  Methods and algorithms that work with data on nonlinear manifolds are
collectively summarized 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. 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. Under the Euclidean metric, the maximum was
conjectured to be at 1. For both 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 for
the Stiefel manifold that the global bounds of 5/4 (canonical metric) and 1
(Euclidean metric) hold 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 under the canonical metric, the global curvature maximum is attained
at tangent plane sections that are spanned by rank-two matrices, while the
extreme curvature cases of the Euclidean Stiefel manifold occur for rank-one
matrices. Numerical examples are included for illustration purposes.
