Isometry groups of simply connected nonunimodular Lie groups of dimension four
Youssef Ayad
TL;DR
The paper classifies the full isometry groups for all left-invariant Riemannian metrics on simply connected nonunimodular Lie groups of dimension four by leveraging the type $(R)$ solvable structure, automorphism actions, and a metric classification up to automorphism. It shows that, for each group, $ ext{Isom}(G,g)$ splits as $G times ext{Aut}(G)_g$ with the isotropy part determined by the metric class, yielding explicit semidirect product forms such as $G times ext{D}(4)$, $G times ext{O}(2)$, or $G$ alone depending on the representative metric. The results cover decomposable cases like $G_{2.1} imesb R^2$, $G_2$, and $G_{3.2} imesb R$, as well as indecomposable groups $G_{4.2}^{ullet}$, $G_{4.3}$, $G_{4.4}$, $G_{4.5}^{ullet}$, $G_{4.7}$, and $G_{4.8}^{ullet}$, with detailed lists of isometric automorphism groups for each metric class. The work relies on a recent classification of four-dimensional left-invariant metrics and on the reduction to automorphism classes, yielding a comprehensive map from metric data to symmetry groups. This advances the understanding of symmetry in four-dimensional nonunimodular geometries and provides a resource for applications in geometric analysis and mathematical physics.
Abstract
For each left-invariant Riemannian metric on simply connected nonunimodular Lie groups of dimension four, we determine the full group of isometries.