Ricci flow on three-dimensional, unimodular metric Lie algebras

We give a global picture of the Ricci flow on the space ofthree-dimensional, unimodular, nonabelian metric Lie algebras considered up toisometry and scaling. The Ricci flow is viewed as a two-dimensional dynamicalsystem for the evolution of structure constants of the metric Lie algebra withrespect to an evolving orthonormal frame. This system is amenable to directphase plane analysis, and we find that the fixed points and specialtrajectories in the phase plane correspond to special metric Lie algebras,including Ricci solitons and special Riemannian submersions. These results areone way to unify the study of Ricci flow on left invariant metrics onthree-dimensional, simply-connected, unimodular Lie groups, which hadpreviously been studied by a case-by-case analysis of the different Bianchiclasses. In an appendix, we prove a characterization of the space ofthree-dimensional, unimodular, nonabelian metric Lie algebras modulo isometryand scaling.

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Published 1 January 2010