On Locally Symmetric $3$-Dimensional Riemannian Lie Groups

TL;DR

This paper classifies all 3-dimensional connected, locally symmetric Riemannian Lie groups using Milnor bases and polynomial equations, revealing that only E_0(2) admits locally symmetric metrics that are not globally symmetric.

Contribution

It provides a complete classification of 3D locally symmetric Riemannian Lie groups and identifies the unique case of E_0(2) with non-symmetric locally symmetric metrics.

Findings

Classified all 3D locally symmetric Riemannian Lie groups.

Identified E_0(2) as the only non-symmetric locally symmetric case.

Used polynomial equations of structure constants for classification.

Abstract

In this paper, we use the powerful tool Milnor bases to classify all the $3-$dimensional connected and locally symmetric Riemannian Lie Groups by solving system of polynomial equations of structure constants of each Lie algebra . Moreover, we showed that $E_0(2)$, is the only Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.