On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups
| Autor: | Katrin Fässler, Enrico Le Donne |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Dimension (graph theory) Quasi-isometric isometric 53C23 01 natural sciences differentiaaligeometria 0103 physical sciences Simply connected space Mathematics::Metric Geometry 0101 mathematics Isometric 20F65 bi-Lipschitz Mathematics Transitive relation Original Paper Lie groups Riemannian manifold 010102 general mathematics 22D05 ryhmäteoria Lie group Bi-Lipschitz Classification Lipschitz continuity metriset avaruudet quasi-isometric classification Differential geometry geometria 010307 mathematical physics Geometry and Topology Mathematics::Differential Geometry Counterexample |
| Zdroj: | Geometriae Dedicata |
| ISSN: | 1572-9168 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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