Positive biorthogonal curvature on $S^2\times S^2$

Abstract:

We prove that $S^2\times S^2$ satisfies an intermediate condition between $\operatorname {Ric}>0$ and $\operatorname {sec}>0$. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the $2$-Grassmannian, is strictly positive. This can be done with an arbitrarily small lower bound on the distance between the planes considered. Although they have positive Ricci curvature, these metrics do not have nonnegative sectional curvature. Such metrics also have positive biorthogonal curvature, meaning that the average of sectional curvatures of any two orthogonal planes is positive.