TITLE: Corollary to Preissman's theorem
QUESTION [7 upvotes]: Preismann's theorem states (ref. Petersen's "Riemannian Geometry", chapter 6):

On a compact manifold with negative sectional curvature, any abelian subgroup of the fundamental group is cyclic.

A corollary is stated:

No compact product manifold admits a metric with negative curvature.

I don't understand how the corollary follows from the thoerem. Of course it would if, on a product manifold, there are always non-cyclic abelian subgroups of the fundamental group. Is this true? (It seems like it shouldn't be, taking for example the product of simply connected manifolds.)

REPLY [4 votes]: Consider a product $M\times N$ with negative sectional curvature. 
If $M$ has nonnegative sectional curvature, it would be a contradiction.  
So abelian subgroup of fundamental group of $M$ is $0$ or ${\bf Z}$, since it has no torsion. 
Hence $\pi_1(M\times N)=\pi_1(M)\times \pi_1(N)$ has abelian subgroup in ${\bf Z}^2$. 
From given theorem, so abelian subgroup of 
at least one factor of $M\times N$, say $M$, is $0$. Hence it is simply connected so that $M={\bf R}^n$. It contradicts to compactness.