Isometry groups of simply connected nonunimodular Lie groups of dimension four

TL;DR

This paper determines the complete isometry groups for all left-invariant Riemannian metrics on simply connected nonunimodular four-dimensional Lie groups, providing a comprehensive classification of their symmetries.

Contribution

It explicitly computes the isometry groups for each such Lie group and metric, filling a gap in the understanding of their geometric symmetries.

Findings

Full isometry groups identified for all cases

Classification of symmetries for nonunimodular four-dimensional Lie groups

Enhanced understanding of geometric structures on these Lie groups

Abstract

For each left-invariant Riemannian metric on simply connected nonunimodular Lie groups of dimension four, we determine the full group of isometries.