On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry $F^4$

TL;DR

This paper classifies Ricci solitons on a specific 4-dimensional Lie group with a left-invariant metric, showing they are all expanding and non-gradient, and explores harmonic maps and vector fields in this geometric setting.

Contribution

It provides a complete classification of Ricci solitons on $(F^4,g)$ and characterizes harmonic vector fields and maps in this geometry.

Findings

All Ricci solitons are expanding and non-gradient.

Characterization of harmonic vector fields on $(F^4,g)$.

Existence results for harmonic maps into $(F^4,g)$.

Abstract

In this paper, we consider a left-invariant Riemannian metric $g$ on the Lie group $F^4$. We classify Ricci solitons on $(F^4,g)$ and show that all such solitons are expanding and non-gradient. Moreover, we study the existence of harmonic maps from compact Riemannian manifolds into $(F^4,g)$. Finally, we characterize a class of harmonic vector fields on $(F^4,g)$.