Let
A
{\mathcal {A}}
be a von Neumann algebra and
P
A
{\mathcal {P}}_{\mathcal {A}}
the manifold of projections in
A
{\mathcal {A}}
. There is a natural linear connection in
P
A
{\mathcal {P}}_{\mathcal {A}}
, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of
C
n
\mathbb {C}^n
. In this paper we show that two projections
p
,
q
p,q
can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of
A
{\mathcal {A}}
), if and only if
p
∧
q
⊥
∼
p
⊥
∧
q
,
\begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*}
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
∧
q
⊥
=
p
⊥
∧
q
=
0
p\wedge q^\perp = p^\perp \wedge q=0
. If
A
{\mathcal {A}}
is a finite factor, any pair of projections in the same connected component of
P
A
{\mathcal {P}}_{\mathcal {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
N
⊂
M
{\mathcal {N}}\subset {\mathcal {M}}
are II
1
_1
factors with finite index
[
M
:
N
]
=
t
−
1
[{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1}
, then the geodesic distance
d
(
e
N
,
e
M
)
d(e_{\mathcal {N}},e_{\mathcal {M}})
between the induced projections
e
N
e_{\mathcal {N}}
and
e
M
e_{\mathcal {M}}
is
d
(
e
N
,
e
M
)
=
arccos
(
t
1
/
2
)
d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2})
.