On the topology of scalar-flat manifolds

Details On the topology of scalar-flat manifolds On the topology of scalar-flat manifolds

2000-06-20

arxiv.org/abs/math/0006149v3

Bull. London Math. Soc. 33, (2001), pp. 203-209

Mathematics

Differential Geometry

revised version of preprint 45, SFB 478, Muenster; to appear in Bulletin of the LMS; 10 pages; no figures; MSC added

Scientific paper

Let $M$ be a simply-connected closed manifold of dimension $\geq 5$ which does not admit a metric with positive scalar curvature. We give necessary conditions for $M$ to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of $M$. As a consequence any simply-connected scalar-flat manifold of dimension $\geq 5$ with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.

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