Using the 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 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.

中文翻译：

---

在简单连接的 5 流形上具有正双正交曲率的黎曼度量的存在

使用 Bettiol 的最近工作，我们表明，$$S^2\times S^3$$ 上几乎正截面曲率的 Wilking 度量的一阶共形变形产生了一系列具有严格正截面曲率平均值的度量在 2-Grassmanian 中以最小距离分隔的任何一对 2-平面的。Smale 的结果使我们能够得出结论，每个具有无扭转同源性和微不足道的第二 Stiefel-Whitney 类的闭合简单连接的 5-流形都承认黎曼度量具有任何一对正交 2-平面的截面曲率的严格正平均值。