The index of symmetry for a left-invariant metric on a solvable three-dimensional Lie group

TL;DR

This paper computes the index of symmetry for all solvable three-dimensional Lie groups with a left-invariant metric, completing the classification when combined with previous work on unimodular cases.

Contribution

It determines the index of symmetry for all solvable three-dimensional Lie groups with a left-invariant metric, extending previous results to the non-unimodular case.

Findings

Index of symmetry is positive for some metrics on each solvable Lie group.

Index is never equal to 2 for these groups.

Complete classification when combined with prior unimodular results.

Abstract

We find the index of symmetry for all solvable three-dimensional Lie groups with a left-invariant metric. When combined with the work of Reggiani on unimodular three-dimensional Lie groups, the index of symmetry is then known for all three-dimensional Lie groups with a left-invariant metric. Similar to the unimodular case, for every solvable three-dimensional Lie algebra there is at least one left-invariant metric for which the index of symmetry is positive. Also, the index is never equal to 2.