On geodesics and the cut locus in the oriented Grassmann manifold

Abstract

The Stiefel and Grassmann manifolds have been studied extensively and play a prominent role in many applications. Their geodesic structure is well understood; in particular, minimal geodesics and the cut locus in the Grassmann manifold admit classical descriptions. The oriented Grassmann manifold is less frequently treated explicitly. In this note we review the relevant Stiefel and Grassmann results for Hilbert spaces, including the infinite-dimensional case, give direct proofs adapted to our notation, and derive an explicit minimal-geodesic criterion and a cut-locus characterization for the oriented Grassmann manifold.