6533b863fe1ef96bd12c7963
RESEARCH PRODUCT
On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups.
Katrin FässlerKatrin FässlerEnrico Le Donnesubject
Pure mathematicsDimension (graph theory)Quasi-isometricisometric53C2301 natural sciencesdifferentiaaligeometria0103 physical sciencesSimply connected spaceMathematics::Metric Geometry0101 mathematicsIsometric20F65bi-LipschitzMathematicsTransitive relationOriginal PaperLie groupsRiemannian manifold010102 general mathematics22D05ryhmäteoriaLie groupBi-Lipschitz; Classification; Isometric; Lie groups; Quasi-isometric; Riemannian manifoldRiemannian manifoldLipschitz continuityClassificationmetriset avaruudetquasi-isometricBi-LipschitzclassificationDifferential geometrygeometria010307 mathematical physicsGeometry and TopologyMathematics::Differential GeometryCounterexampledescription
AbstractThis 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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-01-01 | Geometriae dedicata |