From Petersen book Corollary 5.6.12:
If $M$ is closed simply connected manifold with constant curvature $k$ then $k>0$ and $M=S^n$. Thus, $S^p \times S^q$, $CP^n$ do not admit any constant curvature metrics.
How is the second part of the Corollary proved?
I mean with $S^2 \times S^1$ which has $S^2 \times R$ as a universal covering space i understand it, since $S^2 \times R$ is not a space form $S^n_k$, but what about the other cases?