Riemannian geometry of finite rank positive operators

Abstract

A riemannian metric is introduced in the infinite dimensional manifold Σ_n of positive operators with rank n<∞ on a Hilbert space H. The geometry of this manifold is studied and related to the geometry of the submanifolds Σ_p$ of positive operators with range equal to the range of a projection p (rank of p =n), and P_p of selfadjoint projections in the connected component of p. It is shown that these spaces are complete in the geodesic distance.