\(\mathrm{Sp}(2) / \mathrm{U}(1)\) and a positive curvature problem (Q745031)

A compact Riemannian homogeneous space \(G/H\), with a bi-invariant orthogonal decomposition \(\mathfrak{g}=\mathfrak{h}+\mathfrak{m}\) is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in \(T_{eH}(G/H)\) spanned by a linearly independent commuting pair in \(\mathfrak{m}\). The authors prove that the coset space \(\mathrm{Sp}(2) / \mathrm{U}(1)\), in which \(\mathrm U(1)\) corresponds to a short root, admits positively curved metrics for commuting pairs. This coset space was in the list of \textit{L. Bérard Bergery}'s classification of odd-dimensional positively curved Riemannian homogeneous spaces [J. Math. Pures Appl. (9) 55, 47--68 (1976; Zbl 0289.53037)], but B. Wilking recently proved that \(\mathrm{Sp}(2) / \mathrm{U}(1)\) cannot be positively curved in the general sense. His unpublished result and his proof are given in Theorem 5.1 of the present paper. Therefore, this is the first example distinguishing the set of compact coset spaces admitting positively curved metrics, from that for metrics positively curved only for commuting pairs.