# Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces

## TL;DR

This paper constructs and analyzes a minimal foliation of the symmetric space SL(3,C)/SU(3) by Damek-Ricci spaces, revealing curvature properties and explicit geometric formulas for the hypersurfaces involved.

## Contribution

It explicitly describes a minimal foliation of a symmetric space by Damek-Ricci spaces and provides formulas for their Ricci curvature and sectional curvature properties.

## Key findings

- One hypersurface is minimally embedded and isometric to a 7D Damek-Ricci space.
- All hypersurfaces for certain parameters admit both negative and positive sectional curvature.
- The symmetric space admits a minimal foliation with leaves isometric to the Damek-Ricci space.

## Abstract

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $\alpha \in [0,\pi/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $\alpha = 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the Ricci curvature of these hypersurfaces and show that all hypersurfaces for $\alpha \in (0,\frac{\pi}{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.