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

TL;DR

This paper demonstrates that all simply connected 5-manifolds with torsion-free homology and trivial second Stiefel-Whitney class admit Riemannian metrics with positive biorthogonal curvature, extending curvature existence results.

Contribution

It proves the existence of metrics with positive biorthogonal curvature on a broad class of simply connected 5-manifolds, using conformal deformations and recent geometric results.

Findings

Existence of metrics with positive biorthogonal curvature on all such 5-manifolds.

Application of conformal deformation of Wilking's metric.

Use of Smale's theorem to classify manifolds with the curvature property.

Abstract

Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale's allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel-Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes.