### 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 language | English (US) |
---|---|

Pages (from-to) | 927-961 |

Number of pages | 35 |

Journal | Communications in Analysis and Geometry |

Volume | 18 |

Issue number | 5 |

State | Published - Dec 2010 |

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### ASJC Scopus subject areas

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

### Cite this

*Communications in Analysis and Geometry*,

*18*(5), 927-961.

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

Research output: Contribution to journal › Article

*Communications in Analysis and Geometry*, vol. 18, no. 5, pp. 927-961.

}

TY - JOUR

T1 - Ricci flow on three-dimensional, unimodular metric Lie algebras

AU - Glickenstein, David A

AU - Payne, Tracy L.

PY - 2010/12

Y1 - 2010/12

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=79957656990&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79957656990

VL - 18

SP - 927

EP - 961

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

SN - 1019-8385

IS - 5

ER -