# Curvatures of Stiefel manifolds with deformation metrics

arXiv: https://arxiv.org/abs/2105.01834 · HTML: https://mlsanity.com/paper/2105.01834/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 16 references that are in the map.
- First posted on arXiv 5 May 2021.

## Builds on (most cited first, 2 of 2)

- [Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics](https://mlsanity.com/paper/2009.10159/) (2020)
- [Riemannian Geometry with differentiable ambient space and metric operator](https://mlsanity.com/paper/2105.01583/) (2021)

## Built on by (most cited first, 1 of 1)

- [Parallel transport on matrix manifolds and Exponential Action](https://mlsanity.com/paper/2408.06054/) (2024)

## Abstract (arXiv)

We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H"uper, 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.