Some compact conformally flat manifolds with non-negative scalar curvature
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 case that \(M\) has finite fundamental group. If \(M\) has infinite fundamental group, then the result follows from the nonnegative Ricci curvature splitting theorem of Cheeger and Gromoll together with a characterization of conformal flatness due to Kulkarni. The proof of Theorem 2 uses the Berger Holonomy Theorem together with a characterization of locally symmetric Riemannian manifolds that are locally conformally flat. The results on 4-manifolds follow from special properties of the curvature tensor in dimension 4.
scientific article; zbMATH DE number 417123
scientific article