In the paper Conformal Deformation of a Riemannian metric to a constant scalar curvature of Richard Schoen (J. Differential Geom. 20(2) (1984) 479-495, doi:10.4310/jdg/1214439291), in the first page, it says that
"Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of $S^1\times S^{n-1}$."
My question is: Is there any other conformally flat manifold with positive scalar curvature which is not in these forms? Or the manifolds Schoen mentioned exhaust the list of all conformally flat manifolds of positive scalar curvature?
Best Answer
Products $M^m \times S^{n-m}$ will be conformally flat, where $M^m$ is a compact manifold of curvature $-1$ and $S^{n-m}$ has curvature $1$. If $n>2m$ then the scalar curvature of the product will be positive (positive curvature of the sphere dominates the negative curvature of the hyperbolic manifold, so the scalar curvature of the product will be positive). You can also (frequently) deform these metrics a bit so that they no longer (locally) split as products and still have positive scalar curvature and remain conformally flat. Thus, you have counter-examples in all dimensions $n\ge 5$. See e.g. this paper by R.Mazzeo and N.Smale http://intlpress.com/JDG/archive/1991/34-3-581.pdf for further discussion.