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On the Injectivity Radius of the Stiefel Manifold: Numerical investigations and an explicit construction of a cut point at short distance

Jakob Stoye, Ralf Zimmermann

TL;DR

The paper addresses the injectivity radius of the Stiefel manifold under the canonical metric, a key quantity determining unique geodesic-based shortest paths. It combines a curvature-based bound with Jacobi-field analysis to locate cut points, and it provides an explicit cut-point construction on $St(4,2)$ where the first conjugate point occurs at $t\approx 2.8691$. Numerical experiments indicate the actual injectivity radius is near $2.869$, rather than the curvature bound $\approx 2.81$, suggesting conjugate points govern sharpness. The results corroborate conjectures about the canonical metric and deepen understanding of Stiefel geometry, with code available for reproducibility.

Abstract

Arguably, geodesics are the most important geometric objects on a differentiable manifold. They describe candidates for shortest paths and are guaranteed to be unique shortest paths when the starting velocity stays within the so-called injectivity radius of the manifold. In this work, we investigate the injectivity radius of the Stiefel manifold under the canonical metric. The Stiefel manifold $St(n,p)$ is the set of rectangular matrices of dimension $n$-by-$p$ with orthogonal columns, sometimes also called the space of orthogonal $p$-frames in $\mathbb{R}^n$. Using a standard curvature argument, Rentmeesters has shown in 2013 that the injectivity radius of the Stiefel manifold is bounded by $\sqrt{\frac{4}{5}}π$. It is an open question, whether this bound is sharp. With the definition of the injectivity radius via cut points of geodesics, we gain access to the information of the injectivity radius by investigating geodesics. More precisely, we consider the behavior of special variations of geodesics, called Jacobi fields. By doing so, we are able to present an explicit example of a cut point. In addition, since the theoretical analysis of geodesics for cut points and especially conjugate points as a type of cut points is difficult, we investigate the question of the sharpness of the bound by means of numerical experiments.

On the Injectivity Radius of the Stiefel Manifold: Numerical investigations and an explicit construction of a cut point at short distance

TL;DR

The paper addresses the injectivity radius of the Stiefel manifold under the canonical metric, a key quantity determining unique geodesic-based shortest paths. It combines a curvature-based bound with Jacobi-field analysis to locate cut points, and it provides an explicit cut-point construction on where the first conjugate point occurs at . Numerical experiments indicate the actual injectivity radius is near , rather than the curvature bound , suggesting conjugate points govern sharpness. The results corroborate conjectures about the canonical metric and deepen understanding of Stiefel geometry, with code available for reproducibility.

Abstract

Arguably, geodesics are the most important geometric objects on a differentiable manifold. They describe candidates for shortest paths and are guaranteed to be unique shortest paths when the starting velocity stays within the so-called injectivity radius of the manifold. In this work, we investigate the injectivity radius of the Stiefel manifold under the canonical metric. The Stiefel manifold is the set of rectangular matrices of dimension -by- with orthogonal columns, sometimes also called the space of orthogonal -frames in . Using a standard curvature argument, Rentmeesters has shown in 2013 that the injectivity radius of the Stiefel manifold is bounded by . It is an open question, whether this bound is sharp. With the definition of the injectivity radius via cut points of geodesics, we gain access to the information of the injectivity radius by investigating geodesics. More precisely, we consider the behavior of special variations of geodesics, called Jacobi fields. By doing so, we are able to present an explicit example of a cut point. In addition, since the theoretical analysis of geodesics for cut points and especially conjugate points as a type of cut points is difficult, we investigate the question of the sharpness of the bound by means of numerical experiments.
Paper Structure (11 sections, 9 theorems, 52 equations, 4 figures)

This paper contains 11 sections, 9 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Notation
  4. Background theory
  5. Geometric concepts on manifolds
  6. Quotient manifolds
  7. The Stiefel Manifold
  8. On Cut Points and a bound of the Injectivity Radius
  9. Numerical experiments on sharpness of the injectivity radius bound
  10. Explicit construction of a cut point under the canonical metric
  11. Summary

Key Result

Theorem 7

Let $\mathcal{G}$ be a Lie group and let $\mathcal{H}$ be a closed subgroup of $\mathcal{G}$. Then the left coset space $\mathcal{G}/\mathcal{H}$ is a manifold of dimension $\dim\mathcal{G} - \dim\mathcal{H}$ with a unique smooth structure such that the quotient map $\pi:\mathcal{G}\to\mathcal{G}/\m turns $\mathcal{G}/\mathcal{H}$ into a homogeneous $\mathcal{G}$-space.

Figures (4)

  • Figure 1: Number of reached cut points relative to the number of examined geodesics for different ranks of tangent vectors $\Delta$. (variables: $n\in\left[4,100\right]$, $p\in\left[2,15\right]$ and $\mu\in\left[2.8,3.1\right]$).
  • Figure 2: Number of reached cut points relative to the number of examined geodesics for different lengths $\mu$ and a zoom in on the right. (variables: $n\in\left[4,100\right]$, $p\in\left[2,7\right]$ and $\mu\in\left[2.86,2.95\right]$ discretized in step sizes of $0.005$).
  • Figure 3: Number of reached cut points relative to the number of examined geodesics for different lengths $\mu$. The geodesics are defined by starting velocities from maximal sectional curvature planes. (variables: $n\in\left[4,100\right]$, $p\in\left[2,7\right]$ and $\mu\in\left[2.86,2.95\right]$ discretized in step sizes of $0.005$).
  • Figure 4: Smallest length $\mu$ of a geodesic to which endpoints a shorter geodesic is found (variables: $n\in\left[4,50\right]$, $p = 2$, $\mu$ started at $3.2$, stopped after $250$ iterations)

Theorems & Definitions (22)

  • Definition 1: Tangent space
  • Definition 2: Length of a curve
  • Definition 3: Geodesics
  • Definition 4: Riemannian exponential
  • Definition 5: Riemannian logarithm
  • Definition 6: Injectivity radius
  • Theorem 7: cf. leesmoothmanifold
  • Remark 1: cf. ONeill1983
  • Definition 8: cf. docarmo
  • Definition 9: Jacobi Field (cf. docarmo)
  • ...and 12 more