Curvatures of Stiefel manifolds with deformation metrics
This work provides foundational mathematical insights into the geometry of Stiefel manifolds, which are incremental but important for applications in optimization and machine learning.
The authors computed curvatures for a family of metrics on Stiefel manifolds, identifying parameter values that make these manifolds Einstein and analyzing sectional curvature ranges, including exact results for matrices with 2 columns.
We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H{ü}per, Markina and Silva Leite, which includes the well-known embedded and canonical metrics on Stiefel manifolds as special cases. The metrics could be identified with the Cheeger deformation metrics. We identify parameter values in the family to make a Stiefel manifold an Einstein manifold and show Stiefel manifolds always carry an Einstein metric. We analyze the sectional curvature range and identify the parameter range where the manifold has non-negative sectional curvature. We provide the exact sectional curvature range when the number of columns in a Stiefel matrix is $2$, and a conjectural range for other cases. We derive the formulas from two approaches, one from a global curvature formula derived in our recent work, another using curvature formulas for left-invariant metrics. The second approach leads to curvature formulas for Cheeger deformation metrics on normal homogeneous spaces.