Methods and algorithms that work with data on nonlinear manifolds are collectively summarised under the term `Riemannian computing'. In practice, curvature can be a key limiting factor for the performance of Riemannian computing methods. Yet, curvature can also be a powerful tool in the theoretical analysis of Riemannian algorithms. In this work, we investigate the sectional curvature of the Stiefel and Grassmann manifold from the quotient space view point. On the Grassmannian, tight curvature bounds are known since the late 1960ies. On the Stiefel manifold under the canonical metric, it was believed that the sectional curvature does not exceed 5/4. For both of these manifolds, the sectional curvature is given by the Frobenius norm of certain structured commutator brackets of skew-symmetric matrices. We provide refined inequalities for such terms and pay special attention to the maximizers of the curvature bounds. In this way, we prove that the global bound of 5/4 for Stiefel holds indeed. With this addition, a complete account of the curvature bounds in all admissible dimensions is obtained. We observe that `high curvature means low-rank', more precisely, for the Stiefel and Grassmann manifolds, the global curvature maximum is attained at tangent plane sections that are spanned by rank-two matrices. Numerical examples are included for illustration purposes.

翻译：处理非线性流形上数据的方法与算法被统称为“黎曼计算”。在实践中，曲率是限制黎曼计算方法性能的关键因素。同时，曲率也能成为黎曼算法理论分析中的有力工具。本文从商空间视角研究了Stiefel流形与Grassmann流形的截面曲率。对于Grassmann流形，自20世纪60年代末起其紧致曲率界便已为人所知；而关于标准度量下的Stiefel流形，此前研究者认为其截面曲率不超过5/4。这两类流形的截面曲率均由特定结构化的反对称矩阵交换子括号的Frobenius范数定义。我们对此类项给出了精细不等式，并特别关注了曲率界的极大值点。通过这一方法，我们严格证明了Stiefel流形5/4的全局上界确实成立。基于此补充，我们获得了所有允许维度下曲率界的完整刻画。进一步发现“高曲率意味着低秩”——具体而言，对Stiefel与Grassmann流形，全局曲率最大值出现在由秩为二的矩阵张成的切平面截面上。文中附有数值示例以作说明。