Ricci flow on three-dimensional, unimodular metric Lie algebras
Ricci flow on three-dimensional, unimodular metric Lie algebras
复制标题
三维单模度量李代数上的 Ricci 流
DOI：
10.4310/cag.2010.v18.n5.a3
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
Tracy L. Payne
中科院分区：
文献类型：
--
作者：
David Glickenstein;Tracy L. Payne
我们给出了三维、单模、非阿贝尔度量李代数空间上的利玛窦流的全局图景，考虑到等距和缩放。里奇流被视为一个二维动力系统，用于度量李代数的结构常数相对于演化正交框架的演化。该系统适合直接相平面分析，我们发现相平面中的不动点和特殊轨迹对应于特殊的度量李代数，包括利奇孤子和特殊的黎曼淹没。这些结果是统一对三维、单连通、单模李群的左不变度量的 Ricci 流研究的一种方法，该研究之前是通过对不同 Bianchi 类进行个案分析来研究的。在附录中，我们证明了三维、单模、非阿贝尔度量李代数模等距和缩放空间的表征。
We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving orthonormal frame. This system is amenable to direct phase plane analysis, and we find that the fixed points and special trajectories in the phase plane correspond to special metric Lie algebras, including Ricci solitons and special Riemannian submersions. These results are one way to unify the study of Ricci flow on left invariant metrics on three-dimensional, simply-connected, unimodular Lie groups, which had previously been studied by a case-by-case analysis of the different Bianchi classes. In an appendix, we prove a characterization of the space of three-dimensional, unimodular, nonabelian metric Lie algebras modulo isometry and scaling.