Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof 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 ∇ (that is invariant under the group of automorphisms of V ) is defined on every connected component M of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0 ≤ r ≤ ∞). 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 ∇ is the Levi-Civita and the Kähler connection of M. Moreover, M turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed.