Calculus of Variations and Geometric Measure Theory

K. Fässler - E. Le Donne

On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups

created by ledonne on 20 Dec 2018


Submitted Paper

Inserted: 20 dec 2018
Last Updated: 20 dec 2018

Year: 2018

ArXiv: 1811.02253 PDF


This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric' is not a transitive relation.

Tags: GeoMeG