Some geometrical properties of Berger Spheres

TL;DR

This paper explores the geometric characteristics of Berger Spheres, focusing on Ricci solitons, harmonicity of invariant vector fields, and energy functional critical points, providing explicit calculations and non-existence results.

Contribution

It identifies all critical points of the energy functional for invariant vector fields on Berger Spheres and proves the non-existence of harmonic map-defining vector fields.

Findings

All critical points for the energy functional are determined.

No invariant vector fields define harmonic maps.

Explicit energy values for critical points are calculated.

Abstract

Our aim in this paper is to investigate some geometrical properties of Berger Spheres i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields. We determine all vector fields which are critical points for the energy functional restricted to vector fields. We also see that do not exist any vector fields defining harmonic map, and the energy of critical points is explicitly calculated.