Harmonic vector fields on extended 3-dimensional Riemannian Lie groups

TL;DR

This paper establishes conditions for harmonic vector fields on warped product manifolds and applies these to classify harmonic vector fields on extended 3D Riemannian Lie groups, including non-left-invariant examples.

Contribution

It provides new harmonicity conditions for vector fields on warped products and characterizes harmonic vector fields on extended 3D Riemannian Lie groups, including non-invariant cases.

Findings

Harmonicity conditions derived for vector fields on warped products.

Classification of harmonic vector fields on $ eal imes_f G$ where $G$ is a 3D Lie group.

Examples of harmonic vector fields that are not left-invariant.

Abstract

Given two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$, we give harmonicity conditions for vector fields on the Riemannian warped product $B\times_fF$, with $f:B \longrightarrow ]0,+\infty[$, using a characteristic variational condition. Then, we apply this to the case $B=\mathbb{R}$ and $F$ is a three-dimensional connected Riemannian Lie group $G$ equipped with a left-invariant metric, to determine harmonic vector fields on $\mathbb{R}\times_fG$. We give examples of harmonic vector fields on $G$ which are not left-invariant and determine harmonic vector fields on $\mathbb{R}\times_fG$. We conclude with some examples of vector fields on $\mathbb{R}\times_fG$ which are harmonic maps.