Some geometrical properties of the Oscillator group

TL;DR

This paper explores the geometric properties of the oscillator group with a bi-invariant Lorentzian metric, focusing on Ricci solitons, harmonic vector fields, and energy-critical vector fields, providing explicit calculations and classifications.

Contribution

It characterizes homogeneous Ricci solitons, harmonic vector fields, and energy-critical vector fields on the oscillator group with a bi-invariant Lorentzian metric, offering explicit formulas and classifications.

Findings

Classification of homogeneous Ricci solitons

Identification of harmonic invariant vector fields

Explicit energy calculations for critical vector fields

Abstract

We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also determined, and the energy of critical points is explicitly calculated.