Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

TL;DR

This paper classifies certain real hypersurfaces with constant principal curvatures in complex space forms, revealing their geometric structure and existence conditions, especially distinguishing between complex projective and hyperbolic spaces.

Contribution

It provides a complete classification of non-Hopf real hypersurfaces with constant principal curvatures in complex space forms, identifying their geometric types and non-existence in projective spaces.

Findings

Non-existence in complex projective spaces.

Existence as tubes around minimal submanifolds in hyperbolic spaces.

Characterization as homogeneous hypersurfaces.

Abstract

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones.