Conformal Minimal Foliations on Semi-Riemannian Lie Groups

TL;DR

This paper classifies conformal minimal foliations of codimension two on semi-Riemannian Lie groups generated by specific subgroups, leading to new examples of harmonic morphisms on these groups.

Contribution

It provides a classification of such foliations for key subgroups, constructing new families of Lie groups with these structures and associated harmonic morphisms.

Findings

Classified conformal minimal foliations for specific subgroups.

Constructed new Lie groups with these foliations.

Produced local complex-valued harmonic morphisms.

Abstract

We study left-invariant foliations ${\mathcal F}$ on semi-Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations ${\mathcal F}$ when the subgroup $K$ is one of the important $\text{SU}(2)$, $\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SU}(2)$, $\text{SU}(2)\times\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SO}(2)$, $\text{SL}_{2}(\mathbb R)\times\text{SU}(2)$. This way we construct new multi-dimensional families of Lie groups $G$ carrying such foliations in each case. These foliations ${\mathcal F}$ produce local complex-valued harmonic morphisms on the corresponding Lie group $G$.