  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
$\nabla$ (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<\infty), G-invariant Riemann and K\"ahler structures are defined on M
which in this way becomes a totally geodesic symmetric holomorphic manifold.
