Stolz's theorem

Statement

Let $M$ be a differential manifold. If $M$ admits a Riemannian metric which admits a spin structure, then $M$ also admits a Riemannian meteric of positive scalar curvature if and only if $\alpha (M)=0$.

Here, $\alpha (M)$ is defined as Fill this in later

Relation with other results

Gromov-Lawson theorem

Further information: Gromov-Lawson theorem on spin structure

The Gromov-Lawson theorem on spin structure, which is a corollary of their theorem on surgery, says that if $M$ does not have a Riemannian metric admitting a spin structure, then $M$ must have a Riemannian metric of positive scalar curvature.

References

• Simply connected manifolds of positive scalar curvature by Stephan Stolz, The Annals of Mathematics, 2nd Ser., Vol. 136, No. 3 (Nov., 1992), pp. 511-540
• Positive scalar curvature with symmetry by Bernhard Yanke