On the Global Structure of Hopf Hypersurfaces in Complex Space Form

TL;DR

This paper characterizes the global structure of Hopf hypersurfaces in complex space forms, showing they are tubes over algebraic varieties or geodesic spheres depending on the ambient space.

Contribution

It establishes a classification of compact Hopf hypersurfaces in complex projective and hyperbolic spaces as tubes over algebraic varieties or geodesic spheres.

Findings

In complex projective space, Hopf hypersurfaces are tubes over irreducible algebraic varieties.

In complex hyperbolic space, they are geodesic hyperspheres.

Provides a global geometric characterization of Hopf hypersurfaces.

Abstract

It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed C^(2n-1) regular Hopf hypersurface is a geodesic hypersphere