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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?

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  • $\begingroup$ $S^2\times S^1$ is not simply connected. $\endgroup$ Jul 15, 2018 at 10:23

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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.

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  • $\begingroup$ Yes, my typo, but i still cannot get it. $\endgroup$
    – Allotrios
    Jul 15, 2018 at 10:32
  • $\begingroup$ @Allotrios: I have expanded my answer. $\endgroup$ Jul 15, 2018 at 10:33

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