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 the above statement not proved ?

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 ?