Fässler Katrin, Le Donne Enrico
Department of Mathematics, University of Fribourg, Fribourg, Switzerland.
Present Address: Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland.
Geom Dedic. 2021;210(1):27-42. doi: 10.1007/s10711-020-00532-8. Epub 2020 Apr 28.
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.
本笔记关注的是配备左不变黎曼度量的三维及以下连通李群的几何分类。一方面，综合文献中的结果，我们对这类群直至拟等距的完全分类进行了综述，并将拟等距分类与双李普希茨分类进行了比较。另一方面，我们研究了这样一个问题：两个拟等距等价的李群，若配备合适的左不变黎曼度量，是否能成为等距的。我们证明对于三维单连通群情况是这样，但对于多连通群一般不成立。该反例还表明“能成为等距的”不是一个传递关系。
