Addendum to the paper "Hypersurfaces with Isometric Reeb Flow in Complex hyperbolic Two-Plane Grassmannians"

TL;DR

This paper classifies real hypersurfaces with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians, identifying them as either tubes over certain submanifolds or specific horospheres.

Contribution

It provides a complete classification of such hypersurfaces, extending previous results by characterizing their geometric structures in complex hyperbolic Grassmannians.

Findings

Hypersurfaces are either tubes over totally geodesic submanifolds or horospheres.

Centers at infinity of horospheres are singular and of a specific type.

Classification completes understanding of Reeb invariant shape operators in this setting.

Abstract

We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is singular and of type $JN \in {\mathfrak J}N$ for a unit normal vector field $N$ of $M$.