COMPACT LOCALLY CONFORMALLY

FLAT RIEMANNIAN MANIFOLDS

Qing-Ming Cheng*

Dedicated to Professor Katsuhiro Shiohama for his 60th birthday

Abstract. In this paper, we have two purposes. Firstly, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product S^{n−1}(c)×S^{1} if its Ricci curvature is nonnegative. Secondly, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied

∑_{i,j} R_{ij}^{2} ≤ \frac{r^{2}}{n − 1} ,

where ∑_{i,j} R_{ij}^{2} is the squared norm of the Ricci curvature tensor.

1. Introduction

A locally conformally flat structure on a higher dimensional Riemannian manifold is a natural generalization of the conformal structure of a Riemannian surface. The study of the global properties of manifolds with such structures was initiated by Kuiper, who classified compact locally conformally flat manifolds with abelian fundamental groups. In higher dimensions, not every manifold admits a locally conformally flat structure and it is a difficult problem to give a good classification of locally conformally flat Riemannian manifolds. But, a well known result in Riemannian geometry states that every compact n-dimensional Riemannian manifold can be deformed to a Riemannian manifold with constant scalar curvature by a conformal transformation (see [11], [16], [17], 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 [12]). Recently, Schoen and Yau also investigated the global properties on compact connected locally conformally flat n-dimensional Riemannian manifolds by using the Yamabe equation.

On the other hand, in [15], Tani proved that any compact connected oriented locally conformally flat n-dimensional Riemannian manifold M^{n} with constant scalar curvature is isometric to a space form if the Ricci curvature of M^{n} is positive. By making use of the Generalized Maximum Principle due to Omori [10] and Yau [18], Goldberg [7] and Wu and the author [5] proved that every complete connected locally conformally flat Riemannian manifold M^{n} with constant scalar curvature r is a space form if the Ricci curvature tensor of M^{n} satisfies the inequality

Key words and phrases. locally conformally flat, sphere, Riemannian product, scalar curvature and Ricci curvature.

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

2000 Mathematics Subject Classification 53C20

sup ∑_{ij} R_{ij}^{2} < \frac{r^{2}}{n − 1},

where ∑_{ij} R_{ij}^{2} is the squared norm of the Ricci curvature tensor of M^{n}. In particular, when n = 3, Cheng, Ishikawa and Shiohama [3] completely classified complete connected locally conformally flat 3-dimensional Riemannian manifolds with both r ≥ 0 and S = ∑_{ij} R_{ij}^{2} being constant, that is, they proved the following:

Theorem. Let M^{3} be a complete connected locally conformally flat 3-dimensional Riemannian manifold with both r ≥ 0 and S = ∑_{ij} R_{ij}^{2} being constant. Then M^{3} is isometric to a space form of constant sectional curvature or M^{3} is isometric to a Riemannian product M^{2}(c) × N^{1} where N^{1} = R^{1} or S^{1} for some c.

When r < 0, Cheng, Ishikawa and Shiohama [3] and [4] also gave several characterizations on such Riemannian manifolds.

In this paper, firstly, we shall extend the result due to Tani [15] to case that Ricci curvature is nonnegative. Secondly, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r and satisfying

∑_{ij} R_{ij}^{2} ≤ \frac{r^{2}}{n − 1},

that is, we obtain the following:

Theorem 1. Let M^{n} be a compact connected oriented locally conformally flat ndimensional Riemannian manifold with constant scalar curvature. If the Ricci curvature of M^{n} is nonnegative, then M^{n} is isometric to a space form or a Riemannian product S^{n−1}(c) × S^{1}.

Theorem 2. Let M^{n} be a compact connected oriented locally conformally flat ndimensional Riemannian manifold with nonnegative scalar curvature r. If

∑_{ij} R_{ij}^{2} ≤ \frac{r^{2}}{n − 1},

then either the universal covering of M^{n} is diffeomorphic to a sphere or M^{n} is isometric to a Riemannian product S^{n−1}(c) × S^{1}. Where ∑_{ij} R_{ij}^{2} is the squared norm of the Ricci curvature tensor.

Remark. It should be remarked that in our Theorem 2, we do not assume that the scalar curvature of M^{n} is constant.

Acknowledgement. I would like to express my gratitude to the referee for his valuable suggestion

2. Preliminaries

Throughout this article, we consider compact connected oriented locally conformally flat Riemannian manifolds. 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. 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},  (2.1)

R_{ij,klm} − R_{ij,kml} = ∑_{t} R_{tj,k}R_{tilm} + ∑_{t} R_{itk}R_{tjlm} + ∑_{t} R_{ijt}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

R_{ijkl} = \frac{1}{n − 2}(R_{ik}δ_{jl} − R_{il}δ_{jk} + R_{jl}δ_{ik} − R_{jk}δ_{il})  (2.2)

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

and

(R_{ij,k} − \frac{1}{2(n − 1)}δ_{ij}r_{k}) − (R_{ik,j} − \frac{1}{2(n − 1)}δ_{ik}r_{j}) = 0.  (2.3)

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

R_{ij,k} = R_{ik,j}.  (2.4)

3. Proofs of Theorems

Proof of Theorem 1. We choose a local frame of orthonormal vector fields e_{1}, e_{2}, ⋯, e_{n} adapted to the Riemannian metric of M^{n} such that R_{ij} = λ_{i}δ_{ij}. The λ_{i}’s for i = 1, 2, ⋯, n are the eigenvalues of Ricci curvature tensor (R_{ij}) of M^{n}. Since the scalar curvature r of M^{n} is constant, from (2.4), we have

R_{ij,k} = R_{ik,j}.  (3.1)

From (2.1) and (2.2), we calculate the Laplacian of ∑_{i,j} R_{ij}^{2}.

