Inserted: 4 nov 2010
Last Updated: 11 feb 2022
Journal: Comptes rendus - Mathématique
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.
Keywords: Riemannian Geometry, cutlocus, stiefel manifold, hilbert space, grassmann manifold, minimal geodesic, space of curves, sobolev active contours