Isometry groups of simply connected unimodular 4-dimensional Lie groups

TL;DR

This paper classifies the full isometry groups of all left invariant Riemannian metrics on four-dimensional simply connected unimodular nilpotent or solvable Lie groups, providing a comprehensive understanding of their geometric symmetries.

Contribution

It provides a complete description of the isometry groups for all such metrics on four-dimensional unimodular Lie groups, a previously uncharted classification.

Findings

Explicit isometry groups for each metric type

Complete classification of symmetries in 4D unimodular Lie groups

Foundation for further geometric analysis of these groups

Abstract

We describe the full group of isometries of each left invariant Riemannian metric on the simply connected unimodular nilpotent or solvable $(R)$-type Lie groups of dimension four.