Geodesics in infinite dimensional Stiefel and Grassmann manifolds

TL;DR

This paper investigates the geometric structure of infinite-dimensional Stiefel and Grassmann manifolds, establishing the existence of minimal geodesics and characterizing cut loci by reducing the problem to finite dimensions.

Contribution

It extends the understanding of geodesic properties and cut loci in infinite-dimensional manifolds by leveraging finite-dimensional reductions.

Findings

Existence of minimal geodesics between any two points.

Characterization of the cut locus in these manifolds.

Reduction of infinite-dimensional problems to finite-dimensional cases.

Abstract

Let $V$ be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of $p$ vectors in $V$, and let $\Gr(p,V)$ be the Grassmann manifold of $p$ dimensional subspaces of $V$. We study the distance and the geodesics in these manifolds, by reducing the matter to the finite dimensional case. We then prove that any two points in those manifolds can be connected by a minimal geodesic, and characterize the cut locus.