The index of symmetry of three-dimensional Lie groups with a left-invariant metric

TL;DR

This paper calculates the index of symmetry for 3D unimodular Lie groups with left-invariant metrics, revealing that all such groups can have metrics with positive symmetry index and exploring their geometric quotients.

Contribution

It provides a complete determination of the index of symmetry for 3D unimodular Lie groups and analyzes their geometric structures and fibrations.

Findings

Every 3D unimodular Lie group admits a left-invariant metric with positive index of symmetry.

The geometry of quotients by the foliation of symmetry is characterized.

Conditions under which the group fibers over a 2D space of constant curvature are identified.

Abstract

We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.