Hopf hypersurfaces in complex Grassmannians of rank two

TL;DR

This paper classifies Hopf and $q$-umbilical real hypersurfaces in complex Grassmannians of rank two, establishing nonexistence results and describing the behavior of principal curvatures.

Contribution

It proves the nonexistence of mixed foliate hypersurfaces and classifies contact and $q$-umbilical hypersurfaces in these spaces, advancing understanding of their geometric structure.

Findings

Nonexistence of mixed foliate real hypersurfaces.

Reeb principal curvature is constant along Reeb vector field integral curves.

Classification of contact and $q$-umbilical hypersurfaces.

Abstract

In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb vector field. As a result the classification of contact real hypersurfaces is obtained. We also introduce the notion of $q$-umbilical real hypersurfaces in complex Grassmannians of rank two and obtain a classification of such real hypersurfaces.