On a problem of Hopf (Q5942023)

The question of Hopf is, whether on the product \(S^2\times S^2\) of two 2-dimensional spheres metrics of strongly positive curvature exist. The paper contains two theorems and two lemmas and a series of concrete and interesting results. Two theorems are proved, asserting, the existence of points and planes of vanishing curvature. From our point of view of special interest is theorem 2: Suppose that \(\alpha\) and \(\beta\) are two regular functions on the topological product \(M=S_1^2\times S^2_2\) of two spheres, that \(d\sigma^2=ds^2_1+ds^2_2\) is the Riemannian product of two metrics \(ds^2_i\) on \(S^2_i\) \((i=1,2)\) and that \(ds^2=d\sigma^2+\alpha^2d\beta^2\). Then, there exist a point \(x\in M\) and a plane \(\pi\in T_xM\) such that the sectional curvature of \(ds^2\) for this plane vanishes: \(K_x(\pi)=0\). The list of references has seven items.