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arXiv · 1608.06438

Hopf hypersurfaces in spaces of oriented geodesics

Nikos Georgiou·Brendan Guilfoyle
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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.

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Nikos Georgiou, Brendan Guilfoyle. 2016-08-23. Hopf hypersurfaces in spaces of oriented geodesics. https://doi.org/10.1007/s00022-017-0400-4

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