## Geometry & Topology

### Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

Michael Jablonski

#### Abstract

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that one may determine the existence of a such a metric by analyzing algebraic properties of the Lie algebra and infinitesimal deformations of any initial metric.

Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics.

#### Article information

Source
Geom. Topol., Volume 15, Number 2 (2011), 735-764.

Dates
Revised: 13 January 2011
Accepted: 13 March 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732304

Digital Object Identifier
doi:10.2140/gt.2011.15.735

Mathematical Reviews number (MathSciNet)
MR2800365

Zentralblatt MATH identifier
1217.22005

#### Citation

Jablonski, Michael. Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups. Geom. Topol. 15 (2011), no. 2, 735--764. doi:10.2140/gt.2011.15.735. https://projecteuclid.org/euclid.gt/1513732304

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