  We study the geometry of families of hypersurfaces in Eguchi-Hanson space
that arise as complex line bundles over curves in $S^2$ and are
three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf
tori for closed curves. They are negatively curved, asymptotically flat spaces,
and we compute the complete three-dimensional curvature tensor as well as the
second fundamental form, giving also some results concerning their geodesic
flow. We show the non-existence of $\L^p$-harmonic functions on these
hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the
infima of the essential spectra of the Laplace and of the square of the Dirac
operator in the case of closed curves. For circles we also compute the
$\L^2$-kernel of the Dirac operator in the sense of spectral theory and show
that it is infinite dimensional. We consider further the Einstein Dirac system
on these spaces and construct explicit examples of T-Killing spinors on them.
