[Paper Review] On conformally flat manifolds with constant positive scalar curvature

Giovanni Catino|arXiv (Cornell University)|Aug 5, 2014

Geometric Analysis and Curvature Flows6 references4 citations

TL;DR

This paper classifies compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature under an optimal $L^{n/2}$-type integral pinching condition on the traceless Ricci tensor. It proves that such manifolds are isometrically covered by either the round sphere $\SS^n$, the product $\SS^1 \times \SS^{n-1}$ with the product metric, or a rotationally symmetric Derdziński metric on $\SS^1 \times \SS^{n-1}$, establishing a rigidity result via curvature estimates and Codazzi tensor analysis.

ABSTRACT

We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $\mathbb{S}^{n}$ with the round metric, $\mathbb{S}^{1} imes \mathbb{S}^{n-1}$ with the product metric or $\mathbb{S}^{1} imes \mathbb{S}^{n-1}$ with a rotationally symmetric Derdziński metric.

Motivation & Objective

• To classify compact conformally flat $n$-dimensional Riemannian manifolds with constant positive scalar curvature under a sharp integral pinching condition.
• To identify the precise geometric structures that arise when equality holds in the proposed integral inequality.
• To extend rigidity results for conformally flat manifolds by incorporating curvature estimates involving the traceless Ricci tensor.
• To characterize the role of harmonic curvature and Codazzi tensors in classifying such manifolds.
• To unify known examples—round sphere, product $\SS^1 \times \SS^{n-1}$, and Derdziński metrics—under a single optimal pinching condition.

Proposed method

• Derives a pointwise curvature inequality using the Weitzenböck formula for the traceless Ricci tensor $E$ on conformally flat manifolds.
• Applies a weighted cutoff function $f_\varepsilon$ to handle the singularity when $|E| = 0$, enabling the use of integration by parts.
• Uses the conformal flatness condition to express the Riemann curvature tensor in terms of $E$ and scalar curvature $R$, simplifying curvature contractions.
• Employs the sharp inequality $E_{ij}E_{ik}E_{jk} \geq -\frac{n-2}{\sqrt{n(n-1)}}|E|^3$ to bound the curvature term from below.
• Takes the limit $\varepsilon \to 0$ to obtain the integral inequality $\int_M |E|^{\frac{n-2}{n}}\left(R - \sqrt{n(n-1)}|E|\right) \leq 0$.
• Applies the classification result of Derdziński (Theorem 3.2) for manifolds with harmonic curvature and two distinct Ricci eigenvalues to conclude the classification.

Experimental results

Research questions

• RQ1 What compact conformally flat $n$-manifolds with constant positive scalar curvature satisfy an optimal integral pinching condition on the traceless Ricci tensor?
• RQ2 Which geometric structures arise when equality holds in the derived integral inequality?
• RQ3 How do Codazzi tensors and harmonic curvature constrain the global geometry of such manifolds?
• RQ4 Can the class of rotationally symmetric Derdziński metrics be characterized via a curvature pinching condition?
• RQ5 What is the role of eigenvalue multiplicity in the Ricci tensor for classifying conformally flat manifolds with positive scalar curvature?

Key findings

• The integral inequality $\int_M |E|^{\frac{n-2}{n}}\left(R - \sqrt{n(n-1)}|E|\right) \leq 0$ holds for all compact conformally flat $n$-manifolds with constant positive scalar curvature.
• Equality holds in the integral inequality if and only if the manifold is isometrically covered by $\SS^n$ with the round metric, $\SS^1 \times \SS^{n-1}$ with the product metric, or $\SS^1 \times \SS^{n-1}$ with a rotationally symmetric Derdziński metric.
• When $E \equiv 0$, the manifold is Einstein and, due to conformal flatness, has constant positive sectional curvature, hence is covered by $\SS^n$.
• When $E$ has eigenvalues of multiplicity $n-1$ and $1$ at each point, and the Ricci tensor is not parallel, the manifold is covered by a Derdziński metric on $\SS^1 \times \SS^{n-1}$.
• The classification is sharp: the pinching condition is optimal and characterizes the three model geometries.
• The result establishes a rigidity phenomenon where curvature estimates interpolate between Einstein and non-Einstein cases, with equality characterizing specific symmetric structures.