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

Halima Boukhari, Hadjer Okbani, Ahmed Mohammed Cherif

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 $(F^4,g)$.

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

Abstract

In this paper, we consider a left-invariant Riemannian metric on the Lie group . We classify Ricci solitons on and show that all such solitons are expanding and non-gradient. Moreover, we study the existence of harmonic maps from compact Riemannian manifolds into . Finally, we characterize a class of harmonic vector fields on .
Paper Structure (4 sections, 9 theorems, 39 equations)

This paper contains 4 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Existence of Ricci solitons on $(F^4, g)$
  3. Application to Harmonic Maps
  4. Harmonic Vector Fields on $(F^4,g)$

Key Result

Theorem 2.1

A vector field $\xi$ on the Thurston geometry $(F^4, g)$ is a Ricci soliton if and only if for some constants $c_{1},...,c_5 \in \mathbb{R}$. Moreover, the Ricci soliton $(F^4, g, \xi, \lambda)$ is expansive with $\lambda = -6$.

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 5 more