  A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for
which one of the principal directions of the second fundamental form is the
(para-)complex dual of the normal vector.
  We consider particular Hopf hypersurfaces in the space of oriented geodesics
of a non-flat space form of dimension greater than 2. For spherical and
hyperbolic space forms, the oriented geodesic space admits a canonical
Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a
natural notion of a Hopf hypersurface exists.
  The particular hypersurfaces considered are formed by the oriented geodesics
that are tangent to a given convex hypersurface in the underlying space form.
We prove that a tangent hypersurface is Hopf in the space of oriented geodesics
with respect to this canonical (para-)Kaehler structure iff the underlying
convex hypersurface is totally umbilic and non-flat.
  In the case of 3 dimensional space forms, however, there exists a second
canonical complex structure which can also be used to define Hopf
hypersurfaces. We prove that in this dimension, the tangent hypersurface of a
convex hypersurface in the space form is always Hopf with respect to this
second complex structure.