\frac{1}{2}Δ ∑_{i,j} R_{ij}^{2}  (3.2)
= ∑_{i,j,k} R_{ij,k}^{2} + ∑_{i,j,k} R_{ij}R_{ij,kk}
= ∑_{i,j,k} R_{ij,k}^{2} + ∑_{i,j,k} R_{ij}(R_{ki,kj} + ∑_{t} R_{ti}R_{tkjk} + ∑_{t} R_{kt}R_{tijk})
= ∑_{i,j,k} R_{ij,k}^{2} + ∑_{i,j,k} R_{ij}(∑_{t} R_{ti}R_{tkjk} + ∑_{t} R_{kt}R_{tijk})
= ∑_{i,j,k} R_{ij,k}^{2} + ∑_{i,k} λ_{i}(λ_{i}R_{ikik} + λ_{k}R_{kiik}).

Since the Ricci curvature is nonnegative, we have λ_{i} ≥ 0 for any i. Without loss of generality, we assume λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{n}. Hence, from (2.2), we have

∑_{i,k} λ_{i}(λ_{i}R_{ikik} + λ_{k}R_{kiik})  (3.3)
= ∑_{i,k} λ_{i}(λ_{i} − λ_{k})R_{ikik}

= ∑_{i,k} λ_{i}(λ_{i} − λ_{k})\frac{1}{n − 2}(λ_{i} + λ_{k} − \frac{r}{n − 1}).

Since r = ∑_{i} R_{ii} = ∑_{i} λ_{i}, we have

∑_{i,k} λ_{i}(λ_{i} − λ_{k})\frac{1}{n − 2}(λ_{i} + λ_{k} − \frac{r}{n − 1})  (3.4)

= \frac{2}{(n − 2)(n − 1)} ∑_{i,j,k,j<k} λ_{i}(λ_{i} − λ_{j})(λ_{i} − λ_{k}).

If i < j < k or j < k < i, we have λ_{i}(λ_{i} − λ_{j})(λ_{i} − λ_{k}) ≥ 0. If j < i < k, we have

λ_{i}(λ_{i} − λ_{j})(λ_{i} − λ_{k}) + λ_{k}(λ_{k} − λ_{j})(λ_{k} − λ_{i})  (3.5)
= (λ_{i} − λ_{k})^{2}(λ_{k} + λ_{i} − λ_{j}) ≥ 0.

Hence, from (3.3), (3.4) and (3.5), we infer

∑_{i,k} λ_{i}(λ_{i}R_{ikik} + λ_{k}R_{kiik}) ≥ 0.  (3.6)

From (3.6) and Stokes formula, we obtain by integrating (3.2)

∑_{i,j,k} R_{ij,k}^{2} = 0

on M^{n}. Therefore, ∑_{i,j} R_{ij}^{2} is constant. From the Theorem 3 due to Goldberg in [6], we know that M^{n} is isometric to a space form or isometric to S^{n−1}(c) × S^{1} since M^{n} is compact. □

In order to prove the Theorem 2, we prepare the following lemma which will be used for the proof of Theorem 2.

Lemma 1 (cf. p.55 in [2]). Let a_{1}, ⋯, a_{n} be real numbers. If

∑_{i=1}^{n}(a_{i})^{2} < \frac{(∑_{i=1}^{n} a_{i})^{2}}{n − 1} ,

then, we have either a_{j} > 0 for all j, or a_{j} < 0 for all j.

Proof. Since

∑_{i=1}^{n}(a_{i})^{2} < \frac{(∑_{i=1}^{n} a_{i})^{2}}{n − 1}

holds, we have

(n − 2)(a_{n})^{2} − 2(∑_{i=1}^{n−1} a_{i})a_{n} + {(n − 2) ∑_{i=1}^{n−1}(a_{i})^{2} − 2 ∑_{i<j≤n−1} a_{i}a_{j}} < 0.

Denote the left hand side of the above inequality by −a < 0. Since a_{n} is real, the discriminant of the quadratic equation on t

(n − 2)t^{2} − 2(∑_{i=1}^{n−1} a_{i})t + {(n − 2) ∑_{i=1}^{n−1}(a_{i})^{2} − 2 ∑_{i<j≤n−1} a_{i}a_{j} + a} = 0

is nonnegative, namely,

∑_{i=1}^{n−1}(a_{i})^{2} < \frac{(∑_{i=1}^{n−1} a_{i})^{2}}{n − 2} .

