The Grassmann manifold of a Hilbert space

Abstract

The present paper surveys the geometric properties of the Grassmann manifold Gr(H ) of an infinite dimensional complex Hilbert space H . Gr(H ) is viewed as a set of operators, identifying each closed subspace S ⊂ H with the orthogonal projection PS onto S . Most of the results surveyed here were stated by G. Corach, H. Porta and L. Recht: submanifold structure, homogeneous reductive structure, local minimality of geodesics. Some recent results concerning the existence and uniqueness of a geodesic joining two given projections, which were obtained by the present author, are also presented.