Calculus of Variations and Geometric Measure Theory

P. Harms - A. C. G. Mennucci

Geodesics in infinite dimensional Stiefel and Grassmann manifolds

created by mennucci on 04 Nov 2010
modified on 11 Feb 2022


Published Paper

Inserted: 4 nov 2010
Last Updated: 11 feb 2022

Journal: Comptes rendus - Mathématique
Year: 2012

ArXiv: 1209.2878 PDF
Links: DOI


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