  Let $Gr$ be a component of the Grassmann manifold of a $C^*$-algebra,
presented as the unitary orbit of a given orthogonal projection $Gr=Gr(P)$.
There are several natural connections in this manifold, and we first show that
they all agree (in the presence of a finite trace in $\mathcal A$, when we give
$Gr$ the Riemannian metric induced by the Killing form, this is the Levi-Civita
connection of the metric). We study the cut locus of $P\in Gr$ for the spectral
rectifiable distance, and also the conjugate tangent locus of $P\in Gr$ along a
geodesic. Furthermore, for each tangent vector $V$ at $P$, we compute the
kernel of the differential of the exponential map of the connection. We exhibit
examples where points that are tangent conjugate in the classical setting, fail
to be conjugate: in some cases they are not monoconjugate but epinconjugate,
and in other cases they are not conjugate at all.
