2001/10/11 by J. M. Isidro, José M. Isidro, Isidro, J. M. +3
Mathematics · #48G20 #72H51 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.DG #math.FA #msc:48G20 #msc:72H51
paper · pdf · doi:10.48550/arxiv.math/0110115
17 pages, Latex 2e, to appear in Expositiones Mathematicae
arxiv created 2001/10/11 · openalex publication_date 2001/10/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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. Keywords: JBW-algebras, Grassmann manifolds, Riemann manifolds. AMS 2000 Subject Classification: 48G20, 72H51.