Ricci flow on three-dimensional, unimodular metric Lie algebras

David A Glickenstein, Tracy L. Payne

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We give a global picture of the Ricci flow on the space of threedimensional, 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-bycase 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.

Original languageEnglish (US)
Pages (from-to)927-961
Number of pages35
JournalCommunications in Analysis and Geometry
Volume18
Issue number5
StatePublished - Dec 2010

Fingerprint

Ricci Flow
Lie Algebra
Metric
Three-dimensional
Isometry
Scaling
Riemannian Submersion
Phase Plane Analysis
Ricci Soliton
Phase Plane
Invariant Metric
Orthonormal
Modulo
Dynamical system
Fixed point
Trajectory

ASJC Scopus subject areas

  • Statistics and Probability
  • Geometry and Topology
  • Analysis
  • Statistics, Probability and Uncertainty

Cite this

Ricci flow on three-dimensional, unimodular metric Lie algebras. / Glickenstein, David A; Payne, Tracy L.

In: Communications in Analysis and Geometry, Vol. 18, No. 5, 12.2010, p. 927-961.

Research output: Contribution to journalArticle

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