Harmonic Morphisms on Lie groups and Minimal Conformal Foliations of Codimension two

TL;DR

This paper investigates harmonic morphisms and minimal conformal foliations on Lie groups with semi-Riemannian metrics, showing conditions under which these foliations are minimal or totally geodesic, especially in the Riemannian case.

Contribution

It establishes that conformal foliations generated by semisimple subgroups are minimal and characterizes when they are totally geodesic in the Riemannian setting.

Findings

Foliation F is minimal when generated by a semisimple subgroup K.

In the Riemannian case, if the metric on K is biinvariant, F is totally geodesic.

Locally, leaves of F are fibers of complex-valued harmonic morphisms.

Abstract

Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This means that locally the leaves of F are fibres of a complex-valued harmonic morphism. In the Riemannian case, we prove that if the metric restricted to K is biinvariant then F is totally geodesic.