On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups
- Fässler, Katrin Department of Mathematics, University of Fribourg, Fribourg, Switzerland - Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
- Donne, Enrico Le Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
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28.04.2020
Published in:
- Geometriae Dedicata. - 2021, vol. 210, no. 1, p. 27-42
English
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.
- Faculty
- Faculté des sciences et de médecine
- Department
- Département de Mathématiques
- Language
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- English
- Classification
- Mathematics
- License
- License undefined
- Identifiers
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- RERO DOC 328451
- DOI 10.1007/s10711-020-00532-8
- Persistent URL
- https://folia.unifr.ch/unifr/documents/308565
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