Geometry & Topology

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

Michael Jablonski

Full-text: Open access

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
Received: 26 September 2010
Revised: 13 January 2011
Accepted: 13 March 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 22E25: Nilpotent and solvable Lie groups 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

Keywords
Einstein metric Ricci soliton solvsoliton nilsoliton solvable nilpotent Lie group left-invariant metric

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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