TITLE: Positively curved metrics on $S^2\times S^2$
QUESTION [6 upvotes]: As you know, the Hopf conjecture is about the existence of positively curved metric on $S^2\times S^2$. Hsiang-Kleiner have shown that there exists no positively curved metric admitting $S^1$-action on $S^2\times S^2$. 
My question is simple. If $(S_1=S^2,g)$ and $(S_2=S^2,h)$ are positively curved, 
 then for any positive function $f: S_1 \rightarrow \mathbb{R}$, is a warped metric $g+ fh$ not positively curved ? Or is this statement not proved ?

REPLY [5 votes]: An alternative proof of this result is to realize that the injection into the first factor $i:S^2\to S^2\times S^2$ gives you a "soul" of the warped product. By that I mean that the composition of the Riemannian submersion (for the warped metric) $S^2\times S^2\to S^2, (p,q)\to p$ with $i$ provides you with an analog of the Sharafutdinov map for open nonnegatively curved manifolds. Then you can adapt the proof of Perelman's proof of the soul conjecture to rule out positive curvature, as well as gain some rigidity for nonnegative curvature. The details appear in the last section of (sorry for the shameless self-promotion)
V. Berestovskii and L. Guijarro, A metric characterization of Riemannian submersions, Annals of Global Analysis and Geometry 18 (2000), pp.577-588