# Metric of non-negative scalar curvature

Let $$(M,g)$$ be a closed Riemannian manifold, if $$Scal^g>0$$, we know that in a metric space of $$M$$, there is a neighborhood of $$g$$, such that all metrics in this neighborhood have the positive scalar curvature. In other words, we can choose a generic metric with P.S.C.-property.

Q If we replace the PSC with Non-negative scalar curvature, can we get a similar property? Is there any/some research about such a question?

• Similar, maybe. Exactly the same, no. Just look at the two dimensional case with the flat metric on $\mathbb{T}^2$ and apply Gauss-Bonnet. – Willie Wong Oct 12 '18 at 15:12
• Expanding on Willie's comment: there is work of Schoen--Yau and Gromov--Lawson on obstructions to the existence of a metric with positive scalar curvature. One consequence is that if a metric on $\mathbb{T}^n$ has nonnegative scalar curvature, then that metric is flat. So, at least in certain cases, "nonnegative scalar curvature" is a rigid property of a metric. – Jeffrey Case Oct 12 '18 at 17:38

In general, there's probably no hope of getting anything too similar for non-negative curvature. Willie Wong gave an excellent example of how the Gauss-Bonnet restricts the scalar curvature (a deformation of the flat metric on the torus must have negative scalar curvature unless it is totally flat). In fact, a small deformation of a manifold with non-negative sectional curvature might not even have non-negative scalar curvature. To see this, take a round 2-sphere and flatten its north pole until it becomes flat at that pole. This can be done so that the induced metric $$g$$ on this flattened sphere has positive sectional curvature everywhere except for the north pole. If we then consider a small conformal deformation of the metric $$\tilde g= e^{2f} g$$, then $$\tilde g$$ has negative scalar curvature at the north pole whenever $$\triangle\left( e^{(n-2)f/2} \right) <0$$ at the north pole. There is nothing special about conformal deformations here, they just allow for easy computation of the scalar curvature.