The rectifiable metric on the set of closed subspaces of Hilbert space

Abstract:

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 $\{ p:{\text {rank}}\;p = \alpha ,{\text {rank}}(1 - p) = \beta \}$, 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 $\varepsilon$-chains. If $\left \| {p - q} \right \| < 1$, then ${d_r}(p,q) = {\sin ^{ - 1}}(\left \| {p - q} \right \|)$, but if $\left \| {p - q} \right \| = 1$, ${d_r}(p,q)$ can have any value in $\left [ {\frac {\pi } {2},\pi } \right ]$ (assuming $\alpha$ and $\beta$ are infinite). If ${d_r}(p,q) \ne \frac {\pi } {2}$, a minimizing path joining $p$ and $q$ exists; but if ${d_r}(p,q) = \frac {\pi } {2}$, a minimizing path exists if and only if ${\text {rank}}(p \wedge (1 - q)) = {\text {rank}}(q \wedge (1 - p))$.