The manifold of finite rank projections in the algebra L(H) of bounded linear operators

Abstract

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V:=L(H). Using the algebraic structure of V, a torsionfree affine connection $\nabla$ (that is invariant under the group of automorphisms of V) is defined on every connected component of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0< r < \infty). We prove that M admits a Riemann structure if and only if M consists of projections that have the same finite rank r or the same finite corank, and in that case $\nabla$ is the Levi-Civita and the K\"ahler connection of M. Moreover, M turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed. Keywords: JBW-algebras, Grassmann manifolds, Riemann manifolds. AMS 2000 Subject Classification: 48G20, 72H51.