Positive biorthogonal curvature on S^2 x S^2

TL;DR

This paper constructs metrics on S^2 x S^2 with positive biorthogonal curvature, an intermediate condition between positive Ricci and sectional curvature, showing new geometric properties of this manifold.

Contribution

It introduces metrics with positive biorthogonal curvature on S^2 x S^2, bridging the gap between positive Ricci and sectional curvature conditions.

Findings

Existence of metrics with positive biorthogonal curvature on S^2 x S^2.

Metrics can have arbitrarily small separation between planes while maintaining positivity.

Such metrics have positive Ricci curvature but not nonnegative sectional curvature.

Abstract

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. 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; and 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.