Curvatures of Stiefel manifolds with deformation metrics

Abstract

We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H{\"u}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.