  For a given Hilbert space $\mathcal H$, consider the space of self-adjoint
projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable
structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by
$$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \,
Pf=f , \, \|f\|=1\, \}. $$ We establish the smooth action on $\mathcal R$ of
the group of unitary operators of $\mathcal H$, therefore $\mathcal R$ is an
homogeneous space. Then we study the metric structure of $\mathcal R$ by
endowing it first with the uniform quotient metric, which is a Finsler metric,
and we establish minimality results for the geodesics. These are given by
certain one-parameter groups of unitary operators, pushed into $\mathcal R$ by
the natural action of the unitary group. Then we study the restricted bundle
$\mathcal R_2^+$ given by considering only the projections in the restricted
Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow
$\mathcal R_2^+$ with a natural Riemannian metric that can be obtained by
declaring that the action of the group is a Riemannian submersion. We study the
Levi-Civita connection of this metric and establish a Hopf-Rinow theorem for
$\mathcal R_2^+$, again obtaining a characterization of the geodesics as the
image of certain one-parameter groups with special speeds.
