  The set $D_{A_0}$, of pairs of orthogonal projections $(P,Q)$ in generic
position with fixed difference $P-Q=A_0$, is shown to be a homogeneus smooth
manifold: it is the quotient of the unitary group of the commutant $\{A_0\}'$
divided by the unitary subgroup of the commutant $\{P_0, Q_0\}'$, where
$(P_0,Q_0)$ is any fixed pair in $D_{A_0}$. Endowed with a natural reductive
structure (a linear connection) and the quotient Finsler metric of the operator
norm, it behaves as a classic Riemannian space: any two pairs in $D_{A_0}$ are
joined by a geodesic of minimal length. Given a base pair $(P_0,Q_0)$, pairs in
an open dense subset of $D_{A_0}$ can be joined to $(P_0,Q_0)$ by a {\it
unique} minimal geodesic.
