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 space of oriented geodesics 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 if and only if the underlying convex hypersurface is totally umbilic. In the case of three dimensional space forms 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 Hopf if and only if the underlying convex hypersurface is totally umbilic.
Georgiou, N., Guilfoyle, B. Hopf hypersurfaces in spaces of oriented geodesics. J. Geom. 108, 1129–1135 (2017). https://doi.org/10.1007/s00022-017-0400-4