  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.
