THE RECTIFIABLE METRIC ON THE SET OF CLOSED SUBSPACES OF HILBERT-SPACE

Consider the set of selfadjoint projections on a fixed Hilbert space. It is well known that the connected components, under the norm topology, are the sets {p: rank p = alpha, rank(1 - p) = beta} , where alpha and beta are appropriate cardinal numbers. On a given component, instead of using the metric induced by the norm, we can use the rectifiable metric d(r) which is defined in terms of the lengths of rectifiable paths or, equivalently in this case, the lengths of epsilon-chains. If \\p - q\\ < 1 , then d(r)(p, q) = sin-1(\\p - q\\) , but if \\p - q\\ = 1, d(r)(p, q) can have any value in [pi/2, pi] (assuming alpha and beta are infinite). If d(r)(p, q) not-equal pi/2, a minimizing path joining p and q exists; but if d(r)(p, q) = pi/2, a minimizing path exists if and only if rank(p AND (1 - q)) = rank(q AND (1 - p)).