Harmonic vector fields on extended 3-dimensional Riemannian Lie groups

Abstract

Given two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$, we give harmonicity conditions for vector fields on the Riemannian warped product $B{\times }_{f}F$, with $f:B⟶]0,+\infty [$, 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 $\mathbb{R}{\times }_{f}G$. We give examples of harmonic vector fields on $G$ which are not left-invariant and determine harmonic vector fields on $\mathbb{R}{\times }_{f}G$. We conclude with some examples of vector fields on $\mathbb{R}{\times }_{f}G$ which are harmonic maps.