Existence of Riemannian metrics with positive biorthogonal curvature on simply connected 5-manifolds

We show that the closed simply connected 5-manifold S 3 × S 2 admits Riemannian metrics with strictly positive averages of sectional curvatures of any 2-planes tangent at a given point and which are separated by the smallest distance in the Grassmanian of 2-planes. These metrics have positive Ricci curvature yet there are 2-planes of negative sectional curvature. We use these metrics to show that every closed connected simply connected 5-manifold with vanishing second Stiefel-Whitney class and torsion-free homology admits a Riemannian metric with strictly positive average of sectional curvatures of any pair of orthogonal 2-planes. We show that the symmetric space metric on the Wu manifold satisfies such lower curvature bound.