On the Injectivity Radius of the Stiefel Manifold: Numerical investigations and an explicit construction of a cut point at short distance (2403.03782v2)

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 $\sqrt{\frac{4}{5}}\pi$. 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.