Some compact conformally flat manifolds with non-negative scalar curvature
type: Item
instance of: scholarly article
title: Some compact conformally flat manifolds with non-negative scalar curvature
zbMATH Open document ID: 0792.53035
DOI: 10.1007/BF01263660
author: M. Helena Noronha
published in: Geometriae Dedicata
publication date: 1993-09-30T00:00:00Z
review text: A manifold \(M\) is locally conformally flat if \(M\) can be covered by coordinate charts \(\{U_ \alpha,\varphi_ \alpha\}\) whose overlap maps are conformal diffeomorphisms between open sets in Euclidean space. A Riemannian metric \(g\) on \(M\) is compatible with the conformally flat structure if the coordinate maps \(\{\varphi_ \alpha :U_ \alpha\to \varphi_ \alpha(U_ \alpha) \subseteq \mathbb{R}^ n\}\) are conformal. In this article the author studies locally conformally flat manifolds with a compatible metric of nonnegative scalar curvature. The two main results are Theorem 1. Let \(M^ n\), \(n \geq 3\), be a compact, locally conformally flat manifold with nonnegative Ricci curvature. Then one of the following holds: a) \(M\) is conformally equivalent to a quotient of the standard sphere \(S^ n\) by a group of fixed point free conformal diffeomorphisms b) \(M\) is a quotient of \(S^{n-1} \times \mathbb{R}\) or \(\mathbb{R}^ n\) with the standard metrics by a group of fixed point free isometries. Theorem 2. For \(k > 1\) let \(M\) be a \(2k\)-dimensional compact, locally conformally flat manifold with nonnegative scalar curvature. Then either the \(k\)-th Betti number is zero or \(M\) is covered by \(\mathbb{R}^{2k}\) or \(S^ k \times H^ k\), where \(S^ k\) and \(H^ k\) denote the sphere and the real hyperbolic space with the standard metrics of constant sectional curvature. The author also obtains some results for 4-dimensional compact, locally conformally flat manifolds whose scalar curvature is identically zero: a) \(M\) has negative Euler characteristic unless \(M\) is flat b) The Riemannian connection associated to this metric is an absolute minimum for the Yang-Mills functional. c) If the largest eigenvalue of the Ricci operator is no larger than the absolute value of the smallest eigenvalue, then \(M\) is locally symmetric and covered by or \(\mathbb{R}^ 4\) or \(S^ 2\times H^ 2\) with the standard metrics. For Theorem 1 one uses a result of Kuiper in the c…
reviewed by: Patrick Eberlein
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 417123
zbMATH Keywords: conformally flat manifolds
zbMATH Keywords: nonnegative scalar curvature
zbMATH Keywords: Betti number
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Yang-Mills functional
zbMATH Keywords: Ricci operator
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01263660
OpenAlex ID: W1978361633
description: scientific article; zbMATH DE number 417123