A multi-parameter family of metrics on stiefel manifolds and applications

Markus Schlarb; Markus Schlarb

doi:10.3934/jgm.2023008

Journal of Geometric Mechanics

2023, Volume 15, Issue 1: 147-187. doi: 10.3934/jgm.2023008

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A multi-parameter family of metrics on stiefel manifolds and applications

Markus Schlarb ^,

Institute of Mathematics, Julius-Maximilians-Universität Würzburg, Germany

Received: 16 November 2022 Revised: 11 January 2023 Accepted: 16 January 2023 Published: 01 February 2023
53C30, 53B30, 70H03

The real (compact) Stiefel manifold realized as set of orthonormal frames is considered as a pseudo-Riemannian submanifold of an open subset of a vector space equipped with a multi-parameter family of pseudo-Riemannian metrics. This family contains several well-known metrics from the literature. Explicit matrix-type formulas for various differential geometric quantities are derived. The orthogonal projections onto tangent spaces are determined. Moreover, by computing the metric spray, the geodesic equation as an explicit second order matrix valued ODE is obtained. In addition, for a multi-parameter subfamily, explicit matrix-type formulas for pseudo-Riemannian gradients and pseudo-Riemannian Hessians are derived. Furthermore, an explicit expression for the second fundamental form and an explicit formula for the Levi-Civita covariant derivative are obtained. Detailed proofs are included.
- constrained Lagrangian systems,
- pseudo-Riemannian gradients,
- pseudo-Riemannian Hessians,
- pseudo-Riemannian submanifolds,
- Riemannian optimization,
- second fundamental form,
- sprays,
- Stiefel manifolds
Citation: Markus Schlarb. A multi-parameter family of metrics on stiefel manifolds and applications[J]. Journal of Geometric Mechanics, 2023, 15(1): 147-187. doi: 10.3934/jgm.2023008

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Abstract

The real (compact) Stiefel manifold realized as set of orthonormal frames is considered as a pseudo-Riemannian submanifold of an open subset of a vector space equipped with a multi-parameter family of pseudo-Riemannian metrics. This family contains several well-known metrics from the literature. Explicit matrix-type formulas for various differential geometric quantities are derived. The orthogonal projections onto tangent spaces are determined. Moreover, by computing the metric spray, the geodesic equation as an explicit second order matrix valued ODE is obtained. In addition, for a multi-parameter subfamily, explicit matrix-type formulas for pseudo-Riemannian gradients and pseudo-Riemannian Hessians are derived. Furthermore, an explicit expression for the second fundamental form and an explicit formula for the Levi-Civita covariant derivative are obtained. Detailed proofs are included.

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