curvature of $S^p \times S^q$ is not constant

Question

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?

Answer 1

You're missing a part of the statement. This is the statement in the book (Petersen's Riemannian Geometry, third edition):

Corollary $5.6.12.$ If $M$ is a closed simply connected manifold with constant curvature $k$, then $k > 0$ and $M = S^n$. Thus, $S^p\times S^q$, $\mathbb{CP}^n$ do not admit any constant curvature metrics.

Now the conclusion follows immediately as $S^p\times S^q$ (for $p, q > 1$) and $\mathbb{CP}^n$ are closed simply connected manifolds which are not $S^n$.

If $M = S^1\times S^1$, then $M$ admits a metric of constant curvature zero. For $M = S^p\times S^q$ with $p = 1$ or $q = 1$ but not both, then as you said, the universal cover is not a space form so $M$ does not admit a constant curvature metric.