Whether there is a Riemannian metric on $S^2\times S^2$ with positive scalar curvature?

Question

Whether there is a Riemannian metric on $S^2\times S^2$ with positive scalar curvature ?

If there is a such metric, how make the section curvature is positive under some suitable flow ?

Answer 1

Let $g$ be the standard metric on $\mathbb S^2$. Then the product metric $g\times g$ on $\mathbb S^2 \times \mathbb S^2$ has positive scalar curvature (and non-negative sectional curvatures). Indeed pick any $g_1, g_2$ with positive curvatures would do the job.

I hope you have luck finding such a flow, as that would prove (or disprove?) the Hopf conjecture.

Some references: Survey by W. Ziller, survey by B. Wilking