Classification of left invariant Riemannian metrics on nonunimodular 4-dimensional Lie groups

TL;DR

This paper classifies left invariant Riemannian metrics on 4-dimensional nonunimodular Lie groups, providing a comprehensive understanding of their geometric structures up to automorphism.

Contribution

It offers a complete classification of such metrics on nonunimodular Lie groups, extending the understanding of their algebraic and geometric properties.

Findings

Classification up to automorphism achieved for all nonunimodular Lie groups

Equivalent classification of inner products on associated Lie algebras

Provides a foundation for further geometric analysis of these groups

Abstract

We classify, up to automorphism, left invariant Riemannian metrics on 4-dimensional simply connected nonunimodular Lie groups. This is equivalent to classifying, up to automorphism, inner products on 4-dimensional nonunimodular Lie algebras.