Curvatures of Stiefel Manifolds with Deformation Metrics

We compute curvatures of a family of metrics on Stiefel manifolds, introduced recently by Hper, Markina and Silva Leite.We derive the formulas from two approaches, one using curvature formulas for left-invariant metrics on homogeneous spaces, computed for the case of Cheeger/Jensen deformation metrics of a quotient space of a compact Lie group; another from a global curvature formula derived in our recent work.Allowing more than one deformation parameter, we compute Ricci curvature for a large family of diagonal metrics explicitly and obtain new Einstein metrics.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 expect the method developed here generalizes to other homogeneous spaces.