Real Hypersurfaces in Complex Hyperbolic Two-Plane Grassmannians with commuting Ricci tensor

TL;DR

This paper classifies real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting Ricci tensor, revealing they are either tubes over certain submanifolds or specific horospheres.

Contribution

It provides a complete classification of Hopf hypersurfaces with commuting Ricci tensor in complex hyperbolic two-plane Grassmannians, including explicit geometric descriptions.

Findings

Hypersurfaces are either tubes over totally geodesic submanifolds or horospheres.

Derived a new formula for the Ricci tensor in this setting.

Achieved a full classification of such hypersurfaces.

Abstract

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new formula for the Ricci tensor of $M$ in $SU_{2,m}/S(U_2{\cdot}U_m)$. Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$ with commuting Ricci tensor. Each can be described as 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.