The manifold of finite rank projections in
the algebra L(H) of bounded linear operators.
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.
(This paper has appeared in Expo. Math. 20 (2002),
97 - 116)
Author: J. M. Isidro < jmisidro@zmat.usc.es >
M. Mackey < michael.mackey@ucd.ie >