  Let ${\cal A}$ be a von Neumann algebra and ${\cal P}_{\cal A}$ the manifold
of projections in ${\cal A}$. There is a natural linear connection in ${\cal
P}_{\cal A}$, which in the finite dimensional case coincides with the the
Levi-Civita connection of the Grassmann manifold of $\mathbb{C}^n$. In this
paper we show that two projections $p,q$ can be joined by a geodesic, which has
minimal length (with respect to the metric given by the usual norm of ${\cal
A}$), if and only if $$ p\wedge q^\perp\sim p^\perp\wedge q, $$ where $\sim$
stands for the Murray-von Neumann equivalence of projections. It is shown that
the minimal geodesic is unique if and only if $p\wedge q^\perp= p^\perp\wedge
q=0$. If ${\cal A}$ is a finite factor, any pair of projections in the same
connected component of ${\cal P}_{\cal A}$ (i.e., with the same trace) can be
joined by a minimal geodesic.
  We explore certain relations with Jones' index theory for subfactors. For
instance, it is shown that if ${\cal N}\subset{\cal M}$ are {\bf II}$_1$
factors with finite index $[{\cal M}:{\cal N}]=t^{-1}$, then the geodesic
distance $d(e_{\cal N},e_{\cal M})$ between the induced projections $e_{\cal
N}$ and $e_{\cal M}$ is $d(e_{\cal N},e_{\cal M})=\arccos(t^{1/2})$.
