Scalar-flat closed manifolds not admitting positive scalar curvature metrics
In this article the author uses the results of Gromov-Lawson and S. Stolz to characterize (in dimensions \(\geq 5\)) those closed connected manifolds that admit a Riemannian metric of nonnegative scalar curvature but no metric of positive scalar curvature. Let \((Z)\) denote the class of such manifolds. Let (P) (respectively (N)) denote the classes of closed, connected manifolds that admit a metric of positive scalar curvature (respectively, admit no metric of nonnegative scalar curvature). The author observes that if \(M\) is a manifold belonging to (Z), then any metric on \(M\) with nonnegative scalar curvature must have scalar curvature identically zero. The author also points out that if \(M\) is a closed spin manifold with infinite fundamental group and \(\widehat{A}(M) \neq 0\), then \(M\) belongs to \((N)\) by the work of K. Ono and V. Mathai. The main result of this article is the following Theorem. Let \(M\) be a closed simply connected manifold with \(\dim M \geq 5\). Then \(M\) belongs to the class (Z) if and only if \(M\) is the Riemannian product of manifolds \(M_ 1 \times \dots \times M_ \ell\) such that (1) For all \(i\), \(\pm M_ i\) admits a Ricci-flat Kähler metric or a Riemannian metric with \(Spin(7)\) holonomy (in both cases \(M_ i\) is necessarily spin) (2) The Lichnerowicz-Hitchin obstruction \(\alpha(M)\) is nonzero. The author also obtains the following Corollary. Let \(M^ n\) be a connected closed manifold with finite fundamental group such that its universal cover is spin. If \(\# \pi_ 1(M)\cdot | \widehat{A}(M)| > 2^{n/4}\), then \(M\) does not admit a metric of nonnegative scalar curvature. A key ingredient in the proof of the Theorem is the observation that if a spin manifold \(M\) admits a parallel spinor, then it has special holonomy.
scientific article; zbMATH DE number 447092