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On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups
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...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer Netherlands
2020
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7810620/ https://www.ncbi.nlm.nih.gov/pubmed/33505086 http://dx.doi.org/10.1007/s10711-020-00532-8 |
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author | Fässler, Katrin Le Donne, Enrico |
author_facet | Fässler, Katrin Le Donne, Enrico |
author_sort | Fässler, Katrin |
collection | PubMed |
description | 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. |
format | Online Article Text |
id | pubmed-7810620 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2020 |
publisher | Springer Netherlands |
record_format | MEDLINE/PubMed |
spelling | pubmed-78106202021-01-25 On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups Fässler, Katrin Le Donne, Enrico Geom Dedic Original Paper 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. Springer Netherlands 2020-04-28 2021 /pmc/articles/PMC7810620/ /pubmed/33505086 http://dx.doi.org/10.1007/s10711-020-00532-8 Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
spellingShingle | Original Paper Fässler, Katrin Le Donne, Enrico On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups |
title | On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups |
title_full | On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups |
title_fullStr | On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups |
title_full_unstemmed | On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups |
title_short | On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups |
title_sort | on the quasi-isometric and bi-lipschitz classification of 3d riemannian lie groups |
topic | Original Paper |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7810620/ https://www.ncbi.nlm.nih.gov/pubmed/33505086 http://dx.doi.org/10.1007/s10711-020-00532-8 |
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