Sp(2)/U(1) and a positive curvature problem

作者：Xu Ming; Wolf Joseph A^*

来源：Differential Geometry and its Applications, 2015, 42: 115-124.

DOI：10.1016/j.difgeo.2015.08.002

• Riemannian homogeneous space
• Sectional curvature
• Positively curved for commuting pairs

A compact Riemannian homogeneous space G/H, with a bi-invariant orthogonal decomposition g = h + 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 m. In this paper, we will prove that on the coset space Sp(2)/U(1), in which U(1) corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this Sp(2)/U(1) cannot be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.

• 出版日期2015-10
• 单位天津师范大学