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Journal of Lie Theory 32 (2022), No. 2, 563--600
Copyright Heldermann Verlag 2022

Curvatures of Stiefel Manifolds with Deformation Metrics

Du Nguyen
1 Gardiner Street, Darien, CT 06820, U.S.A.

We compute curvatures of a family of metrics on Stiefel manifolds, introduced recently by Hüper, 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.

Keywords: Lie group, homogeneous space, optimization, Riemannian geometry, Riemannian curvature, Einstein manifold, Stiefel manifold.

MSC: 22E70, 53C30, 17B81, 65K10, 49Q12, 53C25, 68T05.

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