2001/10/30 by Jose M. Isidro, Isidro, Jose M.
Mathematics · #48G20 #72H51 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #math.DG #math.FA #msc:48G20 #msc:72H51
paper · pdf · doi:10.48550/arxiv.math/0110315
12 pages, Latex 2e, to appear
arxiv created 2001/10/30 · arxiv updated 2009/11/30
Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the algebraic structure of Z, a torsionfree affine connection ∇ (that is invariant under the group G= Aut (Z) of automorphisms of Z) is defined on each of these connected components and the geodesics are computed. In case M consists of elements that have a fixed finite rank r, (0<r<∞), G-invariant Riemann and Kähler structures are defined on M which in this way becomes a totally geodesic symmetric holomorphic manifold.