} (3.29)

= ∑_{i,j,k,m} R_{ij,k}R_{mj,3}(λ_{m} + λ_{i} − \frac{r}{2})(δ_{m3}δ_{ik} − δ_{mk}δ_{i3})

= − ∑_{i,j} R_{ij,3}^{2}(λ_{i} + λ_{3} − \frac{r}{2}),

(3.30) ∑_{i,j,k,m} R_{ij,k}R_{ij,m}R_{m33k}

= ∑_{i,j,k,m} R_{ij,k}R_{ij,m}(λ_{m} + λ_{3} − \frac{r}{2})(δ_{m3}δ_{3k} − δ_{mk}δ_{33})

= ∑_{i,j} R_{ij,3}^{2}(2λ_{3} − \frac{r}{2}) − ∑_{i,j,k} λ_{i}R_{ij,k}^{2} + \frac{r}{2} ∑_{i,j,k} R_{ij,k}^{2} − λ_{3} ∑_{i,j,k} R_{ij,k}^{2},

(3.31) ∑_{i,j,k} R_{ij,k}(λ_{i} − λ_{j})R_{ij3k,3}

= ∑_{i,j,k} R_{ij,k}(λ_{j} − λ_{i})(R_{ik,3}δ_{j3} − R_{i3,3}δ_{jk} + δ_{ik}R_{j3,3} − δ_{i3}R_{jk,3})

= 2λ_{3} ∑_{i,j} R_{ij,3}^{2} − 2 ∑_{i,j} λ_{i}R_{ij,3}^{2},

(3.32)  ∑_{i,j,k} R_{ij,k}(λ_{i} − λ_{3})R_{i33j,k}

= ∑_{i,j,k} R_{ij,k}(λ_{i} − λ_{3})(R_{i3,k}δ_{3j} − R_{ij,k}δ_{33} + δ_{i3}R_{3j,k} − δ_{ij}R_{33,k})

= ∑_{i,j} λ_{i}R_{ij,3}^{2} − λ_{3} ∑_{i,j} R_{ij,3}^{2} − ∑_{i,j,k} λ_{i}R_{ij,k}^{2} + λ_{3} ∑_{i,j,k} R_{ij,k}^{2}.

Therefore, by summing up (3.25), (3.29), (3.30), (3.31) and (3.32), we have at p,

(3.33) ∑_{i,j,k} R_{ij,k}(R_{33,ijk} − R_{ij,k33})

= −5 ∑_{i,j} R_{ij,3}^{2}(λ_{i} + λ_{3} − \frac{r}{2}) + 2 ∑_{i,j} R_{ij,3}^{2}(2λ_{3} − \frac{r}{2}) − 2 ∑_{i,j,k} λ_{i}R_{ij,k}^{2}

+ r ∑_{i,j,k} R_{ij,k}^{2} − 2λ_{3} ∑_{i,j,k} R_{ij,k}^{2} + 2λ_{3} ∑_{i,j} R_{ij,3}^{2} − 2 ∑_{i,j} λ_{i}R_{ij,3}^{2}

+ ∑_{i,j} λ_{i}R_{ij,3}^{2} − λ_{3} ∑_{i,j} R_{ij,3}^{2} − ∑_{i,j,k} λ_{i}R_{ij,k}^{2} + λ_{3} ∑_{i,j,k} R_{ij,k}^{2}

= −12(μ + \frac{r}{3})A_{1} + 3rA_{1} − 3\{4μA_{2} + \frac{r}{3}(6A_{1} + 4A_{2})\}

+ (2μ − \frac{r}{3})(6A_{1} + 4A_{2}) + r(6A_{1} + 4A_{2})

= −3rA_{1} − (4μ + \frac{4}{3}r)A_{2}.

Here, we used the following relations at p

(3.34)  ∑_{i,j} R_{ij,3}^{2} = 2R_{11,3}^{2} = 2A_{1},
(3.35)  ∑_{i,j,k} R_{ij,k}^{2} = 6A_{1} + 4A_{2},
(3.36)  ∑_{i,j,k} λ_{i}R_{ij,k}^{2} = ∑_{i,j,k} μ_{i}R_{ij,k}^{2} + \frac{r}{3} ∑_{i,j,k} R_{ij,k}^{2}

= 4μA_{2} + \frac{r}{3}(6A_{1} + 4A_{2})

(3.37)  ∑_{i,j} λ_{i}R_{ij,3}^{2} = 2(μ + \frac{r}{3})A_{1}.

Therefore, from Lemma 3.5, Lemma 3.6 and (3.33), we conclude

(3.38) 3(R_{11,23}^{2} + R_{22,13}^{2}) + (R_{11,13}^{2} + R_{22,23}^{2}) + \frac{10A_{1}^{2} + 12A_{1}A_{2}}{9μ^{2}}

+ 9 ∑_{i,j} μ_{i}R_{ij,3}^{2} + \frac{9}{2} ∑_{i,j} μ_{i}^{2}R_{ii,33} + \frac{4}{3μ^{2}} (2A_{1}^{2} − A_{2}^{2}) + \frac{4A_{1}}{μ} R_{11,11}

= −3rA_{1} − (4μ + \frac{4}{3}r)A_{2}.

Since we have at p by Lemma 3.3

(3.39) ∑_{i} μ_{i}^{2}R_{ii,33} = μ^{2}(R_{11,33} + R_{22,33}) + 4μ^{2}R_{33,33}
= 2μA_{1},

it folows from (3.38) that μ^{2}{3(R_{11,23}^{2} + R_{22,13}^{2}) + (R_{11,13}^{2} + R_{22,23}^{2})} (3.40)

= −\frac{10A_{1}^{2} + 12A_{1}A_{2}}{9} − 27A_{1}μ^{3} − \frac{4}{3}(2A_{1}^{2} − A_{2}^{2})

− 3rμ^{2}A_{1} − (4μ + \frac{4}{3}r)μ^{2}A_{2} − 4μA_{1}R_{11,11},

that is, Lemma 3.7 holds. □ Lemma 3.8. At the point p, we have

2μ^{2}(R_{11,11}^{2} + R_{11,12}^{2}) + 2(A_{2} − A_{1})μR_{11,11}

= \frac{39A_{1}^{2} − 32A_{1}A_{2} − 48A_{2}^{2}}{18} + (\frac{9}{2}Bμ + 3rμ^{2})A_{1}

+ (\frac{2}{9}r − \frac{10μ}{3})BA_{2} − (\frac{3}{8}r + \frac{9}{8}μ)μ^{2}(6A_{1} + 4A_{2}).

Proof. According to (3.7), we have at p

\frac{3Bμ^{2}}{16} (r^{2} − 3rμ − 3B) (3.41)

= −(\frac{9}{8}μ + \frac{3}{8}r)μ^{2}\{3B(μ − \frac{r}{6})\}

= −(\frac{3}{8}r + \frac{9}{8}μ)μ^{2}(6A_{1} + 4A_{2}).

From Lemma 3.4 and Lemma 3.7, we also obtain at p

2μ^{2}(R_{11,11}^{2} + R_{11,12}^{2}) + 2(A_{2} + A_{1})μR_{11,11} − 4A_{1}μR_{11,11}

= \frac{10A_{1}^{2} + 12A_{1}A_{2}}{9} + 27A_{1}μ^{3} + \frac{4}{3}(2A_{1}^{2} − A_{2}^{2}) + 3rμ^{2}A_{1}

+ (4μ + \frac{4}{3}r)μ^{2}A_{2} − \frac{58A_{1}^{2} + 112A_{1}A_{2} + 48A_{2}^{2}}{36}

− 4BμA_{2} + \frac{3Bμ^{2}}{16} (r^{2} − 3rμ − 3B).

