Harmonic Morphisms and Minimal Conformal Foliations on Lie Groups

TL;DR

This paper studies harmonic morphisms and minimal conformal foliations on Lie groups with specific subgroup structures, showing conditions under which these foliations are Riemannian, minimal, or totally geodesic.

Contribution

It establishes that certain conformal foliations on Lie groups are Riemannian and minimal, and characterizes when they are totally geodesic based on the metric properties.

Findings

Foliation $$ is Riemannian and minimal.

If the metric on $K$ is biinvariant, then $$ is totally geodesic.

Leaves of $$ are fibers of harmonic morphisms.

Abstract

Let $G$ be a Lie group equipped with a left-invariant Riemannian metric. Let $K$ be a semisimple and normal subgroup of $G$ generating a left-invariant conformal foliation $\F$ of on $G$. We then show that the foliation $\F$ is Riemannian and minimal. This means that locally the leaves of $\F$ are fibres of a harmonic morphism. We also prove that if the metric restricted to $K$ is biinvariant then $\F$ is totally geodesic.