Harmonicity and Minimality of vector fields on four-dimensional Lorentzian lie groups

TL;DR

This paper investigates harmonic and minimal vector fields on four-dimensional Lorentzian Lie groups with Einstein metrics, classifying harmonic maps and analyzing energy and minimality properties of these vector fields.

Contribution

It provides a classification of harmonic vector fields and analyzes their energy and minimality on four-dimensional Lorentzian Lie groups with Einstein metrics.

Findings

All vector fields are critical points for the energy functional in some cases.

Explicit calculations of energy for harmonic vector fields are provided.

The minimality of these critical points is studied and characterized.

Abstract

We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields defining harmonic maps, and calculate explicitly the energy of these vector fields. Then we study the minimality of critical points for the energy functional.