By induction, we know that the Lemma 1 is true. □

Proof of Theorem 2. From the assumption of Theorem 2, we have

∑_{ij} R_{ij}^{2} ≤ \frac{r^{2}}{n − 1}.

If there exist a point p ∈ M^{n} such that

∑_{ij} R_{ij}^{2} < \frac{r^{2}}{n − 1}

holds, we have

∑_{i} λ_{i}^{2} < \frac{(∑_{i} λ_{i})^{2}}{n − 1}

because of R_{ij} = λ_{i}δ_{ij}. From the Lemma 1, we know either λ_{i} > 0 for all i at the point p, or λ_{i} < 0 for all i at the point p. Since r = ∑_{i} λ_{i} ≥ 0 holds, we obtain λ_{i} > 0 for any i at the point p. Hence, M^{n} has positive Ricci curvature at the point p. From the theorem due to Aubin [1], we infer M^{n} has a metric such that the Ricci curvature on this metric is positive on M^{n}. From Myer’s theorem, we know that the first fundamental group of M^{n} is finite. From the theorem 3.9.1 in [p.118, 8], we obtain that the Betti number b_{i} = 0 for 0 < i < n. Therefore, M^{n} is a rational homology sphere. Let M̃ denote the universal covering of M^{n}. Because M^{n} is compact and the first fundamental group of M^{n} is finite, we know that the universal covering M̃ is also compact and its first fundamental group is also finite. Hence, from a well known result of Kuiper in [9], M̃ is diffeomorphic to a sphere and the metric is conformally equivalent to the standard metric in the sphere globally.

Next we shall consider the case that

∑_{ij} R_{ij}^{2} = \frac{r^{2}}{n − 1}  (3.7)

holds on M^{n}. From the proof of the Lemma 1, we know λ_{i} ≥ 0 for any i on M^{n}. If at some point, λ_{i} > 0 for all i holds, from the above argument, we know that the universal covering M̃ of M^{n} is diffeomorphic to a sphere. Hence, at least one of λ_{i}’s is zero. We can assume 0 = λ_{1} ≤ λ_{2} ⋯ ≤ λ_{n} on M^{n} without loss of generality. Since

∑_{i} λ_{i}^{2} = \frac{(∑_{i} λ_{i})^{2}}{n − 1} ,

we have

(n − 1) ∑_{i=2}^{n} λ_{i}^{2} = (∑_{i=2}^{n} λ_{i})^{2}.

For any nonnegative real numbers x_{i}, i = 1, ⋯, k,

(∑_{i=1}^{k} x_{i})^{2} ≤ k ∑_{i=1}^{k}(x_{i})^{2}

and equality holds if and only if x_{1} = x_{2} = ⋯ = x_{k}. This inequality and the description of the case of equality are given by the Cauchy-Schwarz inequality. Hence, we obtain λ_{2} = ⋯ = λ_{n} on M^{n}. If λ_{2} = 0 on M^{n}, then λ_{i} = 0 for all i on M^{n}. Thus, M^{n} is flat because M^{n} is locally conformally flat. Since M^{n} is compact, it is impossible. If λ_{2} ≠ 0 at some point, we know that there exists a neighborhood of p such that λ_{2} ≠ 0 because λ_{i} is a differentiable function on M^{n}. Next we shall prove that the λ_{2} = ⋯ = λ_{n} is constant.

In fact, since M^{n} is locally conformally flat, from (2.3) we have

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

Since λ_{1} = 0 and λ_{2} = ⋯ = λ_{n}, we have r = λ_{i} for i ≥ 2. From R_{ij} = λ_{i}δ_{ij}, we conclude, for any i ≥ 2, j ≥ 2, l ≥ 2 and t ≥ 2,

∇_{k}λ_{i}δ_{ij} − \frac{1}{2}∇_{k}λ_{l}δ_{ij} = ∇_{j}λ_{i}δ_{ik} − \frac{1}{2}∇_{j}λ_{t}δ_{ik}.  (3.8)

By taking k = 1, i = j = l = t ≥ 2, we have, from (3.8), ∇_{1}λ_{i} = 0. For any k ≠ i and taking i = j = l ≥ 2, we infer from (3.8) ∇_{k}λ_{i} = 0. By letting i = k = j = l ≥ 2 and t ≠ i, we obtain ∇_{i}λ_{i} = 0. Hence, for any k, we have ∇_{k}λ_{i} = 0 for i ≥ 2. Thus, we infer that λ_{i} for all i are constant. Hence, on M^{n}, λ_{i} > 0 for any i ≥ 2 is constant because λ_{i} for all i are continuous. Therefore, the scalar curvature r and the squared norm of the Ricci curvature tensor of M^{n} are constant from (3.7). From the theorem 3 due to Goldberg in [6], we know M^{n} is isometric to the Riemannian product S^{n−1}(c) × S^{1} since r ≠ 0 and (3.7) hold. Thus, we finished the proof of Theorem 2. □

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Department of Mathematics, Faculty of Science, Josai University, Saitama, Sakado 350-0295, Japan cheng@math.josai.ac.jp