why positive scalar curvature manifolds

Question

I am studying scalar curvature and I have seen that many mathematicians studied obstruction against positive scalar curvature (for example Stolz, Schick, Roe, J. Rosenberg, Hanke and many others). Unfortunately, I couldn't find out why it is important that a manifolds has a positive scalar curvature metric. Does anyone knows why?

Answer 1

There is a result by Kazdan & Warner which classifies which functions can be scalar curvatures of some metric. The result states that compact manifolds of dimension 3 or greater come in three varieties:

1. There is no restriction on the possible scalar curvatures; every function is the scalar curvature of some metric.
2. A function is the scalar curvature of some metric if and only if it is negative somewhere.
3. A function is the scalar curvature of some metric if and only if it is either identically zero or negative somewhere.

In particular on every compact manifold, any negative function is the scalar curvature of some metric, but there are certain manifolds, the second and third types above, which do not admit any metric of strictly positive scalar curvature. So negative scalar curvature is not very interesting, but positive scalar curvature is a nontrivial restriction! Nonnegative also arises naturally in general relativity, where any spacetime with nonnegative local mass density (physically a very reasonable condition) has nonnegative scalar curvature. Furthermore, if we foliate a spacetime with maximal slices (spacelike submanifolds with vanishing mean curvature), then if the slices satisfy the dominant energy condition, they must also have nonnegative scalar curvature.

As a final note I'll make a slightly more advanced remark. We can be more precise, at least in certain cases, about the restriction from Kazdan-Warner mentioned above (in fact this result predates their work). Let $M$ be a smooth manifold whose second Stiefel-Whitney class vanishes (this is a purely topological condition). If $M$ admits a metric with positive scalar curvature, then its "A genus" must vanish, a stronger topological condition. This was proved by Lichnerowicz by using a Bochner technique for a differential operator (the "Dirac operator") on sections of a certain vector bundle (the "spinor bundle") over $M$ which we can be sure exists due to the vanishing of the second Stiefel-Whitney class. This implies that the analytical index of the Dirac operator is zero, which in turn implies by the Atiyah-Singer index theorem that the topological index of the manifold is zero, which gives the conclusion.

Hope this helps.