Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:


* The scalar curvature at a point relates the volume of an infinitesimal ball centered at that point to the volume of the ball with the same radius in Euclidean space.
* There are no topological obstructions to negative scalar curvature.
* On a compact spin manifold of positive scalar curvature, the index of the Dirac operator vanishes (equivalently, the $\hat{A}$ genus vanishes).


The third item is of course part of a larger story - one can use higher index theory to produce more subtle positive scalar curvature obstructions (e.g. on non-compact manifolds) - but all of these variations on the compact case.


I am also aware that the scalar curvature is an important invariant in general relativity, but that is not what I want to ask about here. This is what I would like to know:


1. Are there any interesting theorems about metrics with constant scalar curvature? For example, are there topological obstructions to the existence of constant scalar curvature metrics, or are there interesting geometric consequences of constant scalar curvature?
2. Can anything be said about manifolds with scalar curvature bounds (other than the result I quoted above about spin manifolds with positive scalar curvature), analogous to the plentiful theorems about manifolds with sectional curvature bounds? (Thus additional hypotheses like simple connectedness are allowed)
3. Is anything particular known about positive scalar curvature for non-spin manifolds?



Some questions about scalar curvature
The Kazdan-Warner theorem goes a long way toward answering the first and second questions.


(For notes typed up by Kazdan, see <http://www.math.upenn.edu/~kazdan/japan/japan.pdf>.)


Here's what is says (taken almost verbatim from the notes, page 93): Divide the class of all closed manifolds (edit: of dimension > 2. See comments) into 3 types:


I. Those which admit a metric of nonnegative scalar curvature which is positive somewhere.


II. Those which don't but admit a metric of 0 scalar curvature.


III. All other closed manifolds.


The theorem is that if $M$ is in class I, then any $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric.


If $M$ is in class II, then $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric iff it's identically 0 or negative somewhere.


If M is in class III, then $f:M\rightarrow\mathbb{R}$ is the scalar curvature of some metric iff it's negative somewhere.


In particular, every closed manifold has a metric of constant negative scalar curvature. Those in class I or II have a metric of 0 scalar curvature, and those in class I have a metric of constant positive scalar curvature.


