Hopf hypersurfaces in spaces of oriented geodesics

TL;DR

This paper studies special hypersurfaces called Hopf hypersurfaces in the space of oriented geodesics of non-flat space forms, revealing conditions under which they are Hopf with respect to canonical (para-)Kaehler structures.

Contribution

It characterizes when tangent hypersurfaces to convex hypersurfaces are Hopf in the space of oriented geodesics, especially highlighting the role of totally umbilic and non-flat conditions.

Findings

Tangent hypersurfaces are Hopf iff the underlying convex hypersurface is totally umbilic and non-flat.

In 3D space forms, tangent hypersurfaces are always Hopf with respect to a second canonical complex structure.

The study connects geometric properties of hypersurfaces in space forms to their behavior in the space of oriented geodesics.

Abstract

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.