Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms

TL;DR

This paper investigates Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms, proving their Hopf curvature is constant and characterizing their structure, extending previous geometric results.

Contribution

It establishes the constancy of Hopf curvature and characterizes Hopf hypersurfaces as tubes over submanifolds in pseudo-Riemannian complex and para-complex spaces.

Findings

No umbilic hypersurfaces exist in these spaces.

Hopf hypersurfaces have constant Hopf curvature.

Some Hopf hypersurfaces are locally tubes over submanifolds.

Abstract

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by $J$ be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field $N$ is said to be \em Hopf \em if the tangent vector field $JN$ is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the \em Hopf \em curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.