Together with (3.41), we get at p

2μ^{2}(R_{11,11}^{2} + R_{11,12}^{2}) + 2(A_{2} − A_{1})μR_{11,11}

= \frac{39A_{1}^{2} − 32A_{1}A_{2} − 48A_{2}^{2}}{18} + (\frac{9}{2}Bμ + 3rμ^{2})A_{1}

+ (\frac{2}{9}r − \frac{10μ}{3})BA_{2} − (\frac{3}{8}r + \frac{9}{8}μ)μ^{2}(6A_{1} + 4A_{2}).

This completes the proof of Lemma 3.8. □

Proof of Theorem 3.0. We suppose that there exists a complete locally conformally flat 3-dimensional Riemannian manifold which satisfies the assumptions in Theorem 3.0. According to the Lemma 1 in [1], we know that, at each point in M^{3},

−\frac{B^{\frac{3}{2}}}{\sqrt{6}} ≤ ∑_{i} μ_{i}^{3} ≤ \frac{B^{\frac{3}{2}}}{\sqrt{6}} ,

where the equality holds if and only if at least two of μ_{i} are equal. From the assumptions of Theorem 3.0, we have at a point p,

∑_{i} μ_{i}^{3} = \frac{B^{\frac{3}{2}}}{\sqrt{6}} .

Hence, we can assume μ_{1} = μ_{2} = μ and μ_{3} = −2μ. Since ∑_{i} μ_{i}^{3} = −6μ^{3}, we have

μ = −\begin{matrix}√ & \overline{B} \\ √ & \overline{6} \end{matrix} < 0.

For any real numbers a > 0, x and y, the following inequality holds:

(3.42)  2xy ≤ ax^{2} + \frac{1}{a}y^{2}.

By making use of the inequality (3.42) and Lemma 3.8, we have at p

−\frac{1}{2}(A_{1} − A_{2})^{2} ≤ \frac{39A_{1}^{2} − 32A_{1}A_{2} − 48A_{2}^{2}}{18} + (\frac{9}{2}Bμ + 3rμ^{2})A_{1}

+ (\frac{2}{9}r − \frac{10μ}{3})BA_{2} − (\frac{3}{8}r + \frac{9}{8}μ)μ^{2}(6A_{1} + 4A_{2}).

This implies

(3.43)  0 ≤ \frac{48A_{1}^{2} − 50A_{1}A_{2} − 39A_{2}^{2}}{18} + (\frac{9}{2}Bμ + 3rμ^{2})A_{1}

+ (\frac{2}{9}r − \frac{10μ}{3})BA_{2} − (\frac{3}{8}r + \frac{9}{8}μ)μ^{2}(6A_{1} + 4A_{2}).

From (3.7), we have at p

\frac{−48A_{1}^{2} + 50A_{1}A_{2} + 39A_{2}^{2}}{18} (3.44)

= \frac{−8(6A_{1} + 4A_{2})A_{1} + \frac{39}{4}(6A_{1} + 4A_{2})A_{2} + \frac{47}{2}A_{1}A_{2}}{18}

= −\frac{4}{3}B(μ − \frac{r}{6})A_{1} + \frac{13}{8}B(μ − \frac{r}{6})A_{2} + \frac{47}{36}A_{1}A_{2}.

Therefore, (3.43) and (3.44) yield the following at p

(3.45)  0 ≤ −\frac{47}{36}A_{1}A_{2} + (\frac{113}{24}μ − \frac{7}{72}r)BA_{1} + (−\frac{137}{24}μ + \frac{35}{144}r)BA_{2}.

Since μ < 0, r < 0 and \frac{r^{2}}{3} +(\frac{7}{113})^{2} \frac{2r^{2}}{3} < S < \frac{r^{2}}{3} +(\frac{35}{137})^{2} \frac{r^{2}}{6} , then \frac{113}{24}μ− \frac{7}{72}r < 0

and −\frac{137}{24}μ + \frac{35}{144}r < 0 hold. From A_{1} ≥ 0, A_{2} ≥ 0, B > 0, We infer A_{1} = A_{2} = 0.

But (3.7) and A_{1} = A_{2} = 0 say that B = 0 or μ = \frac{r}{6}. This is impossible. Hence, we can conclude that Theorem 3.0 holds. Thus we have completed the proof of Theorem 3.0.

Proof of Main Theorem. If we suppose that there exists a complete locally conformally flat 3-dimensional Riemannian manifold which satisfies the conditions in the Main Theorem, from the Theorem 3 in the section 2, we have

sup ∑_{i=1}^{3} μ_{i}^{3} = \frac{B^{\frac{3}{2}}}{\sqrt{6}} .

From the proof of Theorem 3 in [1], we know that there exists a sequence {p_{m}} of points in M^{3} such that

lim_{m→∞} ∑_{i=1}^{3} μ_{i}^{3}(p_{m}) = sup ∑_{i=1}^{3} μ_{i}^{3} = \frac{B^{\frac{3}{2}}}{\sqrt{6}} .

Since B is constant, we have that μ_{i} for any i is bounded. From formulas (2.3) and (2.4) in Proposition, we know that R_{ij,k} and R_{ij,kl} are bounded for any i, j, k and l. Thus, by taking a subsequence if necessary, we can assume lim_{m→∞} μ_{i}(p_{m}) = μ_{i}^{∗}, lim_{m→∞} R_{ij,k}(p_{m}) = R_{ij,k}^{∗} and lim_{m→∞} R_{ij,kl}(p_{m}) = R_{ij,kl}^{∗}. Since ∑_{i}(μ_{i}^{∗})^{3} =

sup ∑_{i} μ_{i}^{3} = \frac{B^{\frac{3}{2}}}{\sqrt{6}} holds if and only if at least two of μ_{i}^{∗} are equal. Hence, we can

put μ_{1}^{∗} = μ_{2}^{∗} = μ and μ_{3}^{∗} = −2μ. By replacing the point p which was used in the proofs of Lemmas and Theorem 3.0 with the above limit and making use of the same proofs, we can conclude that the Main Theorem holds. In fact, although we do not know whether lim_{m→∞} R_{ij,klt}(p_{m}) = R_{ij,klt}^{∗} holds, from Ricci formulas in the section 2, we know that lim_{m→∞}(R_{ij,klt} −R_{ij,ktl})(p_{m}) exists. From the proof of Lemma 3.5, we can infer lim_{m→∞} ∑_{i,j,k} R_{ij,k}R_{ij,k33}(p_{m}) exists. Therefore, we have lim_{m→∞} ∑_{i,j,k} R_{ij,k}R_{33,ijk}(p_{m}) also exists because lim_{m→∞} ∑_{i,j,k} R_{ij,k}(R_{33,ijk} − R_{ij,k33})(p_{m}) exists. Thus, formula (3.18) in the proof of Lemma 3.6 can also be obtained when we replace the point p with the above limit. Making use of these facts above, these Lemmas in the section can be proved by the same methods. Hence, we can conclude that the Main Theorem is true.

References

1. Q. M. Cheng, S. Ishikawa & K. Shiohama, Conformally flat 3-manifolds with constant scalar curvature, J. Math. Soc. Japan, 51 (1999), 209-226.

2. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479-495.

3. R. Schoen & S. T. Yau, Conformally flat manifolds, Kleinian groups, and constant scalar curvature, Invent. Math. 92 (1988), 47-71.

4. R. Schoen & S. T. Yau, Differential Geometry, Science Press, Beijing, 1988.

5. N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola. Norm. Sup. Pisa. Cl. Sci. 22 (1968), 265-274.

6. H. Yamabe, On the conformal deformation of Riemannian structures on compact manifolds, Osaka J. Math. 12 (1960), 21-37.

Department of Mathematics, Faculty of Science, Josai University, Saitama, Sakado 350-0295, Japan

Department of Mathematics, Saga University, Saga 840-0027, Japan

Department of Mathematics, Saga University, Saga 840-0027, Japan